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The response of sands under partially drained states with emphasis on liquefaction Eliadorani, Ali Akbar 2000

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The Response of Sands Under Partially Drained States with Emphasis Liquefaction by ALI AKBAR ELI AD ORAM B.Sc., University of Mazanderan, 1987 M.A.Sc, University of Waterloo, 1992 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 2000 ©Ali Akbar Eliadorani, 2000 In p resen t ing this thesis in partial fu l f i lment of the requ i rements for an a d v a n c e d d e g r e e at the Univers i ty of Brit ish C o l u m b i a , I agree that the Library shall m a k e it f reely available fo r re fe rence and study. 1 further agree that p e r m i s s i o n fo r extens ive c o p y i n g o f this thes is f o r scholar ly p u r p o s e s m a y b e granted by the h e a d of my d e p a r t m e n t o r by his o r her representat ives . It is u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n of this thesis for f inancial gain shall not be a l l o w e d w i t h o u t my wr i t ten p e r m i s s i o n . D e p a r t m e n t of Ct'jd IrnflUHfgft T h e Univers i ty of Brit ish C o l u m b i a V a n c o u v e r , C a n a d a D E - 6 (2/88) A B S T R A C T The occurrence of liquefaction and instability in saturated sands has been investigated under small departures from the undrained condition. Volumetric deformations are inevitable in field problems on account of the spatial variations in excess pore pressures generated during the earthquake shaking and their subsequent dissipation after, considered together with the relatively high permeability of sands. The departures from the undrained mode were simulated in the laboratory by loading sand under the strain path control, that simulated, not only zero, but small finite amount of controlled volumetric deformations. It is shown that the undrained state of loading normally assumed during earthquake shaking may not represent the most damaging scenario with regard to instability and liquefaction for certain initial density and stress states. At a given initial stress state, the denser sand may become susceptible to liquefaction and prone to instability if rendered partially drained as opposed to when it is completely undrained. For a given density, this potential for liquefaction and instability increases, both with increase in confining stress and static shear. The zone of contractive deformation in stress space where strains associated with strain softening on triggering of liquefaction occur gets enlarged substantially when small expansive volumetric deformations occur. The loose sand, that is liquefiable when undrained, may sustain this vulnerability, although to a lesser degree, even when compressive volumetric strains occur. Criteria for the occurrence of liquefaction and instability to occur under partially drained states have been developed, based on the known behavior of sand in undrained and fully drained conventional (constant confining stress) shear. The effects of the initial state variables - density, confining stress and static shear on partially drained response are assessed both in the triaxial compression and extension modes of deformations, taking cognizance of the inherent anisotropic nature of water deposited sands. For a given initial state, the sand, if liquefiable when undrained, continues to be so over a substantial range of even compressive volumetric strains. Finally, the similarities and differences between the phenomenon of strain softening associated with liquefaction problems and the conditions for instability under constant shear stress are pointed out. ii The association of strain increment directions with stress and stress increment directions are also explored under partially drained states in the region of effective stress space that does not involve strain softening. 111 TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF FIGURES viii LIST OF SYMBOLS xiv ACKNOWLEDGMENTS xvii CHAPTER 1 INTRODUCTION 1 CHAPTER 2 BACKGROUND 5 2.1 Liquefaction 5 2.1.1 Region of Undrained Strain Softening Deformation 8 2.2 Instability 9 2.2.1 Review of Strain Softening and Instability 14 2.3 Strain Increment, Stress and Stress Increment Directions during Deformation 16 2.4 Summary and Research Objectives 18 CHAPTER 3 EXPERIMENTAL WORK 20 3.1 Testing Equipment 20 3.2 Material Tested 23 3.3 Sample Preparation Technique 25 3.4 Specimen Size 26 3.5 Method of Sample Preparation 26 3.6 Test Errors 29 3.6.1 Internal vs. External Measurements 29 3.6.2 Ram Friction 29 3.6.3 End Restraint 29 3.6.4 Bedding Error 33 3.7 Membrane Penetration 34 3.7.1 Factors Influencing Membrane Penetration 35 3.7.2 Membrane Penetration Volume Corrections with DPVC 37 3.8 Test Procedure 38 3.8.1 Sample Set-Up 38 3.8.2 Consolidation 38 iv 3.8.3 Effect of Aging after Consolidation 40 3.8.4 Shear Tests 40 3.8.5 Instability Tests 40 3.9 Capabilities and Performance of the Apparatus, and Confidence in Test Results 43 3.9.1 Repeatability in Test Results 43 3.9.2 Rate of Loading to Ensure no Excess Pore Pressures in Strain Path Control Element Test 45 3.9.3 Equivalence of Response by Stress and Strain Path Controls 45 3.9.3.1 Conventional drained and undrained triaxial compression . . . . 45 3.9.3.2 K„ - coefficient of earth pressure at rest with and without membrane corrections 48 3.10 Experimental Program 48 3.11 Summary 50 CHAPTER 4 TEST RESULTS AND ANALYSIS 52 4.1 Introduction 52 4.2 Undrained Behavior 53 4.2.1 In Compression 53 4.2.2 Region of Undrained Contractive Deformation 56 4.2.3 In Extension 60 4.2.4 Summary - Basic Characteristics of Undrained Behavior 64 4.3 Fully Drained Behavior 64 4.3.1 During Consolidation 64 4.3.2 In da'r = 0 Triaxial Paths 64 4.3.3 Summary -Basic Characteristics of Fully Drained Behavior 74 4.4 Partially Drained Behavior 76 4.4.1 Loose Sand in Compression 76 4.4.1.1 Influence of Initial Effective Stress Ratio and Confining Stress 85 4.4.1.2 Influence of Relative Density 91 4.4.2 Partially Drained Response in Extension 91 4.4.3 The Region of Strain Softening Deformation under Partially Drained Shear 95 4.4.4 Implication of the Observed Behavior 102 4.5 Criterion for Strain Softening Under Partially Drained Conditions in Compression 104 4.6 Summary 117 CHAPTER 5 INSTABILITY UNDER PARTIALLY DRAINED CONDITIONS 120 5.1 Introduction 120 5.2 Repeatability in Instability Tests 121 5.3 In Compression - Loose Fraser River Sand 123 5.3.1 Initial States Inside the Region of Undrained Strain Softening Deformation 123 5.3.2 Ageing Effect Inside the Region of Undrained Strain softening Deformation 129 5.3.3 Initial States Outside the Region of Undrained Strain Softening Deformation 131 5.3.4 Initial States Inside the Region of Undrained Strain Softening Deformation Region But Below the Cut-off "ab" 133 5.3.5 Initial States Outside the Undrained Strain softening Deformation Region Below the Cut-off "ab" 135 5.4 Instability Under Partially Drained Conditions in Compression - Denser Deposition State 135 5.5 Instability Under Partially Drained Conditions in Extension 138 5.6 Comparison between Instability and Strain Softening 142 5.6.1 Further Comments on Strain Softening and Instability 150 5.7 Summary 152 CHAPTER 6 STRAIN INCREMENT, STRESS AND STRESS INCREMENT DIRECTIONS 156 6.1 Introduction 156 6.2 Stress and Strain Parameters 156 6.3 Loading and Unloading 159 6.4 a, p, 9 Relationships 159 6.4.1 a, P, 9 Relationships for States with Identical Past Stress/Strain History 162 6.4.1.1 Influence of Stress Ratio Level 166 6.4.1.2 Influence of Effective Confining Stress 173 6.4.2 a, p, 9 Relationships for States with Different Stress History 179 6.4.3 Ageing (Rest Period) and its Related Memory Effect 184 6.5 Summary 187 CHAPTER 7 SUMMARY AND CONCLUSIONS 190 7.1 Recommendations for further Studies 194 REFERENCES 195 APPENDIX A MEMBRANE PENETRATION CORRECTION CURVES 203 A-l Factors Influencing Membrane Penetration in more Detail 205 vi APPENDIX B DEPENDENCE OF deyde. ON o'r AND R FOR OTTAWA APPENDIX C DRAINED do'=0 EXTENSION RESPONSE OF FRASER RIVER SAND APPENDIX D PARTIALLY DRAINED TEST RESULTS AT o're = 50, 100, 400 kPa vii LIST OF FIGURES Fig. 1.1 Undrained, folly drained and partially drained effective stress paths and the associated strain paths 2 Fig. 2.1 Typical strain hardening, flat plateau, partial, and full strain softening undrained response 6 Fig. 2.2 Schematic representation of the zone of contractive deformation under undrained triaxial compression 10 Fig. 2.3 Strain softening response characteristics of dense sand under drained conditions and of loose sand under undrained conditions 12 Fig. 2.4 Illustration of stable and unstable behavior under drained and undrained conditions (experimental data adopted from Lade et al., 1987) 13 Fig. 2.5 Inclination of the plastic strain vector - independent of: (a) on the stress increment direction and (b) on prior stress history 17 Fig. 3.1 Schematic illustration of the stress/strain path control triaxial device 21 Fig. 3.2 Grain size distribution curve of Fraser River sand 24 Fig. 3.3 Specimen preparation by water pluviation 27 Fig. 3.4 (a) Comparative undrained response in extension with conventional and frictionless ends and (b) profile of typical specimens with free ends in triaxial compression and extension 31 Fig. 3.5 Comparison of drained compression on dense sand with free and regular ends 32 Fig. 3.6 Membrane penetration correction curves (a) error in volume change (b) dependence of unit membrane penetration on relative density for three sands 36 Fig. 3.7 Consolidation paths along constant R and d a'r = 0 paths 39 Fig. 3.8 Effect of small ageing on sand response 41 Fig. 3.9 Instability test procedure (at point R, the strain path is imposed) 42 Fig. 3.10 Repeatability of undrained compression tests 44 Fig. 3.11 Repeatability of undrained compression response of used and virgin sand . . . . 46 Fig. 3.12 Equivalence of conventional da'r = 0 drained behavior under stress and strain path control 47 viii Fig. 3.13 K„ measurement with and without membrane correction for two sands 49 Fig. 4.1 Undrained response of loosest deposited Fraser River sand (Effect of effective confining pressure) 54 Fig. 4.2 Undrained response of loosest deposited Ottawa sand Lower cut-off "ab" in comparison with Fraser River sand 55 Fig. 4.3 Undrained response of loose Fraser River sand (Effect of effective confining pressure at constant initial stress ratio) 57 Fig. 4.4 Undrained response of loosest deposited Fraser River sand (Effect of initial stress ratio at constant a'rc) 58 Fig. 4.5 Undrained response of loose Fraser River sand Delineation of undrained region of contractive deformation 59 Fig. 4.6 Comparison of undrained compression response of loose (L) and medium-dense (M) sand at K,. =1.0 and K c = 2.0 61 Fig. 4.7 Undrained extension response for loose Fraser River sand 62 Fig. 4.8 Undrained extension response for medium-dense Fraser River sand 63 Fig. 4.9 Strain path ^  - Sj during isotropic and anisotropic consolidation for Fraser River sand 65 Fig. 4.10 Drained da'r = 0 compression response of loose Fraser River sand 66 Fig. 4.11 Drained dc'r = 0 compression response of medium-dense Fraser River sand 67 Fig. 4.12 Relationship between de^ de., R, and a'r during da'f=0 shear - loose Fraser River sand 69 Fig. 4.13 Relationship between dejde^ R, and a'r during da'r=0 shear for medium-dense Fraser River sand 70 Fig. 4.14 Dependence of maximum expansion rate on effective confining pressure in da'r=0 drained compression shear of loose, medium-dense, and dense Fraser River sand 72 Fig. 4.15 Dependence of maximum expansion rate on confining stress in da '=0 drained triaxial extension shear 73 Fig. 4.16 (a) Comparison of maximum expansion rate in do'r=0 triaxial extension (TE) and triaxial compression (TC) of Fraser River and Ottawa sands (b) contraction rates prior to the instant of maximum contraction in TC and TE of Fraser River sand 75 Fig. 4.17 Typical range of imposed partially drained strain paths and the conventional undrained and da'r=0 drained strain paths 77 ix Fig. 4.18(a) Partially drained response of loose Fraser River sand at fixed c'^  (stress-strain response) 78 Fig. 4.18(b) Partially drained response of loose Fraser River sand at fixed o'rc (pore pressure response) 79 Fig. 4.18(c) Partially drained response of loose Fraser River sand at fixed a'rc (effective stress paths) 80 Fig. 4.19 Variation of shear stiffness with the direction of effective stress increment at a given ambient stress state 83 Fig. 4.20 Void ratio changes during partially drained test under a large expansive imposed strain increment ratio of -0.40 for loose sand 84 Fig. 4.21(a) Response in partially drained shear at R,. = 1.5 ; a'rc = 200 kPa 86 Fig. 4.21(b) Response in partially drained shear at R,. = 2.1 ; a'rc = 200 kPa 87 Fig. 4.21(c) Response in partially drained shear at R<. = 2.8 ; o'rc = 200 kPa 88 Fig. 4.22 Partially strain softening response under contractive imposed deyde! 89 Fig. 4.23 Response of partially drained tests at R,. = 3.4 ;a' r c = 200 kPa 90 Fig. 4.24 Partially drained response of medium-dense Fraser River sand atR,= 1.0;a'rc = 200kPa 92 Fig. 4.25 Partially drained response of dense Fraser River sand atR,= 1.0;a'rc = 200kPa 93 Fig. 4.26 Comparison of partially drained response in compression (TC) and extension (TE) response at similar imposed de^ de., 94 Fig. 4.27 Partially drained response in extension for the loose Fraser River sand at R, = 1.0 ; o'rc = 200 kPa 96 Fig. 4.28 Partially drained response in compression at constant (dejde^ ratios 97 Fig. 4.29 Effect of (deydej; on the lines of peak states 98 Fig. 4.30 Lines of peaks under partially drained condition for medium-dense sand 99 Fig. 4.31 Strain softening response only at high R,. states for medium-dense sand with (deydei); = - 0.2 indicating the movement of cutoff "ab" away from origin . . 101 Fig. 4.32 Dependence of lines of peaks on imposed (dejde^ in partially drained extension 103 Fig. 4.33 Key features of (a) do'r = 0 drained and (b) undrained response (c) (dejde^^i as a state variable 105 Fig. 4.34 o'r, R excursion during partially drained shear (Loose Fraser River sand) 107 Fig. 4.35(a) Partially drained stress paths superimposed on Fig. 4.12 from do'r = 0 shear (Loose FRS) 108 Fig. 4.35(b) Partially drained stress paths at R,. =2.8 superimposed on Fig. 4.12 from do', = 0 shear (Loose FRS) Ill Fig. 4.36 o'„ R excursion during partially drained shear (Medium-denseFRS) 112 Fig. 4.37(a) Partially drained stress paths at R,. =1.0 superimposed on Fig. 4.13 from da'r = 0 shear (Medium-dense FRS) 113 Fig. 4.37(b) Partially drained stress paths at R,. = 2.0 superimposed on Fig. 4.13 form do'r = 0 shear (Medium-dense FRS) 115 Fig. 5.1 Repeatability of instability test results 122 Fig. 5.2 Partially drained Instability when initial states are inside region of undrained strain softening deformation 125 Fig. 5.3 Partially drained instability from initial states inside the region of undrained strain softening deformation (Effect of an increase in R inside the region) 127 Fig. 5.4 Instability tests inside the region of undrained strain softening deformation (Effect of ageing time at identical stress states) 130 Fig. 5.5 Partially Instability compression for states outside the undrained region of strain softening deformation 132 Fig. 5.6 Partially instability tests for states inside the region Oab 134 Fig. 5.7 Partially instability tests for states outside the region undrained strain softening deformation with stress ratio of 1.5 136 Fig. 5.8 Partially drained instability tests on denser Fraser River sand - states inside the region of loose undrained strain softening deformation 137 Fig. 5.9 Partially drained instability tests in extension - initial states outside the undrained extension strain softening deformation region) 139 Fig. 5.10 Partially drained instability tests in extension mode on medium-dense sand - initial states outside the undrained strain softening deformation 141 Fig. 5.11 Partially drained instability tests in extension mode (comparison of loose and medium-dense) 143 xi Fig. 5.12 Comparison of strain softening (S) and instability (I) at R,. = 2.8 (Effect of resting period = delay in instability) 145 Fig. 5.13 Comparison of strain softening (S) and instability (I) at R,. = 2.1 at two effective confining pressures 147 Fig. 5.14 Comparison of strain softening (S) and instability (I) at R,. = 2.0 148 Fig. 5.15 Comparison of strain softening (S) and instability (I) at R,. = 1.5 149 Fig. 6.1 Schematic representation of stress, stress increment and strain increment at a point 158 Fig. 6.2 Zones of loading and unloading in p'-q stress space in terms of p' 160 Fig. 6.3 Zones of loading and unloading in p'-q stress space in terms of c'r 161 Fig. 6.4 Initial states o'r, R considered for examining the a, P, 9 relationships 163 Fig. 6.5 Stress/strain path directions at R,. =2.8, = 200 kPa 164 Fig. 6.6 a, p, 9 relationships at R, = 2.8, o're = 200 kPa 165 Fig. 6.7 a, p, 9 relationships at Rc = 2.1, o'rc = 200 kPa 167 Fig. 6.8 a, p, 9 relationships at R, = 1.5, o'rc = 200 kPa 168 Fig. 6.9 a, p, 9 relationships at R, = 1.0, o're = 200 kPa 169 Fig. 6.10 Summary of a, p, 9 relationships at o're = 200 kPa 170 Fig. 6.11 Stress and strain increment vectors for different stress ratios at o'rc = 200 kPa 172 Fig. 6.12 Effect of confining pressure on a, P, 9 relationships at R,. = 1.5 174 Fig. 6.13 Effect of confining pressure on a, p, 9 relationships at R,. = 2.1 176 Fig. 6.14 Strain increment directions along constant stress ratio of 2.1 177 Fig. 6.15 Effect of confining pressure on a, p, 9 relationships at R,. = 2.8 178 Fig. 6.16 Strain increment direction along constant stress ratio of 2.8 180 Fig. 6.17 a, P, 9 relationships for states with different stress history atR^l.5 181 Fig. 6.18 a, P, 9 relationships for states with different stress history at ^  = 2.1 182 Fig. 6.19 a, P, 9 relationships for states with different stress history atRe = 2.8 183 Fig. 6.20 Ageing and memory effect 185 Fig. 6.21 Influence of initial stress ratio on ageing effect 186 xii Fig. 6.22 Similar initial stiffness for a range of imposed strain increment ratios when the ageing effect dominates 188 Fig. A-l Grain size distribution of Fraser River, Ottawa, and Silica sands 207 Fig. A-2 Membrane penetration correction curves (a) Ottawa sand (b) Silica sand 209 Fig. B-l Relationship between deJdEj^, R, and da'=0 during do'=0 shear (loose Ottawa sand) 211 Fig. C-1 Drained dc'r=0 extension response of loose Fraser River sand 212 Fig. C-2 Drained da'T-0 extension response of Fraser River sand at fixed confining pressure (Effect of relative density) 213 Fig. D-1 Partially drained response at R,. = 1.0 ; o'rc = 400 kPa 214 Fig. D-2 Partially drained response at R,. = 1.5 ; o'rc = 400 kPa 215 Fig. D-3 Partially drained response at R,. = 2.1 ; o'rc = 400 kPa 216 Fig. D-4 Partially drained response at R,. = 2.8 ; o'rc = 400 kPa 217 Fig. D-5 Partially drained response at R,. = 1.0 ; o're = 100 kPa 218 Fig. D-6 Partially drained response at R<. = 1.5 ; a'rc = 50 kPa 219 Fig. D-7 Partially drained response at R, = 2.1 ; a'rc = 50 kPa 220 Fig. D-8 Partially drained response at Rc = 2.8 ; o're = 50 kPa 221 xiii LIST OF SYMBOLS I. NOTATIONS A,,, = total surface area of specimen in contact with the membrane D = 1- dejdej^ = dilatancy factor D r i = initial relative density of specimen as prepared (under initial effective stress of 20 kPa) D r c = relative density after consolidation du = excess pore water pressure ec = void ratio after consolidation e; = initial void ratio of specimen as prepared (under initial effective stress of 20 kPa) G s = [(aa - o r) / 2 y] = strain secant shear stiffness K c = c ' l c / a ' 3 c = consolidation effective stress ratio; consolidated along K c path K 0 = coefficient of earth pressure at rest m = normalized unit membrane penetration, the slope of sm vs. log a ' r p' = 1 / 3 (a'j + 2a' 3) = mean effective stress q = al-c3 = deviator stress R = a\ I a ' 3 = effective stress ratio R<. = c ' l c / a ' 3 c = consolidation effective stress ratio; consolidated along hydrostatic then da ' 3 paths s' = V2 (o'. + a' 3) t = ' / 2 (a ' 1 -a ' 3 ) u = pore water pressure V T = total sample volume a = tan"1 (dq /dp') = stress increment direction P = tan"1 (des / dej = strain increment direction y = y 2(de a-de r) A u m a x = maximum excess pore pressure 5 = tan"1 (At/As') = stress increment direction in t-s' space 6! = major principal strain xiv e3 = minor principal strain sa = axial strain 8, = radial strain es = % - 8 3) = deviator shear strain = S j + 2e3 = volumetric strain dsydej = strain increment ratio em = membrane induced volume change per unit area = true skeleton volumetric strain T| = q / p' = effective stress ratio r|c = qc / p'c = consolidation effective stress ratio 9 = tan"1 (r\) = stress direction v = Poisson's ratio a\ - major effective principal stress CT'3 = minor principal stress °"'ic> ° ' 3 c = major and minor principal stresses at the end of consolidation o\ = axial effective stress o'r = radial effective stress o'd = (ax - o3) = deviator stress (pCSR = friction angle at critical stress ratio (pMO = friction angle at maximum obliquity cp P T = friction angle at phase transformation II. ABBREVIATIONS CSR = critical effective stress ratio CT = characteristic threshold CTC = conventional triaxial compression CTE = conventional triaxial extension DPT = differential pressure transducer DPVC = digital pressure/volume controller xv ESP = effective stress path FSS = fully strain softening LVDT = linear variable differential transformer PSS = partial strain softening QSS = quasi steady state SS = steady state SH = strain hardening TSP = total stress path xvi A C K N O W L E D G M E N T S The author is grateful to his supervisor, Professor Y.P. Vaid for his constant support, guidance and encouragement during the course of this study. Grateful appreciation is expressed to Professors P.M. Byrne, J.A. Howie, and Dr. M.K. Lee (B.C. Hydro) for their helpful critical comments and suggestions. The authors also wishes to express his appreciation to other members of the examining committee, Professors M.J. Pender (The University of Auckland), R. Chase, R. Pakalnis, and R. Foschi for their valuable suggestions. The author would also like to thank Professors W.D.L. Finn and R.J. Fannin for their encouragement and interest on the subject. Special thanks go to Professor K. Uchida of Kobe University, Japan for his generosity and helpful discussion at the early stage of the research. The author also wishes to thank Professor S-C R Lo from University of New South Wales, Australia for the discussion on free-ends. The helpful discussion with his colleagues in the geotechnical research group are also appreciated. Former student H. Puebla is also thanked for his friendship and mutual interest on variety of subjects. The help of Civil Engineering Workshop in the development and maintenance of testing device is gratefully acknowledged. The author would like to thank the Natural Science and Engineering Research Council of Canada for their financial support. Finally, the author would like to extend his deepest gratitude to his parents, and especially his wife, Shohreh, for her patience and encouragement, and also his lovely son and daughter, Danyal and Dorsa, for being the understanding children of a graduate student. xvii CHAPTER 1 INTRODUCTION The solution of deformation problems of soils requires a confident characterization of their effective stress-strain behavior. This behavior, in general, is nonlinear, inelastic, and dependent on stress history and stress path. On application of load, the effective stress path experienced by a saturated soil element in an earth structure depends on the induced spatial distribution and changes with time of both total stresses and excess pore pressures. These changes are governed by the boundary conditions on stresses and displacements, coupled with the transient flow triggered by variations in excess pore pressure over the entire loading domain of the geotechnical structure. Since most field problems are of the mixed boundary value type (stresses specified on some boundaries and displacements on others), and because the excess pore pressures cannot dissipate immediately, except at the drainage boundaries, the total stresses on soil elements undergo time dependent changes during dissipation of pore pressure; even if the boundary loads remain constant. Changes in total stress and dissipation of excess pore pressure occur because of the inherently coupled nature of the problem - equilibrium and diffusion (Biot, 1941). Traditionally, response of a saturated soil element to an increase in applied stress has been assumed to lie between the two extremes - fully drained and undrained. While the connotation with the term "undrained" is explicit (as it signifies zero volume change, which implies loading along a prescribed strain path), the term "drained" is often viewed as the behavior recorded, for example, in conventional drained triaxial type of loading (Fig. 1.1a, paths © and ©', assuming an axisymmetric 1 3 state of stress). In fact, the conventional drained triaxial loading is only one of an infinite number of possibilities by which increments of effective stresses can be applied on the soil element, and merely represents loading along a linear effective stress path, do'r = 0, that implies no time lag between the application of total and effective stresses, i.e. do = da'and du = 0. In most field situations, however, the soil elements drain and experience change in volume and pore pressure with time (dey * 0, du * 0) simultaneously. The effective stress path to which they are subjected are then the paths in which do'r * 0, and du * 0. The soil response would be fully drained only if on loading the total and effective stress paths are parallel (pairs ©, ©' and (D, ©'), so that the loading corresponds to do = do' and du = 0 at all times. If they are not, the deformation response should be regarded as partially drained. Undrained response that enforces zero volume change amounts to a special case of this suite of partially drained responses, and since it represents loading along a unique strain path, the effective stress path the soil experiences is unique and independent of the applied total stress path (Henkel, 1960; Vaid and Campanella, 1974). Experimental shake table and field studies on liquefaction by Seed (1987), Liu and Qiao (1984) and Whitman (1985) have demonstrated that drainage, however small, does occur in earth structures both during and soon after even the short duration earthquake loading. This can be attributed to the spatial variation in the excess pore pressures generated and their concurrent dissipation in the loading domain due to the high permeability of sands. In most geotechnical problems, the partially drained scenario is in fact the norm rather than an exception. At certain initial stress states, partial drainage may constitute loading conditions more damaging than either of the conventional drained or undrained assumptions. This was recently demonstrated experimentally in laboratory studies of liquefaction in denser sand, in which very small expansive volumetric strains transformed an undrained strain hardening response into a strain softening type (Vaid and Eliadorani, 1998). Large changes in the direction of the effective stress increment due to a small change in the strain increment direction was identified as the cause behind such a transformation. Since the deformation response of soil at a given ambient state depends upon the direction of the effective stress increment, ignoring partial drainage can give an unrealistic assessment of soil deformation and strength. This implies transformation of the effective stress increment direction from strain hardening to strain softening on account of a very small change in the direction of the strain increment, if the sand was undrained. 4 The objective of this thesis research is to investigate experimentally the partially drained response of sands, with particular emphasis on its impact on the occurrence of liquefaction and instability. Instability is a phenomenon in which the sand suffers runaway deformations at essentially constant shear stresses. This can be caused by a mere increase in pore pressure in certain zones of the earth structure, during the dissipation phase of the excess pore pressure generated by the dynamic event. The study was carried out in a specially designed triaxial device that allowed loading under either stress or strain path control. The relationship between effective stress and strain increments at an ambient effective stress state can be established by laboratory element tests, in which controlled effective stress increments are applied and the induced strain increments recorded (Fig. 1.1b). Alternatively, controlled strain increments may be applied, and the effective stress increment to induce this strain is recorded (Vaid et al., 1999). The former constitutes loading along a controlled stress path, in contrast to the controlled strain path in the latter case. The effects of confining pressure, initial static shear and density on partially drained behavior was systematically studied and compared to the behavior under conventional drained and undrained shear. Criteria were developed for predicting the occurrence of liquefaction (strain softening) in partially drained elements that are strain hardening if undrained. The criteria are based on behavior observed under conventional drained and undrained paths. The associations of the strain increment directions to effective stress and stress increment directions were also examined along the partial drained strain path at effective stress states and the influence of stress/strain history prior to the stress state at which these associations are assessed. CHAPTER 2 BACKGROUND The objective of this chapter is to present a brief review of the current state of knowledge on the topics of liquefaction and instability in saturated sands under undrained loading. This background is necessary as it provides a framework for understanding these phenomena under partially drained conditions. 2.1 Liquefaction The term liquefaction encompasses all phenomena that involve excessive deformations in saturated granular materials. This definition is independent of the initiating disturbance which could be static, dynamic, wave action or shock loading. The range of static undrained behavior of saturated sands under monotonic loading can be schematically represented as in Fig. 2.1. The changes in responses shown in curves © to © are associated with increasing relative density at the same initial effective confining stress. The response indicated by curve © represents a strain softening (also termed contractive) behavior, which shows a large reduction in soil resistance at large strains. It is a strain softening response with unlimited unidirectional strain. Excess pore pressure increases with strain until a certain limiting effective stress ratio is reached. This effective stress ratio corresponds to the maximum deviator stress, which has been called the Critical Stress Ratio (CSR) by Vaid and Chern (1983), collapse surface by Sladen et al. (1985), and instability line by Lade and his co-workers (Lade et al., 1987, 1988; Lade, 1992) marks the initiation of the strain softening behavior. It has been suggested by 5 ( a ' ! + a ' 3 ) / 2 Typical strain hardening, flat plateau, partial, and full strain softening undrained response 7 Vaid and Chern (1985) and Dobry et al. (1985) that the CSR is unique for a given sand in compression loading. However, Sladen et al. (1985) show that it varies somewhat with void ratio and confining stress. The CSR in extension loading is smaller than that in the compression mode. Moreover, unlike the CSR in compression mode, the CSR in extension is a function of the deposition void ratio (Vaid et al., 1989). Once the CSR is reached, the soil exhibits a sudden loss in strength accompanied by a rapid increase in strain and pore pressure. Soil resistance eventually reaches a minimum value at very large strains. This minimum soil resistance was termed steady state strength by Castro (1969). It has also been called the residual strength. The phenomenon associated with loss in strength has been termed liquefaction (Castro, 1969; Casagrande, 1975; Seed, 1979; Vaid and Chern, 1985). The characteristic feature of the type © response is continued deformation at constant void ratio, confining stress and shear resistance. This has been called the steady state (SS) deformation or flow deformation, as it resembles the flow of fluid (Poulos, 1981; Castro, 1975; Vaid and Chern, 1983). However, the shear resistance during such deformation is of a frictional nature, instead of zero, which would be the case for a fluid. The response of a contractive sand (curve ©) which exhibits reduction in strength at some stage of loading, yet regains it with increasing deformation, has been called limited liquefaction by Castro (1969) and Vaid and Chern (1985). As in the liquefaction type of response, strain softening is initiated at the CSR and continues until a minimum undrained resistance is reached (point "n" in Fig. 2. la). This is called the quasi steady state (QSS). Instead of developing unlimited deformation, the soil now begins to regain strength and the excess pore pressures decrease with further straining. i The state where the soil starts to increase its shear resistance and the pore pressure is a maximum (QSS) is reflected by a|sharp turn around in the effective stress path (Fig. 2. lb). Such a state has also been termed Phase Transformation (PT) by Ishihara et al. (1975), and Characteristic Threshold (CT) by Loung (1980). | The PT/QSS state has been shown to occur at the same mobilized effective stress ratio, for a given|sand, regardless of its initial density and stress state (Loung, 1980; Dobry et al., 1985; Vaid and Chern, 1985). The effective stress ratio mobilized at the PT/QSS state has been found to be equal to that at steady state under undrained loading (Chern, 1985). Furthermore, 8 Negussey et al. (1988) have shown that for rounded Ottawa sand, the effective stress ratio mobilized at PT/QSS/SS equals that at the point of maximum contraction in conventional drained shear. Response type (D is a terminal case of strain softening in which the degree of strain softening can be considered as zero. Such a behavior is characteristic of a flat plateau in the stress-strain curve over a certain strain range, before the shear resistance starts to increase and pore pressure to decrease with further straining. The strain hardening behavior of sand with no loss of shear resistance at any stage of loading is typical of curve ®. The soil resistance keeps increasing with increasing deformation. Unlike the limited liquefaction type of response, no sharp turn around is observed in the effective stress path diagram. However, as shown in Fig. 2.1(b), a more gradual turn around is observed at the point where the PT line is crossed. For types ® to © responses, the effective stress path after the PT state proceeds along the undrained failure envelope which represents the line of maximum obliquity (MO) in undrained shear. 2.1.1 Region of Undrained Strain Softening Deformation Using comprehensive studies on two sands, Vaid and Chern (1985) have shown that under undrained triaxial compression (i.e., de^ de! = 0, in which 6% = volumetric strain and dex = major principal strain = dea), saturated sands respond in a strain-softening manner over a range of initial stress and void ratio states. The strain softening, if any, is triggered at the mobilized CSR friction angle which is considerably smaller than that at the condition of maximum obliquity. This state has been found to be independent of the initial (prior to undrained shear) state of sand (defined by ec, a'lc, a'3c, in which ec = void ratio and o'lc, a'3(. = major and minor principal stresses at the end of consolidation) in the triaxial compression test, provided the friction angle mobilized at the effective stress ratio prior to undrained loading was equal to or less than at which the strain softening was triggered. The CSR line thus represents the lower limit of the region of strain softening deformation. The region of stress space between the CSR line and the phase transformation line has been termed the zone of strain softening deformation, wherein the strains associated with strain softening occur. If the initial state of the sand prior to undrained loading fell in this region of strain softening deformation, a state of 9 spontaneous liquefaction exists. The CSR line has been called the instability line by Lade and his co-workers (Lade et al., 1987, 1988; Lade, 1992). Fig. 2.2 schematically illustrates the region of stress space in triaxial compression within which strain softening deformation develops. The region, which is shown shaded, is bounded by the CSR and the PT/SS lines. It does not, however, extend down to the origin of stress space, but terminates by an approximately horizontal cutoff "ab" at a finite value of (a'j - a'3)/2. The magnitude of this (a\ -o'3)/2 depends on the initial placement density, Dri of the sand. The position of "ab" relative to the origin of stress space moves up as Dri increases (Vaid and Chern, 1985). This cutoff between strain softening and strain hardening undrained behavior was also implied, but not specifically identified, by Lade and his co-workers. 2.2 Instability Many flow slides have been observed in coastal and offshore areas (Terzaghi, 1956; Andersen and Bjerrum, 1968; Bjerrum, 1971) as a result of instability in natural deposits. The description of the circumstances surrounding the great majority of these slides indicate that some relatively sudden perturbation, such as lowering of the water level outboard of the slope was observed almost immediately prior to the initiation of the flow slides. Bjerrum et al., (1961) based on anisotropically consolidated triaxial compression tests on loose fine sands, typical of those involved in flow slides, remarked that"... it was surprising to see, however, the small increase in deviator stress required to cause failure in the undrained state...". Instability has been defined as a condition that occurs when a soil element subjected to a small perturbation cannot sustain the current stress state imposed on it, and a runaway deformation results (Lade et al., 1987; Lade, 1989). In general, when the stress state of a soil element reaches the failure state, the instability in soil can occur. In conventional drained triaxial tests at constant confining stress, the state corresponding to the peak deviator stress (also the instant of maximum obliquity a'j/o'j) would signify the instant of instability. This type of instability, which is accompanied by the development of shear bands or shear planes at failure has been studied by a number of investigators (Lade, 1982; Hettler and Vardoulakis, 1984; Vardoulakis and Graf, 1985). Such a response in dense 10 11 sand as illustrated in Fig. 2.3. Point F represents the instant at which the instability occurs. In this case, the resulting strain softening response is equivalent to instability. The instability due to shear plane development will not be discussed in this thesis. Instability may also occur during undrained loading of a loose saturated sand prior to the attainment of the state of maximum drained obliquity (or inside the failure envelope as shown in Fig. 2.3b) This type of instability has been referred to as pre-failure instability (Lade, 1989; Lade and Pradel, 1990; Chu et al., 1993). The physical basis of pre-failure instability has been examined by Lade et al. (1988). Lade et al. referred to instability as the inability to sustain or carry a given load, including the inability to sustain general perturbations in load. In tests conducted by Lade et al., the sand sample was first loaded drained to a certain stress level along the da'r=0 path (point A in Fig. 2.4). After a stress level was reached inside the region of strain softening deformation (above the CSR line), the drainage valve was closed, with vertical load and the cell pressure maintained constant. Closing the drainage valve together with a small perturbation caused an increase in the pore pressure (this will be shown in the next Subsection) which eventually led to instability, as the effective stress path moved towards the drained failure envelope. This type of instability under undrained conditions in the triaxial apparatus is shown by the circle symbol in Fig. 2.4. It is clear that as the effective stress path moves towards the drained failure line (Fig. 2.4a), the deviator stress (shown in Section 2.2.1 mathematically) decreases (see Fig. 2.4b) while the effective stress ratio increases (Fig. 2.4c). The primary objective regarding the study of instability is to investigate factors affecting its occurrence under partially drained conditions. In studies of sand liquefaction, it is common to relate the pre-failure strain softening (as shown in Fig 2.3 for loose sand) to pre-failure instability (Vaid and Chern, 1985; Sladen and Handford, 1987; Kramer and Seed, 1 9 8 8 ) . Lade et al. (1987) have, however, shown experimentally that for granular soils, unstable behavior may not occur even when the stability condition (e.g. Hill's criterion) is violated. Their experimental data is shown in Figs. 2.4(a) by the star symbol. The experiment under drained condition was performed as follows. After reaching the state A fully drained along da'r=0 path (Fig. 2.4), the specimen was now further loaded drained under load control along a stress path in which the instant of failure (o ' a + o ' r ) / 2 (kPa) Strain softening response characteristics of dense sand under drained conditions and of loose sand under undrained conditions 13 n o t s h o w n (to f a i l u re ) Mean nomral stress O U N D R A I N E D : r u n a w a y s t ra in ( U N S T A B L E ) O D R A I N E D d a ' 3 = 0 : N O r u n a w a y s t r a i n ( S T A B L E ) desp i te d a ' jj de ^ < 0 (start s y m b o l ) a d a p t e d f r o m L a d e et a l . ( 1 9 8 7 ) de v c o u l d be +ve o r - v e > I n e q u l a i t y (2.3) i s s a t i s f i e d Fig. 2.4 Shear strain Illustration of stable and unstable behavior under drained and undrained conditions (experimental data adopted from Lade et al., 1987) 14 deviator stress ( 0 , - 0 3 ) and confining pressure (o'3) were reduced in such a way that the specimen compressed vertically. The resulting dilative volumetric response is illustrated in Fig. 2.4(d). As reported by Lade et al., the specimen "exhibited stable behavior", and based on this experimental evidence, they concluded that unstable behavior may not occur even when Hill's stability condition (see Section 2.2.1) is violated. This conclusion needs examination it is normally believed that there are similarities between strain softening and unstable behavior in granular soils. As such a comparison between the instability and strain softening behavior of granular soils is made as a part of these investigations. 2.2.1 Review of Strain Softening and Instability When the shear resistance of a soil element decreases with further straining after the peak value, it is said to be strain softening. This is shown in Fig. 2.3 for dense and loose sands under drained and undrained shear, respectively. Several definitions of strain softening within the context of plasticity theory have been proposed in the past. However, these definitions require the choice of the yield function and the plastic potential as a priori in elasto-plastic constitutive modeling or solids. Valanis (1985) definition of strain softening is preferred instead since this is stated in terms of total strains instead of plastic strains in elasto-plastic formulations. In this way, the difficulties inherent in decomposing the measured total strain into an elastic and a plastic component are avoided. Valanis (1985) proposed that: If£,., is a parametric representation of strain path in strain space and a 'v is the corresponding effective stress path in stress space, a material is said to be strain softening at a point on the strain path if, at this point, dafao (2.1) Valanis definition of strain-softening can be compared with Hill's stability postulate (Bishop & Hill, 1951; Hill, 1959) as: do^O (2.2) Comparing (2.1) and (2.2) would imply that if the measured stress and strain increment response at a point violates Hill's stability postulate, then strain softening would be equivalent to instability by 15 definition. However, as indicated earlier, Lade and co-workers (Lade et al., 1987; Lade, 1989) have shown that Hill's stability postulate for stability is only a sufficient condition but not necessary. Hence, according to Lade and co-workers, instability may not be defined as the violation of Hill's stability condition. It will be shown in the research undertaken herein that (Chapter 5) this conclusion will depend on drainage conditions together with the definition of instability. Only then, a proper comparison of instability and strain softening can be made. It should be noted that the Valanis definition of strain softening for axisymmetric stress conditions of the triaxial test can be rewritten as: da[dei+2da/:ide3=dp/dev + dqdes<0 (2.3) The terms in Inequality (2.3) are as defined in Cam-clay theory (Schofield and Wroth, 1968): Deviator stress q = Oj - a3 Mean effective stress p' = Vz (a^ + 2a'3) Deviator shear strain des = % (de1 - de3) Volumetric strain d^ = de1 + 2ds3 For a fully drained triaxial compression test (i.e. da'3=0 and dq=do',), the Inequality (2.3) becomes: do[de. +2dajde3 =da[de. =dgdel<0 (2.4) If dex>0 and the soil strain softens due to a drained triaxial da'3=0 probe, then (2.4) implies that dq<0. Similarly using de^ O and thus de1/de3=-2 in (2.3) would lead to dq<0 for undrained conditions as well. This condition is illustrated in Fig. 2.4 in where dq is less than zero (Fig. 2.4b). As the experimental data by Lade et al. (1987) indicate, that the response was strain softening in a fully drained triaxial (da'3=0) probe, yet runaway strain was noted. Due to an undrained triaxial probe (da3=0), on the other hand strain softening can manifest with runaway strain, as da'3=-du. Therefore, the most important factor for the occurrence of instability under undrained triaxial conditions is the generation of excess pore pressure (or decrease in o'3). The data illustrated in Fig. 16 2.4 summarized what has been reported in the literature. Although considerable evidence exists that indicates there are similarities between strain softening and unstable behavior of granular soils in undrained shear, further investigations as to their similarities, if any, under partially drained state need to be investigated. 2.3 Strain Increment, Stress and Stress Increment Directions during Deformation Many design problems in soil mechanics pertain to the amount of deformations permissible rather than a safeguard against complete collapse. This requires that incremental stress-strain relations be known for the soil. Incremental elasto-plastic idealization of soil behavior has been frequently used in numerical modeling for computations of deformations under applied loads. The theory of plasticity (Hill, 1950) provides a general framework within which a mathematical model could be developed for describing the stress-strain relations of the specified material of interest. In the development of elasto-plastic stress-strain relationships the assumption is made that the inclination of the plastic strain increment vector is a function only of the current stress state and not of the stress increment direction. This means that if an element of material at a given state of stress in which a loading increment will produce plastic deformation, the inclination of the resulting plastic strain increment vector will be independent of the direction of the stress increment vector. This is schematically illustrated in Fig. 2.5. The state of stress on the yield locus is represented by the current stress tensor a ' g a . If now a stress increment da'jjab or do';/0 is applied, the resulting plastic strain increment vector de^  will point in the same direction for either stress increment, or even for any other stress increment vector (Fig. 2.5 a). Furthermore, the inclination of this plastic strain increment vector, d8ij P , is assumed independent of the past loading history, i.e., it is independent of the previous stress state, say a'™, a'?, or together with the stress path the stress point followed to reach the current stress c'^ (Fig. 2.5b). The association of strain increment directions to stress and stress increment directions at stress states 17 (b) Fig. 2.5 Inclination of the plastic strain vector - independent of: (a) on the stress increment direction and (b) on prior stress history 18 along linear and nonlinear effective stress paths to which the sand is subjected will be investigated. This is of relevance in constitutive modeling of non-linear materials. The association of only total strain increment direction with stress and stress increment directions would be examined. The implication of this investigation on elasto-plastic constitutive modeling (such as those shown in Fig. 2.5) will be indicated. 2.4 Summary and Research Objectives Considering the aforementioned review of the undrained sand behavior, it is clear that • Very little is known of sand response under partially drained states. The partially drained conditions impose effective stress paths during loading substantially different from the conventional undrained and the fully drained triaxial (da' r =0). The soil response is often considered bounded by these two extremes, although in most field situations, partially drained conditions prevail as a norm rather than exception. • Similarly little is known about the partially drained instability phenomenon under constant total stresses. These scenarios frequently occur in dams and embankments during an earthquake where the seismic pore pressure generated dissipates with time after the cessation of shaking. This dissipation takes place under partially drained rather than fully drained conditions. • The strain softening/instability behavior of sand under partially drained conditions has not been examined in relation to a defined undrained strain softening region in which various factors such as confining pressure, initial effective stress ratio, relative density, and a more detailed range of strain increment ratios should be taken into account. This is especially important if the criteria for the occurrence of full and partial strain softening response can be established. One of the important objectives of this study is concentrated in this area. • Experimental investigation are needed to verify the similarity between the occurrence of strain softening (under constant rate of strain testing) and instability (under constant stress 19 loading). From the limited experimental studies reported in the literature (e.g. Lade et al., 1987, 1988; Chu et al., 1993), instability and strain softening have been shown to differ from each other. • The association of strain increment direction with the state of stress and with the stress increment direction under the loading paths frequently considered are limited to either undrained or fully drained (da'r=0) paths. Little research appears to have been carried out to establish these association under partially drained conditions. The experimental program undertaken in this thesis was intended to shed light on above important issues of soil behavior. CHAPTER 3 EXPERIMENTAL W O R K This chapter describes experimental investigations carried out in pursuit of the objectives stated at the end of Chapter 2. The testing apparatus that incorporates loading under strain/stress path control is described, together with the associated instrumentation, test control and data acquisition system. This is followed by a discussion of test errors, and the steps taken to minimize them. The procedures for specimen reconstitution, consolidation and shear loading are outlined. Confident assessment of specimen void ratio is suggested, and resolutions of stress and strain measurements are discussed. Typical experimental results are presented in order to demonstrate test consistency, repeatability and the equivalence of sand behavior under stress and strain path control loadings. Finally an outline of the experimental program is given. 3.1 Testing Equipment The conventional triaxial apparatus has been widely used for the study of soil behavior. There is, in general, a broad familiarity and acceptance of the behavior characterized by the conventional triaxial test. Consequently, the decision was made to investigate the stated objectives (Chapter 1) utilizing the triaxial test. The arrangement for the testing system is shown schematically in Fig. 3 .1. Axial load, confining cell pressure, pore pressure and axial displacement were measured using electronic transducers (load cell, pore pressure, and displacement transducers, respectively) coupled to a data acquisition system interfaced with a microcomputer. During stress path controlled tests, a sensitive differential pressure 20 21 > PH Q O T 3 3 •c I o o •§ ft .S3 «j CO CO co «4H O to 3 O m GO 22 transducer (DPT) was used to electronically monitor volume changes (Campanella and Vaid, 1972; Tatsuoka, 1981). The axial loading system is capable of applying compression or extensional loads to the triaxial specimen under stress or strain controlled conditions. Coupling of a double acting frictionless air piston in series with a constant speed drive (strain drive) allows for a smooth transition from stress controlled to strain controlled loading and vice versa. In addition, the air piston assists to consolidate specimens anisotropically, if needed, and allows compensation for the vertical uplift on the loading ram, if hydrostatic consolidation is desired. Triaxial specimens were approximately 64 mm diameter and 130 mm long. Smooth hard anodized aluminum end platens with centrally located 20 mm diameter porous discs were used in order to minimize end restraint. Nevertheless, some comparative tests using both free and regular ends were performed, before a decision in favor of the fixed ends was made. These results will be presented in the next sections. The selected sample geometry and improved membrane penetration corrections for tests in which the effective confining stress changes (Vaid and Negussey, 1984), together with the use of high resolution transducers, data acquisition and control systems enabled confident and consistent measurements of both axial and volumetric deformation with a resolution in the order of 3 x 10"5. During the strain path control, the volume change of the test specimen was measured/controlled by a Digital Pressure Volume Controller (DPVC) via the specimen top drainage line. The principle of a DPVC in soil testing was detailed by Menzies (1987), and is illustrated in Fig. 3.1. A digital signal from a micro-processor controls the stepping motor which, in turn, targets either a pressure (feed-back from the built-in pressure transducer) or a volume change (by counting the number of steps of the stepping motor). Thus, the volumetric changes of the saturated specimen is controlled continuously by sending signals from the micro-processor to achieve the required volumetric strain. Each pulse corresponded to a volumetric strain of 3 x 10"6 for the size of triaxial specimen used. The DPVC was connected to the specimen top, and the bottom drainage line to the pressure transducer to measure the pore pressure within the specimen. 23 Data acquisition was performed with a National Instrument 16 bit high-speed A/D card. All transducers were selected with careful consideration as to their stability and sensitivity. The high resolution data system eliminated need to amplify transducer signals before these measurements by the A/D converter. Averaging 60 reading for each data channel yielded the highest measurement resolution. The data was scanned continuously at 0.2s interval during the test. The traixial cell was designed with a special continuously air leaking frictionless seal and an external load cell was used. The load cell had a resolution of ±10 grams, which for the specimen area of 30 cm2 represents a resolution of deviator stress of ±0.04 kPa. The axial strain had a resolution of 3 x 10"5. The resolution of cell and pore pressure transducer were at ±0.1 kPa. The data collected were reduced and simultaneously displayed on the computer monitor. The measured axial and radial stresses (ca and or) were corrected for membrane strength, which becomes particularly important at low levels of effective stresses (Kuerbis and Vaid, 1990). 3.2 Material Tested The tests were performed on sand dredged from the Fraser River in British Columbia. This sand was selected since it underlies the heavily populated Fraser Delta, a region of high seismicity, and hence prone to liquefaction and instability. The dredged material was wet washed through a 1.00 mm sieve and the material retained was discarded. Fraser River sand is uniform, grey colored, medium grained with sub-angular to sub-rounded particles. The average mineral composition is 40% quartz, quartzite and chert, 11% Feldspar, 45% unstable volcanic rock fragments and 4% miscellaneous detritus (Garrison et al., 1969). The grain size distribution of the sand is shown in Fig. 3.2. The batch of this sand which was used in these investigations is somewhat different from the sand investigated in previous studies (e.g. Thomas, 1992). This batch has an average particle size (D50) of 0.35 mm and the coefficient of uniformity (Cu) of 1.72. The specific gravity is 2.695, according to ASTM 854, and the maximum and minimum void ratios determined according to ASTMD4253 and D4254 were found to be 0.900 and 0.594 respectively. 24 C/3 T 3 C*3 rt O u T3 CD C/3 l-I o U OtI # 0L# o r # 81 # i i i i i 1 I 1 i i : • i i i i i 9 T3 i / 9 ! / i / £ i ...! L / S i i (/> . • : . . . ' j. * ! I ! ( • ' i i i JO / ;/ T ; ; ; A i i i i i i o © o o o o o o o 00 o o ""4- o -a d > CU on rt <-i PH O I O o 'B '-3 _N g ' r t o <N CI 25 Two other sands; Ottawa sand and Silica sand, were also used, to supplement findings on membrane penetration, presented in a later section of this chapter. Results on Ottawa sand were used for occasional comparison of behavior with Fraser River sand, and where generalization of the findings were sought. The gradation curves for these sands are given in Appendix A (Fig. A-1). Ottawa sand is uniformly graded with an average particle size of 0.4 mm and maximum and minimum void ratios of 0.82 and 0.50, respectively. The Silica sand is also uniformly graded, but coarser, with D 5 0 of 0.9 mm and maximum and minimum void ratios of 0.60 and 0.42, respectively. 3.3 Sample Preparation Technique Using undisturbed samples in the laboratory would be ideal for the study of sand behavior in situ. However, as indicated by Seed et al. (1982), the conventional undisturbed sampling techniques invariably alters the mechanical properties of sands. In-situ freezing is regarded as the most desirable technique for retrieving an undisturbed sample(e.g. Yoshimi etai., 1984,1989). It is, however, very expensive. Therefore, fundamental property characterization of water-deposited fluvial and hydraulic fill sands is frequently carried out using reconstituted specimens. Since water pluviated sands in the laboratory has been shown to posses fabric similar to those of the fluvial and hydraulic fill sands (Oda, 1972), this technique of reconstitution of specimens provides a convenient method of studying in-situ properties of these materials. A close agreement between the behavior of sand retrieved by in-situ freezing and its water pluviated counterpart at the same initial state has been recently reported (Vaid et al., 1996). A test sample is assumed to represent a point in stress space. An ideal sample preparation technique should be able to deliver several samples which are homogenous in density and structure. For meaningful correspondence and repeatability of test results, a high degree of sample reproduction capability is essential. As indicated by Vaid andNegussey (1988), pluviation through water provides an effective method in promoting homogeneity and repeatability of test samples. If the sand is well-graded or silty, particle segregation during sedimentation may occur. For these materials, uniform specimens can be formed by a slurry deposition technique (Kuerbis and Vaid, 1989). The sands used in this research were poorly graded and little fines were present, and therefore specimen reconstitution was done by water pluviation. 26 3.4 Specimen Size A proper choice of sample size can minimizes the relative significance of equipment related experimental errors (discussed in Section 3.6). Aside from adopting a widely accepted height to diameter ratio of 2 to 1, there is often less emphasis on a deliberate choice of sample size. With reference to the experiment limitations (discussed in subsequent sections), the influence of ram friction and errors due to membrane penetration volume changes decrease with increasing sample diameter. End restraint affects larger sample height. As mentioned earlier, a 64 mm diameter specimen with, an approximately 2 to 1 height (130 mm) to diameter ratio was adopted. The actual sample height varied slightly depending on the relative density of interest. Overall, this sample geometry was found favorable in reducing the relative significance of equipment related test errors. 3.5 Method of Sample Preparation A fixed mass of oven dried sand was mixed with water in a flask and was boiled for about 10 to 20 minutes. After cooling it to room temperature, the boiled sand was kept under vacuum until sample formation. Porous stones were also boiled in water and cooled to room temperature before the sample was about to be prepared. All drainage lines connected to the triaxial cell base were saturated with the de-aired water. A height reference was taken by mounting a dial gauge on a removable stand and placing an aluminum dummy sample of known height between the bottom pedestal and the top cap prior to sample preparation. In this way, the height of the specimen could be determined after deposition. The thickness of the rubber membranes used was 0.3 mm. The membrane was sealed to the base pedestal and stretched against the wall of the split mold (former) by a small vacuum suction. The cavity thus formed was filled with de-aired water. The flask containing boiled sand was then inverted into the mold cavity (Fig. 3.3a). Deposition of sand proceeded by continuously moving the tip of the flask in a circular motion over the plan area of the sample cavity, while maintaining the tip submerged below the water surface. As pointed out by Vaid and Negussey (1988), the sand grains reach their terminal velocity within a negligible drop height when pluviation takes place in water. When all of the sand was deposited to a level higher than the targeted height of the specimen, the 27 Leveling' bar Extension container Former Sand specimen (b) E ft->'-r,.ssfs'j libit. Excess sand Fig. 3.3 Specimen preparation by water pluviation 28 excess was siphoned off to form a level surface as shown in Fig. 3.3(b) (Sayao, 1989). By this technique the sand gets deposited in the loosest state and siphoning action would not cause any further loosening of the surface. The top cap with the attached loading ram was then carefully placed on the leveled sand surface. Densification, if required, was done as follows. Since the diameter of the split mold, thickness of the membrane and the dry weight of sand were known, the desired relative density could be obtained by controlling the specimen height. With a gentle seating, pressure on the cap, the specimen was densified by high frequency low amplitude vertical vibration induced by tapping on the top of cell base until the targeted height was indicated by the reference dial gauge. This procedure was effective in preventing tilting of the top cap that causes uneven settlement. Densification with top cap in place has been shown to promote development of uniform densities within test samples and virtual elimination of bedding errors (Vaid, 1983). The membrane was now pulled over the top cap and sealed by another "O" ring. The top drainage line was now connected, and a vacuum of about 20 kPa applied to the sample through the bottom drainage line, in order to provide a small confinement. Volume change due to this suction was recorded by the graduated side reservoir. The split mold was now dismantled, leaving the sample with a confining pressure of about 20 kPa. The base drainage line was shut off to maintain the confinement and the vacuum line disconnected. After filling the cell with de-aired water, the final dial gauge reading was taken to determine the sample height and the sample preparation phase was completed. The initial loosest deposition void ratio (e^  was computed as soon as the top cap was placed before any densification. The assessment of void ratio at this point was made from the known measurements of the former cavity. In this way, any potential error due to circumferential measurement in the computations of void ratios was minimized (Vaid and Sivathayalan, 1996). Because the mean area of the former is constant, volume was computed from the current sample height. The area of former cavity was calibrated from time to time by the use of a graduated reservoir that filled the former cavity to a known height. The volume of water expelled during vacuum confinement, together with the record of specimen height after densification permitted an accurate computation of the sample volume, and hence its void ratio (referred to as e20) at the 29 completion of preparation. 3.6 Test Errors The degree to which the conventional triaxial test approximates idealized assumptions relative to the homogeneity of stress and strain within the test specimen has been of long-standing interest. The following steps were implemented in an attempt to minimize end-restraint effects and promote confidence in the measured test variables. 3.6.1 Internal vs. External Measurements Continuous monitoring of loads, pressures and deformation was achieved by using electronic transducers placed outside the cell chamber. The limitation of external measurements have been pointed out by Atkinson and Evans (1985) and Jardine et al. (1985). These are attributed to the different amounts of bedding, seating and tilting errors. As shown by Sayao (1989), no significant differences between vertical strains derived from external and internal measurements was noted if appropriate care was taken to eliminate bedding, seating and/or tilting errors. The specimen preparation method described above essentially eliminated these errors. Internal transducers invariably pose additional difficulties in the specimen preparation procedure. 3.6.2 Ram Friction The loading ram used in the triaxial cells had a low-friction air-bleed bushing seal. Continuous air bleed bushings can virtually eliminate ram friction (Chan, 1975). The magnitude and significance of ram friction was minimal, at about 10 gm, that justifies external measurement of the axial force. 3.6.3 End Restraint The influence of end restraint on strength has been shown to be minimal when a height to diameter ratio of 2 or larger is used (Taylor, 1948; Bishop and Green, 1965; Lade, 1982). Lubricated rubber interface between the soil and rigid end platen to diminish the end restraint were proposed initially by Rowe and Barden (1964) and Barden and Khayatt (1966), with the objective of promoting uniform deformations within the specimen even at a reduced height to diameter ratio of one. 30 The use of free ends was also attempted in this study. Two types of silicone grease were tried as lubricants. The first was a silicone grease compound KS 63 G, with a green color and a specific density of 1.06 provided by Shin-etsu Chemical Company Ltd. and the second choice was Dow Corning high vacuum silicone grease with light gray (translucent) color and a specific density of 1.1. Enlarged cap and base pedestals were used (7.5 cm in diameter), and the stainless steel surfaces facing the sand sample were polished to a minimum roughness. Lubrication was applied directly onto these surfaces. In order to ensure smooth lubrication layers, each layer was preloaded under a vertical stress of about 700 kPa for few hours. This preloading also reduced the deformation/squeezing out of grease film under the average operating axial stresses (Lo, 1996). Two configuration of lubricated ends were tried. In the first trials, two layers of grease were placed between two layers of latex discs (each 0.3 mm thick) and the polished stainless steel surfaces {platen + grease + latex + grease + latex}. This type of composition lead to an excessive squeeze of grease and caused radial sample displacements at the bottom platen more than at the top. This was particularly severe when the Silicone grease KS 63 G was used. In the second configuration which promoted relatively uniform radial deformations at top and bottom of the samples, one layer of grease with a thickness of approximately 0.05 mm was placed between a latex disc (0.3 mm thick) and the polished stainless steel platens {platen + grease + latex}. This configuration performed better with high vacuum Dow Corning silicone grease as the lubricant. Fig. 3.4 illustrating test results and profiles of initially identical specimen before and after 20% drained axial compression and 10% undrained extension are evidence in support of essentially uniform deformation assumption. These observations are also consistent with the findings of Tatsuoka et al. (1984), in which the recommended free-end configuration was also a layer of 0.05 mm of high vacuum Dow Corning grease and one 0.3 mm thick latex disc. When the sand specimen was denser, little difference in test results was found if ftictionless ends were substituted for regular end platens (Fig. 3.5). Though the use of free-end is desirable, it introduces an unknown amount of bedding errors. Moreover extreme care is required in preparing these ends. Therefore, the choice was made in favor of regular end platens for simplicity of test procedures. End restraint was kept to a minimum by using highly polished platens with central -15 -10 -5 0 Axial strain, e (%) '53 i / / / -30 30 Specimen profile ~~r 30 Diameter (mm) (a) Comparative undrained response in extension with conventional and frictionless ends and (b) profile of typical specimens with free ends in triaxial compression and extension 32 Fig. 3.5 Comparison of drained compression on dense sand with free and regular ends 33 porous discs of only 20 mm diameter. 3.6.4 Bedding Error Although lubricated platens inhibit the development of non-homogeneous deformation, they contribute to serious bedding errors in the measurement of axial deformation (Sarsby et al., 1980). The contribution of bedding errors to axial deformation have been identified to be comparable to that due to membrane penetration corrections in obscuring real volume change of the soil skeleton (Sarsby et al., 1982). The relative influence of bedding error would diminish with the use of larger sample heights. Clearly, with a height to diameter ratio of 2 instead of 1, the bedding errors in the measurement of the axial deformation would be reduced to half. They are, however, still significant and difficult to quantify. The top cap and the loading ram were permanently attached (Fig. 3.1). This contributed to minimizing any bedding errors. Also the small diameter stones not only minimized end restraint but virtually eliminated false deformations due to improper seating on to the end platens. The load cell was positioned above the displacement transducer bracket, such that its compliance did not influence axial deformation measurements. Tilting of the top cap was prevented by the ram movement vertically within a pair of linear ball bushings. Fig. 3.5 compares sand response with regular ends and free ends. It may be noted that the volumetric response is essentially identical for both free and regular ends. Slight difference in response in deviator stress is due to bedding error corrections pertinent to the free ends used. These corrections were found not to lead to repeatable and unique test results. It should also be noted that the results shown in Fig. 3.5 are for very dense sand (Drc = 80%). The preliminary test results in this investigation have shown that as the sand gets denser, the effect of nonuniform deformation becomes more pronounced. Due to this reason, and for the sake of brevity, the result at dense density only is shown. Since the experimental investigation presented in this thesis are mainly focused on loosest deposited specimens, the effect of nonuniform deformation during shear would be less significant. Nevertheless considering the volumetric response, which is of more concern, the results shown are 34 considered satisfactory. The identical volumetric response shown in Fig. 3.5 was also demonstrated by Barden and Khayatt (1966) and Rowe (1971). This implies that the strain increment ratios are affected little if free ends were substituted by regular ends. It also confirms the findings by Bishop and Green (1965) who found that the dilation of soils with either end conditions are practically the same, and those of Ueng et al. (1988) who found insignificant difference in the uniformity of volume changes within the specimen prior to the peak strength with and without lubricated ends. The improved conventional end platens were therefore selected based on the result of the experimental trials described above, together with additional support from similar evidence in the literature. Elimination of bedding errors inherent in the usage of lubricated ends was considered crucial for confident measurements of axial deformations particularly in the region of small strain. 3.7 Membrane Penetration A change in effective confining pressure during a triaxial test causes the membrane enclosing the soil specimen to change its peripheral configuration by either indentation or rebound. This membrane penetration affects both the volume changes in drained and pore water pressures during undrained shear. For example, a conventional undrained test conducted by closing the drainage lines only does not constitute a constant volume test. Such a departure from the undrained state can render the results of the conventional undrained test unconservative in assessing liquefaction potential of sands (Nicholson et al., 1993b). Similarly, the measured volume change in a partially drained test will be either less than or exceed the real volume change of the soil skeleton. As the effective confining stress changes, the measured volume change of the specimen (AVT) is comprised of the two components: ^VT = ^ m A m + \ S V T (3.1) The first component is due to the amount of membrane penetration and the second represents the true or skeleton volume change of the specimen. In Eq. 3.1, em is the membrane induced volume change per unit area, A„ the total surface area of specimen in contact with the membrane, and is the true skeleton volumetric strain. Different methods have been proposed to evaluate the magnitude of membrane compliance. The method used for the Fraser River sand was that suggested 35 by Vaid and Neguessey (1984). This method is briefly summarized in Appendix A. The results illustrating unit membrane penetration em versus logarithm of effective confining pressure, for Fraser River sand at relative densities of 21% (loose), 32% (medium-loose), 55% (medium-dense), and 82% (dense) are given in Fig. 3.6(a). It is clear that an increase in relative density causes a decrease in normalized unit membrane penetration m, the slope of £m vs. log c'r (Fig. 3.6b). The key assumption made in determinating em by the Vaid and Negussey method was that the sand response under hydrostatic unloading is isotropic. This assumption was not made in a recent method suggested by Sivathayalan and Vaid (1998), and it has been shown that the new method yields em values very close to those obtained by the technique adopted in this research. 3.7.1 Factors Influencing Membrane Penetration The extent to which the membrane compliance may influence the magnitude of pore pressure generation and volume change of the sand skeleton depends on the ambient effective confining pressure, mean grain size and relative density of sand, and membrane thickness (Ramana and Raju, 1982; Baldi and Nova, 1984; Nicholson et al., 1993a). There is little quantitative knowledge in the literature on the effect of relative density on the magnitude of em. Interestingly, Frydman et al. (1973) concluded that the relative density of sand is not an important factor, and found little difference between normalized membrane penetration in tests on uniform glass microspheres having relative densities of 30% and 70%, contrary to the results shown in Fig. 3.6(b). The importance of mean grain size on membrane penetration has been, in fact, recognized by previous investigators. The magnitude of normalized unit membrane penetration was also evaluated for two other sands. The grain size distribution for Ottawa sand, and Silica sand, both fairly uniform quartz sands, are shown in Fig. A-1. Fig. 3.6(b) shows that regardless of mean grain size, "m" does decrease with increase in relative density of the sand, and for a given relative density, increases with mean grain size among different sands. At the loose density state however, the effect of mean grain size on "m" is less pronounced than at higher densities states. For the loosest specimens, the ratio of "m" value between coarsest Silica sand and finest Fraser River sand is 3.5, but for the densest state this ratio becomes 7.3. 36 0.000 0 40 80 Dr (%) Fig. 3.6 Membrane penetration correction curves (a) error in volume change (b) dependence of unit membrane penetration on relative density for three sands 37 The unit membrane penetration does not seem to vary linearly with the relative density for all sands (Fig. 3.6b). Therefore, a linear interpolation of "m" between any two densities should not be assumed when the test relative density is straddled by these values. The factors influencing membrane penetration are discussed in more detail in Appendix A. 3.7.2 Membrane Penetration Volume Corrections with DPVC During the actual performance of the undrained and partially drained tests reported in this research, the DPVC (explained earlier) was used for making corrections for membrane compliance values. This allowed predetermined volumes of water to be removed from or injected into the test specimen. The volumetric error depends on the change in the ambient effective confining stress (Eq. A-6). This correction technique was first implemented by Raju and Venkataramana (1980) by manual procedures. But the sudden addition of fluid into the specimen will cause the effective confining stress to change which, in turn, will cause additional volumetric compliance to be corrected. The time involved to make the step by step manual correction, as well as the lack of continuity between increments, was a serious drawback of Raju and Venkataraman's procedure. It also restricted volume correction only at certain discrete time intervals during each test. In this study, the membrane correction was continuously applied and was based on the premise that volumetric error due to compliance was a direct and repeatable function of the change in ambient effective confining stress. The computer program designed for controlling the injection process contained a closed-loop algorithm that continually monitored the effective confining stress and adjusted for the corresponding volumetric correction at any time during the test. To perform the compensated test, the normalized unit membrane penetration value (m) for the test density was entered into the microcomputer. When the test began, the system continuously injected or removed controlled volumes of water to exactly offset the volumetric error as a result of variations in o'r. At no time the correction lagged behind by a change in confining pressure of only about 0.2 kPa. 38 3.8 Test Procedure 3.8.1 Sample Set-Up After assembling the triaxial cell it was centered and securely clamped under the loading frame. A cell pressure of 20 kPa was applied to bring the sample pore pressure to essentially zero. The transducers for the measurements of axial load, pore pressure, cell pressure, axial and volumetric deformations were then set to their initial zero levels, and these initial readings recorded. The top and bottom drainage lines were now connected to the volume change device and the DPVC and the reference scan of all transducers taken. The cell pressure was increased under undrained conditions in 20 kPa increments. These Skempton's B measurements proceeded in several increments of hydrostatic confining pressure. Full saturation of the soil specimen was ensured by insisting on a B value greater than 0.98. 3.8.2 Consolidation The back pressure was set to the desired value and fed into both the drainage pipette and the DPVC. Drainage valves were now opened. In the case of isotropic consolidation, the samples was brought to the desired effective hydrostatic stress by incrementing the cell pressure. Uplift on the loading ram during this phase was compensated by loading with the air piston in order to maintain a hydrostatic state of stress at all times. If initial anisotropic consolidation under a given effective stress ratio R (= a\ / a'3) value was desired, the specimen, which was already confined hydrostatically under a low effective stress (c'ro = 20 to 30 kPa) at the time of set up, was first loaded drained along the conventional triaxial drained path to the targeted K c state (at constant a'ro). Thereafter, consolidation continued along the K,. path to the desired a'rc. This anisotropic consolidation path for bringing sand to the target o'lc and a'3c is labeled path 1 in Fig. 3.7 (the path is referred as the K c path and the target effective stress ratio is referred to as the K c value in this thesis). Alternatively, the target effective stress state o'lc = a'ac and a'3c = a'rc prior to shear loading was reached by first consolidating hydrostatically to a'rc followed by deviator loading under a conventional drained dc'r=0 path until the desired R,. was reached. Such a path is labeled 2 in Fig. 3.7 (R,. path and value). The target R,. consolidation 39 40 stresses were applied as continuous drained loading without any time interruptions. All of the above tasks were carried out with continued recording of associated deformations. Sample dimensions at the consolidated stress state were used in strain calculations during shear. 3.8.3 Effect of Aging after Consolidation Following consolidation, the shearing phase proceeded either as a continuous loading phase or after a rest period (aging) of about 15 minutes under the consolidation stresses. The effect of aging time on deformation response, if any, must be taken into account to ensure reproducibility and consistency of test results for valid comparisons (Mejia et al., 1986, Tatsuoka et al., 1998). The importance of even small aging period on the undrained response of Fraser River sand is illustrated in Fig. 3.8. Two specimens having identical initial effective stress and void ratio state but with different aging periods were tested in undrained compression. Without any aging period (indicated as "C": Continuous loading), the initial stiffness may be noted to be smaller and the sand manifests immediate strain softening, while after an aging period of only 15 minutes prior to shearing (shown with "A": Aged) demonstrates a much suffer initial modulus, and the strain softening did not commence until after experiencing a noticeable increase in deviator stress. 3.8.4 Shear Tests The cell pressure was held constant during shear. Controlled stress path and controlled strain path tests were performed by using either the displacement control in the axial direction or stress control only on all loading boundaries. Only tests carried out for investigating instability were of the latter type. In all others, displacement control was used in the axial deformation direction. This displacement control allowed post peak behaviour to be recorded, in case the sand suffered strain softening. 3.8.5 Instability Tests The specimens were brought to the targeted initial effective stress state by hydrostatic compression to the desired confining stress a'rc (path OC in Fig. 3.9), followed by conventional drained loading under constant radial effective stress (do'=0 path) to the desired effective stress ratio R,. (path CR 0 D r c = 28% R c = 2.8 CT ' r c = 200 kPa ( d E v / d s a ) i = + 0.05 —i , 1 i 0 4 8 12 (CT'a + CT'r)/2 (kPa) Effect of small ageing on sand response 42 43 in Fig. 3.9). Subsequent partially drained shear loading (path RS) was simulated by the strain path control. For simplicity constant dejds^^ (volumetric to axial strain ratio) paths were adopted, and the radial total stress was held constant. Since the prescribed strain increment field (de^  dsj must result (principle of reciprocity is discussed in a later section) in a unique effective stress increment (dc'a and do'r), the sand must develop some pore pressure (because of the constant radial stress constraint), in order to accommodate the required change in radial effective stress. During partially drained loading (path RS), while the boundary stresses or and deviator stress (a,- ar) were held constant, the specimen was subjected to a constant strain increment ratio (deydsj) path using the increment in axial deformation as the control variable. The initial effective stress perturbation occurred at constant total stresses due to the development of pore pressures as a consequence of the arrest of secondary consolidation. 3.9 Capabilities and Performance of the Apparatus, and Confidence in Test Results 3.9.1 Repeatability in Test Results Confidence in the test results depends on their consistency and repeatability. Reproduction of relative density, replication of structure, measurement accuracy and exact duplication of test routine promote repeatability. The test procedures described previously ensured that this was achieved to the highest degree. Although time independence of the deformations of sand compared to that of clay is acceptable, it should not be ignored in laboratory investigation. It was found that for the Fraser River sand, the repeatability of the test results was assured only if the aging period after the end of consolidation to the desired stresses was the same, before the commencement of shear or instability tests. The data from two identical specimens of Fraser River sand under conventional undrained triaxial compression are presented in Fig. 3.10. Excellent repeatability in stress strain and pore pressure responses may be noted. Similar data demonstrating repeatability of instability tests is presented in Chapter 5. Fig. 3.10 Repeatability of undrained compression tests 45 Due to depletion of the limited supply of virgin sand towards the end of testing program, it was found necessary to reuse it for additional tests. In such cases, particle breakdown during first use may alter sand behavior, if the stress level experienced was high. After loading to a maximum stress level of approximately 800 kPa, no discernible particle breakdown was observed. The essential identity of results of undrained compression behavior of virgin and reused sand (Fig. 3.11) may be taken as evidence in support of this contention. 3.9.2 Rate of Loading to Ensure no Excess Pore Pressures in Strain Path Control Element Test For a strain path control test to be a truly element test, volumetric changes in the sand specimen must occur equally throughout its depth. Since water is injected at the top to cause these changes and pore pressures are measured at the bottom, it is necessary to ensure that there is no significant gradient created across the specimen to induce volumetric strains at the bottom. This was ensured by limiting the rate of effective stress change during the strain path test to sufficiently small values. A series of partially drained test were carried out on identical specimens at a fixed (deyde.), each with a different rate of effective stress increase. If no further reduction in the rate than the threshold that just accomplished effective stress transfer by virtually little excess pore pressure gradient across the sample, that rate was adopted for the test described herein. 3.9.3 Equivalence of Response by Stress and Strain Path Controls 3.9.3.1 Conventional drained and undrained triaxial compression Fig. 3.12 compares the results of two conventional drained triaxial compression test (da'r = 0) on identical specimens using stress path control and also the strain path control utilizing DPVC as a feedback control on volumetric strains, to maintain back pressure constant. Essentially identical response by the two testing technique may be noted, as would be expected from principle of reciprocity in stress and strain increment which is valid regardless of the material type. The small differences in volumetric response is due to a slight increase in back pressure, as the water head in pipette increases in conventional drained test. A conventional undrained test can also be performed as a constant volume test (Taylor, 1948) by using the feedback control on the pore pressure in order Fig. 3.11 Repeatability of undrained compression response of used and virgin sand 47 Fig. 3.12 Equivalence of conventional da'r = 0 drained behavior under stress and strain path control 48 to keep specimen the volume constant at all times. 3.9.3.2 K 0 - coefficient of earth pressure at rest with and without membrane corrections K 0 - for the sand can be assessed by subjecting a triaxial test specimen along the strain path de^ de-. = +1 (der = 0), while the corresponding effective stresses are recorded. Fig. 3.13 demonstrates the significance of membrane penetration correction in KD measurements for two different sands, namely, Fraser River sand with a D 5 0 = 0.35 mm and normalized unit membrane penetration (m) of 0.0033 and Silica sand with D 5 0 = 0.90 mm and m = 0.0111 (see Appendix A for more detail). Since the radial effective stress continuously increases during K 0 consolidation, correction for membrane compliance is made by extracting water from the specimen in excess of that required for the strain path applied. In other words, the applied strain path should be such that net skeleton volumetric strain has a one to one correspondence with the axial strain. The volume compensation for membrane penetration effects are computed corresponding to the changes in c'r incurred. The measured K„ values for the Fraser River sand under increasing vertical effective stress, with and without membrane correction are compared in Fig. 3.13(a). When membrane corrections are not applied, somewhat longer transition in the initial trial effective stress ratio R,, = o'J<3\ of approximately 2 was noted in comparison to when it was. Nevertheless, eventually an ultimate (equilibrium) K 0 value is obtained at larger stress levels for both corrected and uncorrected strain paths. On the other hand, Fig.3.13(b), which compares K 0 values for the coarser Silica sand, much higher ultimate (equilibrium) K 0 value is noted when the membrane penetration correction is applied than when it is not. The reason for the importance of making correction for measuring true K 0 are clearly related to the larger mean grain size of Silica sand than the Fraser River sand. 3.10 Experimental Program The experimental program undertaken in these investigations was to fulfill the following objectives. The relationship between stress and strain increments at any effective stress state (a\, o'3) can be established either by applying controlled effective stress increments and recording the resulting strain increments; or by applying controlled strain increments and recording the effective stress increments needed to induce these strains. The principle of reciprocity in stress and strain has already been demonstrated by data in Fig. 3.12. With the addition of strain path control capability, the effective 49 0.8 0.4 H 0.0 Loose Fraser River sand (a) Membrane corrected (K 0 = 0.442) Membrane not corrected (K 0 = 0.431) T -800 (kPa) —I— 1200 400 0.8 Loose Silica sand (b) _ • Membrane corrected (K 0 = 0.484) Membrane not corrected (K 0 = 0.419) 0 1 400 1 1 800 1200 0.4 H 0.0 Fig. 3.13 K 0 measurement with and without membrane correction for two sands 50 stress path dependent behavior of soil under arbitrary nonlinear path could thus be investigated. The tests included truly undrained, truly drained of the linear da'r = 0 type and partially drained with a range of the degree of drainage. The latter type were carried out using the strain path control, that gave rise to linear strain paths, i.e. dsydej = constant. Most of the tests carried out were on loose sand, the state likely to be most susceptible to liquefaction. However, in order to demonstrate serious consequence of even a small amount of drainage in denser sand, tests were also performed at initially denser states. The effective confining pressure ranged from 50 kPa to 400 kPa and the initial effective stress ratios (R,. = o'lc / a'3c) from 1.0 to 2.8. For occasional comparison, the behavior at even higher initial stress ratios was also examined. 3.11 Summary Test equipment and procedures were evaluated and improved to enable precise load application in different strain/stress paths, accurately measure deformations and ensure reproducibility and consistency of test results. Frictionless ends were also tried. Using two configuration of lubricated ends, little differences in test results was found if frictionless ends were substituted for regular but somewhat modified end platens. Excellent repeatability in stress strain and pore pressure responses demonstrated confidence in the data obtained. It was established that the effect of ageing time on deformation response, if any, must be taken into account to ensure reproducibility and consistency of test results for valid comparisons. The importance of even small aging period on the undrained response of Fraser River sand was illustrated. The effects of mean grain size, particles shape (Appendix A), and relative density on the magnitude of membrane penetration and compliance, and thus on pore pressure generation were studied. It was found that for a given sand the unit normalized membrane penetration "m" decreases with increase in relative density. For a given relative density, "m" increases with mean grain size among different sands. The unit membrane penetration does not seem to vary linearly with relative density for all sands. Therefore, a linear interpolation of "m" between any two densities should not be assumed when the test relative density is straddled by these values. It was demonstrated that the shape of soil 51 particles was also the important soil property that controlled volumetric-compliance errors (Appendix A). It is essential to perform sets of specific tests for each soil specimen to obtain representative parameters to model membrane penetration. CHAPTER 4 TEST RESULTS AND ANALYSIS 4.1 Introduction This chapter presents the results of undrained, conventional fully drained (da'r = 0), and partially drained tests under constant rate of strain, both in compression and extension. Based on these results, an attempt is made to establish the conditions governing the occurrence of full liquefaction (full strain softening), limited liquefaction (partial strain softening), and strain hardening under partially drained states. The major part of the work deals with Fraser River sand in the loosest deposited states, the states which are more prone to strain softening. But initially denser states, where partially drained conditions can be less conservative than the completely undrained assumption, are also examined. The relative density (Dr;) of Fraser River sand in its loosest deposited state and after an initial vacuum confinement of about 20 kPa was about 11%. The effective confining pressure at the end of consolidation (a'rc) and prior to shear, ranged between 20 and 400 kPa; and the initial effective stress ratio (RJ ranged from 1 to 2.8. Occasionally, behavior at even higher initial stress ratios was examined. In this way, the effects of initial effective confining pressure, effective stress ratio and relative density on the response of sand under partially drained conditions are systematically investigated. Key aspects of the behavior are pointed out and compared with similar characteristic behavior of other sands, if reported in the literature. 52 53 The major principal stress (c^ ) under axisymmetric triaxial compression is the axial stress (cj, while the minor principal stress (c3) is the radial stress (cr). The reverse is true under triaxial extension loading. Throughout this chapter, triaxial test results are presented in terms of stress variables either aa and cr or c1 and o3. This enables a distinction to be made between triaxial compression and extension modes of loading. Effective stress paths are plotted in the modified Mohr diagram with axis t> = ^ ± S'=^L (4.1) 2 2 In compression, t defined in terms of aa and is positive and in extension negative. 4.2 Undrained Behavior 4.2.1 In Compression The behavior of initially hydro statically consolidated loosest deposited Fraser River sand to less than approximately 400 kPa is found to be strain hardening (Fig. 4.1). Nevertheless, as the effective confining pressure increases, the dilative tendency decreases, despite some increase in density at the end of consolidation. The effective stress path (ESP) corresponding to confining pressure of 400 kPa can be regarded as a terminal case of strain hardening (or softening) response in which the degree of strain hardening (or softening) is zero. Tests at confining pressures higher than 400 kPa are reported in a later section (Fig. 4.5 at higher initial stress ratios) and elsewhere (Vaid and Thomas, 1995 at isotropic states). However, in order to illustrate that an increase in the confining pressure transforms undrained response from strain hardening to strain softening, a series of similar tests, though not so comprehensive, was performed on another sand (Ottawa sand). These are presented in Fig. 4.2. It may be noted that for the rounded Ottawa sand the threshold hydrostatic confining pressure that separates the undrained response into contractive and dilative types is much smaller at about 50 kPa. The effect of an increase in effective confining pressure at constant initial effective stress ratio (R,. = aYc '3 ) of 2.1 transforms the response of Fraser River sand from dilative at a'rc = 50 kPa to a 54 Fig. 4.1 Undrained response of loosest deposited Fraser River sand (Effect of effective confining pressure) Fig. 4.2 Undrained response of loosest deposited Ottawa sand Lower cut-off "ab" in comparison with Fraser River sand 56 slightly strain softening at o'rc = 200 kPa, and finally to clearly strain softening type at a'rc = 400 kPa (Fig. 4.3). Undrained behavior at constant a'rc = 200 kPa but increasing R<. from 1.0 to 3.4 is illustrated in Fig. 4.4. The level of initial stress ratio alters the barely contractive response at R,. =1.0 to more and more contractive as R,. increases. At an initial stress ratio of 3.4, the contractive response becomes dilative again. This will be explored further in a later section. Test results in Fig. 4.3 and 4.4 illustrate that both (i) an increase in static shear at constant confining stress and (ii) an increase in confining stress at constant static shear promote strain softening behavior in loose Fraser River sand, despite 5 to 6% increase in relative density. 4.2.2 Region of Undrained Contractive Deformation A selection of effective stress paths for Fraser River sand tests at several initial states which suffered strain softening is shown in Fig. 4.5. The locus of peak states (CSR line) at which strain softening triggered is sloped at about 26 degrees. The region of stress space between this CSR line and the phase transformation (PT) line represents the zone of strain softening deformation, wherein the strains associated with strain softening occur. The PT line corresponds to a friction angle (cpPT) of about 32 degrees. As is well known, stress paths which include initial states both inside and outside the region of contractive deformation develop a knee at the PT state, which marks the termination of strain softening. Straining beyond the PT state is associated with dilation tendencies, and hence decrease in pore pressure. This causes the stress path to climb up along the line of maximum obliquity. The angle of maximum obliquity cpMO, like <pCSR and (pPX, is also unique at about 37° for the Fraser River sand. This feature is highlighted here for future reference. The region of undrained contractive deformation in Fig. 4.5 for the Fraser River sand does not, however, extend down to the origin of stress space, but is terminated by an approximately horizontal cutoff "ab" at a finite value of (aa - or)/2. The almost horizontal cutoff "ab" is defined by the effective stress path of the initially loosest deposited at o'rc = 400 kPa and R,. = 1.0, for which the deviator stress reaches a peak, followed by a flat plateau "ab" (see Fig. 4.1), and no strain softening occurs. The position of this cutoff may change depending on the type of sand, as pointed out in Fig. 4.3 Undrained response of loose Fraser River sand (Effect of effective confining pressure at constant initial stress ratio) Fig. 4.4 Undrained response of loosest deposited Fraser River sand (Effect of initial stress ratio at constant a'rc) Fig. 4.5 Undrained response of loose Fraser River sand Delineation of undrained region of contractive deformation 60 Chapter 2. The region of undrained strain softening deformation for Ottawa sand (Fig. 4.2), for example, appear to terminate at a horizontal cut-off corresponding to the undrained response at an initial o'rc of 50 kPa. The effect of initial relative density on undrained response (Fig. 4.6), as expected, makes the response more strain hardening as the density increases, regardless of level of static shear stress. 4.2.3 In Extension As shown for other sands (e.g., Vaid et al., 1990), the loosest deposited initially hydrostatically consolidated Fraser River sand in undrained extension is strain softening (Fig. 4.7), as opposed to strain hardening in compression (Fig. 4.1), over the same range of confining stresses. This has been attributed to inherent anisotropy in water deposited sands, and demonstrated by several researchers (e.g. earliest by Bishop, 1971). The behavior still remains essentially strain softening even at initially medium-dense states (Fig. 4.8), compared to the compression behavior, which is strongly strain hardening. The regions of strain softening deformation in extension for both loose and medium-dense states are delineated in Fig. 4.7(b) and 4.8(b). They still lie between the two effective stress ratio lines corresponding to the CSR and PT states, similar to that in compression. In extension, however, the slope of the CSR line depends on the initial deposition relative density. For the loose state it corresponds to a mobilized friction angle of about 16 degrees and for the medium-dense state of about 21.6 degrees. Initial relative density may also be noted to affect the location of the horizontal cut-off "cd" of the region of contractive deformation. No such cut-off is seen to exist for the loosest deposited state, and for the medium-dense state it is much closer to the origin of stress space compared to that for the loosest state in compression. The lower CSR in extension than in compression, as explained in Chapter 2, is due to the existence of inherent anisotropy in pluviated sands, but the PT line in extension has the same slope as that in compression. 61 Fig. 4.6 Comparison of undrained compression response of loose (L) and medium-dense (M) sand at K c =1.0 and K c = 2.0 62 •200 of -100 -\ as b PH -200 H 200 (o'a + o' r)/2 (kPa) 400 Fig. 4.7 Undrained extension response for loose Fraser River sand Fig. 4.8 Undrained extension response for medium-dense Fraser River sand 64 4.2.4 Summary - Basic Characteristics of Undrained Behavior Undrained tests in both triaxial compression and extension were used to establish the boundaries of contractive deformation region for Fraser River and Ottawa sands. It was shown that depending on the mean grain size and the shape of sand particles, the location of the region of undrained strain softening deformation may change. This change in location of the horizontal cut-off was also shown to be dependent on the relative density. As the sand becomes denser, the cut-off moves away from the origin of the stress space. A strain hardening response under a given confining stress may change to strain softening type as the initial stress ratio increases. At an initial stress ratio close to and especially larger than the undrained PT state, this transformation may be reversed to a strain hardening response again. This is because the straining beyond the PT state is associated with dilation tendency in the sand. 4.3 Fully Drained Behavior 4.3.1 During Consolidation The drained loading phase of the test specimens from the end of their reconstituting state (a'rc = 20 kPa) until the commencement of shearing phase will be termed "consolidation" herein. During consolidation under hydrostatic stresses (Kc = 1), the strain path in volumetric strain - axial strain space is essentially linear (Fig. 4.9). The ratio eje^ = dejde^ however is >3, reflective of the inherent anisotropy in water deposited sand. For consolidation along constant K c > 1 paths, the linear strain ratio ejz^ is still sustained at K c = 1.5 and 2.1. However, a slight departure from this linearity occurs for consolidation along paths with K c > about 2.1. K c = 2.1 corresponds to approximately the K 0 - consolidation path for the loose Fraser River sand (Vaid et al., 1999). The above response during consolidation phase is also true for the undrained tests as well.. 4.3.2 In do'r = 0 Triaxial Paths The compression behavior at Drc of 30% and 58% is illustrated in Fig. 4.10 (loose) and 4.11 (medium-dense) for initially hydrostatically consolidated states. As would be expected, the initial stiffness increases with increasing confining pressure. The stress strain curves are of the plastic type 65 o Fig. 4.9 Strain path e^, - e. during isotropic and anisotropic consolidation for Fraser River sand Fig. 4.10 Drained da' r = 0 compression response of loose Fraser River sand 67 Fig. 4.11 Drained da'r = 0 compression response of medium-dense Fraser River sand 68 with no significant drop after the occurrence of peak. The change in shear stiffness with confining stress will be discussed further in a later section. The volumetric compression at a given axial strain, like other sands, increases with increasing confining pressure due to suppression in dilatancy (increase in the rate of compression, dcv/de1 or decrease in the rate of expansion). The maximum rate of volume expansion for the loose sand appears to reduce to approximately zero as the initial confining pressure increases close to 473 kPa, the highest level used in the testing program. For the medium-dense sand, however, this rate of volume expansion does not become zero at similar level of confining stress. At a'rc of400 kPa, the stress-strain and volumetric response for the loose and medium-dense sand are qualitatively more similar than that at lower confining stresses (e.g. at c'rc = 20 kPa). Close examination of Figs. 4.10 and 4.11 indicate that at cs'K of 400 kPa, the volumetric strain response are more similar than that at o'rc of 20 kPa. For a given initial state (o'rc, R,. = 1, Drc), the rate of volume change de^de! during shear along the do'r=0 path varies from an initial positive towards zero at the point of maximum contraction (MC). Thereafter it becomes negative, i.e. expansive, until a maximum rate of expansion occurs. The occurrence of peak in R is synchronous with the instant at which the expansive rate becomes a maximum. The relationship between this dejde^ with R and o'r is shown in Fig. 4.12 for the loose Fraser sand at several values of the initial a'rc. This figure was derived from Fig. 4.10 and is presented in two different forms. Fig. 4.12 is basically an expression of Rowe's stress-dilatancy relationship, RvsD = (1- dejde^. However, contrary to the assumption in the Rowe's stress-dilatancy theory, which considers dcydCj dependent only onR, its definite dependence on c'K, in addition, is clearly apparent inFig. 4.12. The initial value of deydej at R=l seems, nevertheless, independent of o'rc level, but with increasing R, this dependence can hardly be ignored. Similar findings regarding the independence of deJdEi at R=l on o'rc were reported by Negussey (1984) and Negussey and Vaid (1990) for the rounded Ottawa sand. Relationships between dejdej^ and R at the medium density state are shown in Fig. 4.13 (based on data in Fig. 4.11). The initial value of dejds1 at R=l at this density too may be regarded essentially Fig. 4.12 Relationship between dejde^ R, and o'r during da'r=0 shear - loose Fraser River sand Fig. 4.13 Relationship between dejde^ R, and a'r during da'r=0 shear for medium-dense Fraser River sand 71 independent of the a'rc level. But now, as R increases, the dependence of de^ de, on o'^  at a given R becomes much more pronounced than in Fig. 4.12 for the loose sand. Although at this density, the dependence of de/de, on a'rc at a given R is only examined at three a'rc, the trend is quite clear by close examination of Fig. 4.11 even without consideration of the data at a'rc = 20 kPa. As will be shown in Fig. 4.14, the effect of a'rc on strain increment ratio is more pronounced on the maximum volume expansion rate (dsyde.),,^  for the medium-dense Fraser River sand for confining pressures of200 and 400 kPa. In order to show that this type of response is true in general, similar trends have been examined for the rounded Ottawa sand (see Fig. B-1 in Appendix B). An alternative expression of the data in Figs. 4.12(a) and 4.13 (a) is given in Figs. 4.12(b) and 4.13(b) in the form of constant dejde1 contours in a'r, R space. It will be shown in a later section that the dependence of deydsj on o'r in addition to R is central to the relationship that exists between fully drained and partially drained shear. The maximum rate of volume expansion (dilation rate) during do'r = 0 shear may be noted to decrease as o'rc increases (Fig. 4.14). The reduction in this rate with confining pressure is very small for the loose sand over the range of confining stresses considered. However, as pointed out earlier for the denser state this rate increases substantially with decrease in a'rc. In extension, the maximum rate of expansion along da'r = 0 shear over the same range of o'rc as in compression is illustrated in Fig. 4.15 for the loose sand. Since in extension the major principal strain is in the radial direction, the expansion rate used is de^ de,. rather than dejde^ in order to maintain consistency with the data on maximum rate of expansion in compression. Likewise in compression, the maximum expansion rate at a given a'rc increases as the relative density increases, and for the loose state this rate undergoes very little change with increase in confining stress. The basic volume change and the stress-strain data in extension is presented in Appendix C (Figs. C-l and C-2) from which Fig. 4.15 has been derived. It should be pointed out that although from a practical point of view strain increment ratio in the form of deydy, where dy = Vz (dea - der), is preferred, due to the direct control on axial strain in triaxial apparatus during strain path testing, the strain increment ratio in the form of deyde., is used 72 Fig. 4.14 Dependence of maximum expansion rate on effective confining pressure in da'r=0 drained compression shear of loose, medium-dense, and dense Fraser River sand 73 XBTII ( T 3 p / A 3 p ) Fig. 4.15 Dependence of maximum expansion rate on confining stress in dc'r=0 drained triaxial extension shear 74 hereinafter. However, the conversion of one to other can be done using the following equation: dev_ 4(dejdea) dy 3-(deJdea) The maximum expansion rates dejde^ for the loose Fraser River sand appears larger in extension than in compression at a given a're (Fig. 4.16). These higher values in extension may be attributed to three reasons: (i) the dilation rate is for the unloading mean effective stresses as opposed to increasing mean normal stresses in compression; (ii) drained shear occurs at constant major principal stress with minor stress decreasing. In compression the scenario is opposite; (iii) the global expansion rate recorded at larger strain level of about 10% are likely to be not representative of the shear zone due to the development of nonuniformities in the extension loading modes. In the domain of R prior to the instant of maximum contraction, the contraction rates in extension are clearly larger than those in compression (Fig. 4.16b). This is what would be expected from the inherent anisotropy in loose sand, that makes it more compressible in horizontal than vertical compression, despite decreasing mean normal stresses. Further evidence in support of the extension mode being more compressible and hence less expansive than compression will be provided in a later section dealing with instability in extension. Data from a limited number of do'r = 0 tests on loose Ottawa sand, in Fig. 4.16 indicate that in compression, the maximum expansion rates in more angular Fraser River sand are higher than those in the rounded Ottawa sand. 4.3.3 Summary - Basic Characteristics of Fully Drained Behavior • During consolidation along constant R ^ 1 paths, a linear strain ratio zjz^ is sustained up to R = 2.1. But a slight departure from this linearity occurs for consolidation along paths with R,. > about 2.1. Rc = 2.1 which corresponds to the K 0 - consolidation path for the loose Fraser River sand. • Contrary to the assumption in the Rowe's stress-dilatancy theory, which implies deydsj dependent only onR, definite dependence dsydsj on a'rc, in addition, was demonstrated. 75 Fig. 4.16 (a) Comparison of maximum expansion rate in da'r=0 triaxial extension (TE) and triaxial compression (TC) of Fraser River and Ottawa sands (b) contraction rates prior to the instant of maximum contraction in TC and TE of Fraser River sand 76 • The maximum rate of volume expansion decreases as the initial confining pressure increases. The rate of reduction is more as the sand gets denser. Data indicated that in compression, the maximum expansion rates in more angular Fraser River sand are higher than those in the rounded Ottawa sand. • It was shown that in the domain of R prior to the instant of maximum contractions, the contraction rates in extension were larger than in compression. Although the maximum expansion rates dejde^ for the loose Fraser River sand appeared larger in extension than in compression at a given a'rc, the higher values in extension could be attributed mainly due to the development of nonuniformities in the extension loading modes. 4.4 Partially Drained Behavior The behavior of loose Fraser River sand in compression is considered first at a given initial hydrostatic stress state (R^ . = 1). This is followed by an assessment of the influence of initial state variables (static shear, confining stress and relative density) on behavior under partially drained states. Finally, similar behavior is investigated in extension. The behavior of sand was investigated under a range of partially drained states, represented by linear strain paths (Fig. 4.17) in volumetric - major principal strain space (deydsj = constant). 4.4.1 Loose Sand in Compression The response of a series of identical specimens at o're = 200 kPaandRc= 1 is illustrated in Fig. 4.18. The stress-strain response at small strains is also included as an inset. The stress-strain curves for this series of tests with different strain increment ratios ranging from +1 to -1.0 are presented in Fig. 4.18(a). The pore water pressures developed in the tests are shown in Fig. 4.18(b). The resulting effective stress paths are shown in Fig. 4.18(c). From Fig. 4.18, it can be seen that at a given axial strain, the larger the strain increment ratio, the higher the deviatoric stress and the larger is the excess pore water pressure change. For the test 77 Fig. 4.17 Typical range of imposed partially drained strain paths and the conventional undrained and da'r=0 drained strain paths 78 («<PD ( J - ° 80 81 conditions considered here (i.e. loose Fraser River sand), a positive strain increment ratio manifests strain hardening behavior and a negative ratio gives rise to strain softening behavior. At this initial state (R,. = 1.0) the undrained behavior (dejde1 = 0) is strain hardening. The partially drained behavior ranges from strain hardening to partially strain softening (PSS), and finally to full strain softening (FSS), depending on the imposed dejdsy ratio. In the domain of positive dejdey (compressive volumetric deformations) an increase in deydej gives rise to: • smaller maximum positive pore pressure generated (Au,^ or minimum reduction in c'r (cr = constant) • for dejde1 = + 1 the effective stress path slopes at less than 45 degrees throughout, and therefore deformation occurs at continuously increasing o'r. This is because of the fact that the effective stress path do'r = 0 which slopes at 45 degrees, corresponds to zero pore pressure generated. If the slope of effective stress path at a stress point is flatter than 45° it represents reduction in pore pressure, whereas a slope > 45° signifies an increase in pore pressure. Conversely, in the domain of negative dejds! (expansive volumetric deformations), reduction in the imposed dejde1 (more negative) results in: • larger maximum positive pore pressure, which in the limit equals the initial confining stress, and a state of zero effective stress (herein called full strain softening) is reached. • The strain until full strain softening decreases as dzjdz^ becomes more negative. Also shown in Fig. 4.18 are the behavior under undrained (dejde1 = 0) and fully drained do'r = 0 (continuously changing deydej) paths for comparison. It may be noted that the partially drained response is not bounded by these two extremes, as often assumed. The region which is bounded by these paths encompasses only those partially drained paths that correspond to imposed dejde1 greater than zero but less than the instantaneous dejdel of the da'r = 0 shear (discussed later) initiated from o'rc = 200 kPa, Rc=l .0. The response under other partially drained paths lies outside this domain. 82 The small strain secant shear stiffness G s = [(aa - cr) / 2 y] at y = 0.05% at the initial state (a'rc = 200 kPa, Rc=1.0) considered is illustrated in Fig. 4.19 as a function of the effective stress increment direction imposed by the partially drained paths. The direction 5 of stress increment is defined by: At Aa'-Aa' tan5 = — = — As' Aaa+Aa'r A 8 = 45° corresponds to the stress increment direction along the da'r=0 drained path, which is associated with a deydEr-0.70 (Fig. 4.12). For the range of 8 explored (45° - about 105°) which correspond to strain paths ranging from de^ de.^  = +0.70 to zero (undrained) and finally expansive at -0.40, the shear stiffness at this Rc=1.0 appears to be essentially independent of the stress increment direction. G s data shown at other R,. values will be discussed later. In Fig. 4.18, the sand is seen to transform from strain hardening when undrained (deydej. = 0) to strain softening type when subjected to a small partial drainage dejdej^ of merely -0.05. Since the axial strain at the trigger of strain softening is small (< 0.2% to 0.3%) regardless of dev/dsl imposed, the associated volumetric expansion is also very small. Even for dejdey as expansive as -0.4, this corresponds to an increase in relative density at trigger of only 1% (Fig. 4.20). The transformation to strain softening behavior under partial expansive drainage cannot, therefore, be attributed to a physical loosening of the sand. Instead, it must be on account of the alteration in the direction 8 of the effective stress increments the sand gets subjected to, when compared to that experienced under the strain hardening undrained conditions. It is also apparent from Fig. 4.18 that the sand must be undergoing a change from the terminally strain hardening (the one with a flat plateau in deviator stress before the commencement of hardening) to the strain softening type for de^ de. between 0.00 and -0.05. This threshold d&JdEy will give rise to the partially drained response with a flat plateau, as was noted for undrained loading from a'rc = 400 kPa (Fig. 4.1). The partially drained condition thus causes an extension in the zone of contractive deformation over that observed under undrained conditions, by moving the plateau "ab" lower. The emergence of strain softening for dejde,^ less than the threshold between -0.05 and 83 Fig. 4.19 Variation of shear stiffness with the direction of effective stress increment at a given ambient stress state Fig. 4.20 Void ratio changes during partially drained test under a large expansive imposed strain increment ratio of -0.40 for loose sand 85 0.00 will first be only partial (PSS or limited liquefaction type). It soon turns into the full strain softening (Aumax/a'rc=l) type with further reduction in de^ de!. Since under dejd^ = -0.20, the response is already of the full strain softening (FSS), and under dejdel = -0.10 it is of the partially strain softening (PSS.) type, the transition of one into the other type must occur at a threshold -2.0 < dejde^ <-0.10. This implies that even very small expansive volumetric strains can transforms the strain hardening undrained sand into a FSS type. 4.4.1.1 Influence of Initial Effective Stress Ratio and Confining Stress With increase in initial stress ratio R^ . from 1.0 to 1.5 at an identical confining stress of <3'rc = 200 kPa, the undrained response is still somewhat strain hardening (Fig. 4.21a). As shown in the figure, the partially drained response to the imposed deydej = constant paths is essentially similar to that under R^ . = 1.0. The transition from strain hardening to strain softening behavior will still occur in the domain of very small expansive volumetric strains. Further increase in initial R^  to 2.1 and 2.8 makes the undrained response strain softening (Figs. 4.21b and 4.21c). At R,. = 2.8 the response is spontaneously strain softening, i.e. it triggers right at the commencement of loading. Therefore, at these higher stress ratios the response under partially drained paths will undergo a transition from strain softening to strain hardening in the domain of small contractive volumetric strains, (Fig. 4.22), the magnitude of de^ de! increasing with increase in R,,. This would be expected because an increase in R,. at constant a'rc has been shown to enhance the strain softening propensity of the sand (Fig. 4.4). Unlike at R,. = 2.8, the undrained response at initial Rc = 3.4 is not strain softening, but shows an initial flat shear stress plateau (Fig. 4.23). This R,. level is larger than R at PT in undrained shear, the domain in which the sand manifests not strain softening but strain hardening response (decrease in pore pressure), which renders it stable when deformed undrained. The small strain shear stiffness G s at o'rc = 200 kPa and R,. > 1.0 are also shown in Fig. 4.19 for comparison with those observed at R^  =1.0. The change in G s with 8 is examined for stress increments that constitute loading i.e. increasing R and Vi(oa - ar). This corresponds to 5 ranging from 45° to 180°. The angle 45° corresponds to the direction of stress vector for loading along 86 0 H . —I 1 1 1 1 0 200 400 600 (a' a + o' r)/2 (kPa) Fig. 4.21(a) Response in partially drained shear at R,. = 1.5 ; a'rc = 200 kPa Fig. 4.21(b) Response in partially drained shear at R,. = 2.1 ; o'rc = 200 kPa Fig. 4.21(c) Response in partially drained shear at Re = 2.8 ; a'rc = 200 kPa Fig. 4.22 Partially strain softening response under contractive imposed deydsj Fig. 4.23 Response of partially drained tests at R,. = 3.4 ; a'rc = 200 kPa 91 da'=0. The shear stiffness for a given 8 may be seen to decrease with increase in R,., the decrease being more dramatic in the range of higher R.. However, at a fixed Rc, G s does not suffer a significant reduction with increase in 8, except at 8 approaching 180°, when G s drops to zero. The range of 8 over which this drop to zero occurs diminishes as R,. decreases. Partially drained behavior at three additional levels of o'rc (50, 100 and 400 kPa) and several initial R,. states is illustrated in Figs. D-l to D-8 (Appendix D). For a given R,., the potential for strain softening when undrained increases with increase in o're and decreases with decrease in o're. Therefore, the partially drained response under a given dejdey is also influenced in the manner similar to that observed at a'K = 200 kPa with decrease or increase inR. Qualitatively the effect of increase in a'rc at constant R<. on the potential for contractiveness is similar to an increase in R,. at constant c'rc (Fig. 4.3). 4.4.1.2 Influence of R e l a t i v e Density Partially drained compression response of medium dense (Drc = 58%) and dense (Drc = 80%) Fraser River sand, hydrostatically consolidated to a'rc = 200 kPa is shown in Figs. 4.24 and 4.25. The response at these denser states is strongly strain hardening compared to at the loose state (Fig. 4.6). The expansive dejde1 needed to cross the thresholds between terminally strain hardening to partially strain softening, as well as between partially to full strain softening response therefore increases over the loose sand values as the relative density increases. For example, under dejdej^ - -0.40, the sand is fully strain softening when loose, partially strain softening when medium-dense and strain hardening when dense. Since the undrained response is already strain hardening imposition of compressive dejdsj^ makes it even more so. 4.4.2 Partially Drained Response in Extension Undrained response of loose sand was shown to be strongly strain softening in extension (Figs. 4.7 and 4.8). Under a strain path of compressive dzjdz^ = +0.20 (dzjdzx = +0.33) the sand is still strain softening in extension, though to a smaller degree than undrained. In compression under similar conditions the sand becomes more strain hardening (Fig. 4.26). Thus the threshold between strain softening and terminally strain hardening behavior will occur in the domain of compressive volume 92 Fig. 4.24 Partially drained response of medium-dense Fraser River sand atRc= 1.0 ; a'rc = 200 kPa 0.0 2.0 4.0 s i (%) 6.0 0 200 400 600 (a' a + a , r ) / 2 (kPa) Fig. 4.25 Partially drained response of dense Fraser River sand at R,= 1.0 ; o'n. = 200 kPa 94 Fig. 4.26 Comparison of partially drained response in compression (TC) and extension (TE) response at similar imposed dejde^ 95 changes in extension as opposed to expansive in compression. The data in Fig. 4.27 suggests that this threshold may be in the neighborhood of dejdej^ = +0.57, for which a small flat plateau in (a. -a3) vs. 8j relationship seems apparent. 4.4.3 The Region of Strain Softening Deformation under Partially Drained Shear At a given dejde^ the effective stress ratio at the instant at which full strain softening is triggered may be seen to be identical in loose Fraser River sand, regardless of the level of confining pressure at Rc =1 (Fig. 4.28). The friction angle mobilized for the imposed de^ /de. of-0.40 is now smaller at 22 degrees as opposed to 26 degrees for the undrained dejde1 of zero. This angle when full strain softening occurs does not appear to change with the level of initial R^ ., regardless of the confining pressure level (Fig. 4.29). If dzjde^ become more negative (expansive) than -0.20, the friction angle mobilized at the peaks gets even more depressed. The more negative the dzjdz^, the lower is the friction angle mobilized at the peak states. On the other hand, if the partially drained response is only partially strain softening, the effective stress ratio at the trigger of strain softening remains essentially identical to that under undrained loading. The reduction in the friction angle at peak due to more and more negative d^Jdz^ causes the cut-off boundary "ab" of the region of strain softening deformation under partially drained loading move closer to the origin. Thus, the zone of contractive deformation under partially drained states gets enlarged towards the origin, beyond that for the undrained case. It is now bounded by the CSR line pertinent to the imposed negative deydej and the undrained PT line. The greater is the imposed negative dzjdzu the larger does this region become. For the denser state (Drc = 58%), undrained response was strain hardening in compression (Fig. 4.6). No CSR line therefore exists for these states. However, when partially drained with expansive strain increment ratio of dzjdz^^ = -0.4 partially strain softening behavior did emerge (Fig. 4.30). The friction angle is about 28 degrees at the instant of trigger, which is again independent of the confining stress and R,., as it was for the loose sand. It is 6° larger than the 22° noted for the loose sand (Fig. 4.29) under identical deyde.. The cutoff "ab" of the zone of contractive deformation for this partially drained denser state is defined approximately by the effective stress path of a specimen at Partially drained response in extension for the loose Fraser River sand at R, = 1.0 ; a're = 200 kPa 0 97 Fig. 4.28 Partially drained response in compression at constant (dejde^ ratios 98 99 100 the initial state of R,. =1, and a'rc ~ 100 kPa. The cutoff for the loose sand for deJdEi = -0.4 extends down to the origin int-s' space (Fig. 4.29). For an imposed dejde1 of -0.2, denser sand at c'rc = 200 kPa but R,. of 2.0 responds in a slightly strain softening manner (Fig. 4.30). This implies that the cutoff "ab" will now move up in relation to that under dejde1 = -0.40. At the same time, the cp' at the trigger of strain softening increases to 9' = 30.5° for strain increment ratio of -0.20. For the very expansive imposed deydsj of -1.00, the 9' angles at trigger reduces to about 25°, which is smaller than the 26° for the partial strain softening loose sand under undrained conditions. From the effective stress path shown for (dejde^ = -1.00 at a'rc = 200 kPa, it is apparent that the cutoff "ab" moves way down towards the origin. The results in Fig. 4.30 thus indicate that as the density increases, the effective stress ratio CSR at the trigger increases, and the region of contractive deformation becomes narrower for a given deydej, until a threshold density for which this region will not exist anymore. This amounts to saying that the sand at any state at this density when sheared undrained would not strain soften, regardless of the confining pressure and stress ratio level. Test results in support of this contention are shown in Fig. 4.31 for R,. values of 1.0, 2.0, and 3.0, at c'rc = 200 kPa. Under a strain increment ratio of -0.2, at R,. = 3.0, slight strain softening occurs. This confirms that for (dejde^-. = -0.20, the cut-off "ab" has moved up and away from origin of t-s' stress space. Furthermore, the undrained response at R;. = 2.0, which is slightly strain softening for the loose sand (Fig. 4.6), transforms into a strain hardening type for the medium-dense state. This also supports the conclusion regarding the effect of density on the position of the CSR line in partially drained shear, i.e. as the density increases the mobilized friction angle at CSR increases, in case strain softening does occur under the imposed expansive dejdel. The upper boundary of the region of contractive deformation under partially drained conditions does not appear to alter in relation to that for the undrained case. However, one difference may exist for initial states above the undrained PT line with partially drained shear at rates more expansive than the maximum expansion rate at the minimum a'rc encountered during shearing. In that case, there would be no strain hardening domain (i.e. between PT and MO lines). Then the response is fully strain softening and excess pore pressure continues to increase until Aumax/a'rc = 1. Partially drained 101 102 paths with (dejde^ less expansive than (dejde,),^ at minimum c'^ during shear would still have a dilation (strain hardening) domain (e.g. see results for (deyde.); < -0.20 for loose Fraser River sand in Fig. 4.18). It is also noted that for partially drained paths with positive strain increment ratios, the effective stress ratio at maximum obliquity reduces below the undrained (deydej = 0) value, the reduction becoming larger with more positive (contractive) de/dej. This is shown in Figs. 4.18, 4.21, and Figs. D-l to D-8 (Appendix D). Because these partially drained paths do not suffer any strain softening, there is obviously no region of partially drained strain softening deformation (i.e. all states are stable). Indeed, these types of paths are the "non-failure" effective stress paths. The region of contractive deformation in partially drained extension is illustrated in Fig. 4.32. It was shown in Figs. 4.26 and 4.27 that loose sand which was strain softening in undrained extension still manifested strain softening response even under a range of contractive dejdej^ (positive strain increment ratios). At these positive strain increment ratios, the strain softening triggered at mobilized friction angles of 18° and 19° respectively under dejde,^ of+0.18 and+0.26. compared to the 17° in the undrained shear (Fig. 4.32). As it was the case for compression loading, the PT state for all contractive strain increment ratios stayed essentially constant at 32°. Likewise, when the response under contractive de^ de. was not strain softening, the effective stress ratios mobilized at maximum obliquity decreased below that for the undrained paths. 4.4.4 Implication of the Observed Behavior The preceding results suggest that the region of contractive deformation under partially drained conditions gets wider with the imposed strain increment ratio becoming more expansive than the maximum dilation rate of the sand in drained da'r=0 shear along R, a'r states encountered during shear. This, by itself, may imply more damaging conditions in the ground. It is generally believed that only those soils that tend to decrease in volume during shear (i.e. contractive soils) suffer strain softening. Those that tend to increase in volume (i.e. dilative soils), on the other hand, will not be susceptible to strain softening, because their undrained strength is greater than the drained. Dilative soils can, however, suffer liquefaction phenomena, i.e. become strain softening as a result of intrusive flow into denser zones caused by the spatial differences in pore pressures generated during the loading event from adjacent loose contractive zones, or even from an equally dilative zone. 103 104 The susceptibility to liquefaction of sand which is strain hardening in undrained shear may be examined if partially drained, based on its physical location relative to a contractive soil as well as the magnitude of intrusive pore pressures generated in the latter. Soils within the shallow zones with low confining pressures will not normally liquefy if undrained. If, however, the underlying contractive layers begin to liquefy, the surficial undrained dilative sand can also become unstable, and with sufficient intrusive pore water, can progressively liquefy culminating into a flow slide. The availability of intrusive pore water is thus of paramount importance in the development of progressive liquefaction of a slope. If the underlying contractive sand does not generate sufficient intrusive pore water, only material immediately adjacent to the loose liquefied zone will become unstable. The overlying dilative soil, not exposed to the intrusive pore water, will remain stable. This also helps explain the common observation of relatively intact blocks of shallow soil riding over the liquefied deeper material in many flow slides. 4.5 Criterion for Strain Softening Under Partially Drained Conditions in Compression The key feature of the results of undrained (dsjdel = 0) and dc'r = 0 fully drained response of Fraser River sand are schematically illustrated in Fig. 4.33. The strain at which maximum contraction occurs increases with increase in confining pressure. The point of maximum contraction signifies an instant of zero dejde^ Before this instant de^ de! is positive (contractive) and after it is negative (expansive), until it attains a maximum expansive value that occurs concurrent with the mobilization of peak in R. In undrained shear, the instants of maximum pore pressure signify the cessation of the tendency to contract and the start of the tendency to expand (Fig. 4.33b). This translates into increase in pore pressures prior to and decrease after the instant of maximum pore pressure. Thus, there seems to be a connection between the states of maximum contraction in dc'r = 0 drained shear and the instants of maximum pore pressure in undrained shear. Either state separates the region of shear deformation into two parts. In one, there is a tendency to undergo volume contraction and in the other a Fig. 4.33 Key features of (a) dc'r = 0 drained and (b) undrained response (c) (de/deOdrained as a state variable 106 tendency for volume expansion. It is now assumed that ds^ /de, at any current stress state (R and c'r) is a state variable. This implies that it is a function of R and o'r only, and is independent of the effective stress path experienced by the sand prior to this stress state. If at this state A, the shearing is switched to or continued along the path dc'r = 0 (Fig. 4.33c) the instantaneous deydej is unique. Specifically, it does not depend on the previous stress history. Thus, deyds! at any stress state R and a'r (such as point A) along a nonlinear stress path can be assessed from the results of a conventional drained da'r = 0 test initiated from o'^  equal to the current c'r. This de^ /de, is called (dejde^^^ hereafter. This feature of the dependence of the (de^ds.)^^ on the current stress state only will be utilized to establish the criterion for the occurrence of strain softening under partially drained conditions, for cases where the undrained response is strain hardening. A selection of partially drained (including undrained) effective stress paths for loose Fraser River sand sheared from the initial state o'rc = 200 kPa and R,. = 1.0 is reproduced in Fig. 4.34. These stress paths in o'r, R space are illustrated in Fig. 4.3 5(a). They represents a range of responses from undrained strain hardening (Fig. 4.34a) to partial strain softening under deydsj = -0.05 and -0.10 (Fig. 4.34b), and finally fully strain softening under deydsj = -0.20 (Fig. 4.34c). Superimposed on Fig. 4.3 5(a) are the contours of constant de^ /de, under da'r = 0 shear taken from Fig. 4.12. A careful examination of Fig. 4.35(a) reveals that: • deJdSi rates along dc'r = 0 paths at all ambient a'^ R states traversed under deyde^ -0.20 are less expansive than the imposed. At no stage do they become more expansive than the rate imposed. Consequently, pore pressure increases all along the strain path, which Aw max ultimately leads to full strain softening, 7~ = 1. • During shear at the imposed dzjdzx = 0 (undrained), the dzjd&x - +0.05 contour from do'r = 0 shear data intersects the imposed c'r, R path at the point of Aum a x (shown by circle symbol in Fig. 4.35a). de^ de, rates along da'r = 0 paths at all a'r, Rvalues (dzjdz^ contours) traversed prior to the instant of Au^ are less expansive than the rate imposed, but 107 Fig. 4.34 a'r, R excursion during partially drained shear (Loose Fraser River sand) 0 100 200 300 400 a ' r (kPa) Fig. 4.35(a) Partially drained stress paths superimposed on Fig. 4.12 from do ' r = 0 shear (Loose FRS) 109 becomes more expansive thereafter. Pore pressure during the imposed strain path consequently increases continuously until Au^ and decrease thereafter. This marks, as is well known, the state of PT during undrained shear typical of the strain hardening type of response. However, the imposed dsjdsl - 0 path along the a'r, R path is not more expansive than all the (dsyde!)^ ,^, contours. It can be seen from Fig. 4.35(a) that the undrained path in a'r, R space terminates at the dejdej^ contour of-0.05. This means that the imposed deyde! is not always expansive than the (dejde^)^^ contours at all o'r, R values and therefore a strain hardening response ensues under partially drained (in this case undrained) shear. • Under dejde^ - -0.05 and -0.10, the scenarios are similar to that under dsjd&x = 0, strain hardening response, but with the differences that now the imposed strain increment ratios are terminally ended at the (dejde^^^ contours of similar values, i.e. (dev/d81) i m p o s e d = -0.10 at dejde1 contour of-0.10 and (dejde^j,^ = -0.05 at dejde1 contour of-0.05. This type of response leads to a partial strain softening behavior. The above can be summarized as follows. The criterion for strain softening under partially drained conditions can be stated as follows. For the full strain softening, the following criterion can be written: id%ld^)impoSed < (^J^drainedatallambienta.R F u l 1 S t r C l m Softening (45) If the two sides of above expression are equal, then a partial strain softening response would be expected under partially drained conditions as was shown in Fig. 4.35(a) for (dev/de1)imposed = -0.05 and -0.10. For partial strain softening response, expression (4.5) may then be written as: (deV/deiXn,poSed = id%ldh)drainedatallambient a„R "* P^rtiall Strain Softening (4 6 ) It is clear that if (4.5) and (4.6) are not satisfied, the partially drained shear would lead to a strain hardening response. 110 The excursion of o'r, R states during partially drained shear at the initial state o'rc = 200 kPa, but higher R,. = 2.8 is shown in Fig. 4.35(b) for loose Fraser River sand. For the imposed deJdey = -0.20 it is clear again that since at all a'n R states, the (dev/d81)drained contours are not more expansive than that imposed dejde^ full strain softening response results. The undrained response which was strain hardening at the initial state a'rc = 200 kPa, R,. = 1.0 is now partially strain softening. According to expression (4.6), the imposed deydsj should be equal to the minimum value of(deJde1)dridned contour during shearing. This is not exactly at zero, but at a value of -0.025. However, comparing this minimum value of (dejde,)^^ contours at this initial stress ratio of 2.8 with that at R<. = 1.0 (dejde! contour = -0.5) indicates that the minimum does increase towards zero. Also, considering some experimental scatter inherent in testing, this difference is ignorable. It is noted that at an imposed positive deydej = +0.10 and +0.20, the initial state a'rc = 200 kPa and R,. = 2.8 responds in a strain hardening manner. This is obvious since the (dejde^p^ = +0.10 shear terminates at a minimum value of (deJdeA^^ contour of approximately +0.01 and for the (dejde^-jjgpo^ = +0.20, the terminal minimum value of (dev/del)dnimi contour is around +0.05. Therefore, since expressions (4.5) and (4.6) are not satisfied, the strain hardening responses occurs. Reasons behind the occurrence of strain hardening, partial strain softening and full strain softening under partially drained states in denser sand (Drc = 58%) are now considered. Data similar to that for the loose sand in Figs. 4.34 and 4.35(a) are presented in Figs. 4.36 and 4.37(a) for the denser sand. It may be again noted that: • Full strain softening under the imposed deydej = -0.60 occurs (star symbol in Figs. 4.36c and 4.37a) because the (deyde.)^^ is always more expansive than those along da'r = 0 path, i.e. (deJdedfa^a, at all ambient o'r, R states traversed during the strain path. Again prior and after the instant of Au^ (star symbol in Fig. 4.37a), the (dejde^p^ is always more expansive than the (deJdeA^^ contour encountered during partially drained shearing. I l l 0 100 200 300 400 rj' r (kPa) Fig. 4.35(b) Partially drained stress paths at 1^  =2.8 superimposed on Fig. 4.12 from do'r = 0 shear (Loose FRS) o'r, R excursion during partially drained shear (Medium-dense FRS) 4.5 a' r (kPa) Fig. 4.37(a) Partially drained stress paths at R<. =1.0 superimposed on Fig. 4.13 from do'r = 0 shear (Medium-dense FRS) 114 • Strain hardening under the imposed dev/de1 = -0.20 is a result of the minimum value of dejdel contours(« -0.35) whichis more expansive than that imposed. Therefore expression (4.5) is not satisfied. Prior to the instant where (deydej)^ ,,^  is more expansive than the corresponding (dE^de^^^ at any c'r, R values, the pore pressure increases. However, the progressive decrease in pore pressure after this instant is mainly due to the fact that (d£ v /d8 1 ) i m p o s e d is now less expansive the corresponding (deJdE^^^ values. It is obvious that for (dejde^p^ values less expansive than -0.20 (including undrained conditions), the strain hardening response becomes more pronounced, as larger differences exist between the (dsydCi)^,,^ values and the minimum value of d&Jdsj^ contours (see the results for (deydeO p^osed of 0.0 and +0.20 in Fig. 4.37a). • Imposition of dejdsy = -0.40 gives rise to a partial strain softening response (circle symbol in Figs. 4.36b and 4.37a). The reasons for such a response to emerge are similar to those described for the partial strain softening of loose Fraser River sand. For initial states at higher stress ratio levels, conclusions similar to those for the loose sand can be drawn. Comparison of results with the imposed (deydSj) = -0.20 at R^  = 1.0 and 2.0 but identical c'rc = 200 kPa indicates that the state with R,. =1.0, which shows a strain hardening response (Fig. 4.36a), transforms into a terminal case of strain softening (flat plateau response as was shown in Fig. 2.1) type (see Fig. 4.30) at R,. = 2.0. This would be expected by looking at the responses under (dcydeOtaposed = -0.20 at Rc = 1.0 (Fig. 4.37a) and at R^  = 2.0 (Fig. 4.37b). The partially drained shear at R^ . = 1.0 terminates at the minimum (dejdej)^^ contour of-0.35 while for the state at R<. = 2.0 this minimum value occurs at a contour of -0.30. Comparison of similar state but with (dejde^ = -0.40 shows similar effect of minimum (deJdBi)^^ contour on partially drained shear responses. This also explains the effect of increase in initial stress ratio on stress-strain response of sands. The criteria established above governing the emergence of the type of partially drained response under deydsj = constant shear of Fraser River sand are contrary to the findings reported by Chu et al. (1992) on dense Sydney sand. Chu et al. suggested a comparison of (dejdej)^^ with the 115 4.5 0 100 200 300 400 o-' r (kPa) Fig. 4.37(b) Partially drained stress paths at R^ . = 2.0 superimposed on Fig. 4.13 form da'r = 0 shear (Medium-dense FRS) 116 maximum expansion rate, (d^de.)maxdnrined, at the starting value of effective confining pressure as the criterion for full strain softening to occur, even for the sand at Drc=80%. Partial, and not full strain softening occurred in denser Fraser River sand at d£v/ds1 = -0.40 despite Chu's et al. criterion being met. This was because at some stage during shearing, (dejdej)^^ became less expansive than that along dc'r = 0 path at the ambient c'n R state along the partially drained effective stress path traversed. At the imposed strain increment ratio of -0.60, which is more expansive at all a'r, R states encountered during partially drained shear, full strain softening did result (Fig. 4.36c). A close examination of the (dejde^^i contours encountered for the imposed de^ de, = -0.20 (diamond symbol in Fig. 4.37a) indicates that o'r, R during shear for this test at the instant of Au,,^ crosses the contour of (dejdej)^^^ = -0.20 and even reaches to the contour of -0.30. Since deJdE1 along da'r = 0 paths at a'r, R state after the occurrence of Au,,,^  is more expansive than (dejde^, no strain softening occurs and (aa - ar) keeps increasing. For very dense Fraser River sand (Drc = 82%), even the imposed expansive strain increment ratio of -0.60 does not cause full strain softening (Fig. 4.25), even though this exceeds the maximum expansion rate under da'r=0 shear at a'rc of200 kPa of -0.42 (Fig. 4.14). This is because as for the medium-dense sand a'r decreases on partially drained shearing, and at some a'r, R during the shear, the maximum dilation (expansive) rate along do'r=0 at the ambient o'r, R, becomes more negative than the imposed. At the instant of Au,^, o'r has been reduced to 40 kPa (Fig. 4.25), and a much larger maximum dilation rate along do'r=0 is expected. The drained behavior along do'r=0 paths at this density was investigated at c're = 200 kPa only. However expectations of a much larger expansive (dejde^^ at a'rc = 40 kPa would be reasonable based on data similar to already observed for medium-dense state (see Fig. 4.14). If the trend of relationship between (dejde^^ and a'r in Fig. 4.14 for the dense sand is assumed similar to that for the medium dense, it does appear that (dejdsy)^ at around o'r = 40 kPa is more expansive than the imposed -0.6. At o'r « 40 kPa in partially drained shear therefore, pore pressure will start decreasing and no strain softening will be possible. Full strain softening, nevertheless occurs at this dense state under (dejde^ = -1.0 (Fig. 4.25). Clearly, at this imposed strain rate, the rate de^ de! along da'r=0 shear at all ambient a'r, R along the partially drained paths must be less expansive than the applied. Partially strain softening response could occur for (dejde^ between -0.6 and -1.0 even in this very dense sand. 4.6 Summary The results presented and discussed in Sections 4.4 and 4.5 can be summarized as follows: 117 • The partially drained situations may constitute loading conditions more damaging than either of the conventional drained or undrained assumptions. Very small expansive volumetric strain can transform an undrained strain hardening response into a strain softening type. A small change in the strain increment direction was identified as the cause for the large alteration of the direction of stress increment. The transformation to strain softening behavior under partial expansive drainage cannot, therefore, be attributed to a physical loosening of the sand. Consequently, ignoring partial drainage can be serious in the estimates of deformations, and even the stability of the earth structure since the deformation response of soil at a given ambient state depends upon the direction of stress increment. • The partially drained response is not bounded by that under undrained (deydSj = 0) and fully drained da'r = 0 (continuously changing dejdej) paths, as often assumed. • The partially drained conditions with negative imposed dejde1 causes an extension in the zone of contractive deformation over that observed under undrained conditions. The more negative the dejde^ the lower is the friction angle mobilized at the peak states. This implies more damaging conditions in the ground. If the partially drained response is only partially strain softening, the effective stress ratio at the trigger of strain softening remains essentially identical to that under undrained loading. • As the density increases, the effective stress ratio CSR at the trigger increases, and the region of contractive deformation becomes narrower for a given dejde^ until a threshold density for which this region will not exist anymore. This amounts to saying that the sand at any state at this density when sheared undrained would not strain soften, regardless of the confining pressure and stress ratio levels. For partially drained shear with fully strain softening responses, there would be no strain 118 hardening domain (i.e. between PT and MO lines). At higher initial stress ratios the response under partially drained paths will undergo a transition from strain softening to strain hardening in the domain of small contractive volumetric strains, the magnitude of dejde1 increasing with increase in R,.. The initial states under undrained conditions with R,. values larger than R at PT but smaller than the maximum obliquity state are within the domain of strain hardening responses. This means that the states within this region are stable when deformation is undrained. The change in shear modulus, G s with the direction of stress increment vectors (8 in t-s' space) showed a decrease in shear stiffness with increasing R,. for a given 8. The decrease being more pronounced in the range of higher R,.. At a fixed Rc, G s does not suffer a significant reduction with increase in 8, except at 8 approaching 180°, when G s drops to zero. The range of 8 over which this drops to zero occurs diminishes as R,. decreases. The criterion for the occurrence of strain softening, partial strain softening and strain hardening was established based on the behavior of sand under undrained and conventional fully drained shear. During shearing, (dejde^)^^ is compared to those along da'r = 0 paths, i.e., (dejde^^^, at the o'r, R states encountered along the partially drained effective stress path. - If the imposed partially drained path is more expansive than (dev/dei)drained a t aU ambient o'r, R states, then pore pressure increases all along the effective stress path, which ultimately leads to full strain softening. - On the other hand, if during partially drained path (deJ'dE1)iiapo!.ed is less expansive than (dEydeJd^j at all ambient a'r, R states, then a strain hardening response is expected. - Finally, if (deydej^p^ is approximately coincident with (deyde!) ,^,, at all ambient o'r, R states, partial strain softening would be expected. 119 The expansive dejdsy needed to cross the thresholds between terminally strain hardening to partially strain softening, as well as between partially to fully strain softening response increases over the loose sand values as the relative density increases. For example, under dsydsj = -0.40, the sand was fully strain softening when loose, partially strain softening when medium-dense and strain hardening when dense. Since the undrained response was already strain hardening imposition of compressive deJdEt makes it even more so. The threshold between strain softening and terminally strain hardening behavior in extension occurred in the domain of compressive volume changes in extension as opposed to expansive in compression. CHAPTER 5 INSTABILITY UNDER P A R T I A L L Y DRAINED CONDITIONS 5.1 Introduction Results of a comprehensive series of partially drained test under constant deviator stress, commencing from various initial stress states (a'ac, a'ro) that give rise to instability are presented in this chapter. The results presented and discussed below constitute typical behavior at selected initial states. The influence of the location of initial state relative to the region of undrained strain softening deformations determined under constant rate of strain tests are examined. Based on the results, the conditions, already established in Chapter 4, that govern the occurrence of true instability (full strain softening), temporary instability (partial strain softening), and stability (strain hardening) are examined. Finally, the relationship between strain softening and instability is explored. As stated in Chapter 2, the pre-failure instability can be studied by loading a soil element fully drained to a preselected effective stress level and then by the closing the drainage valve. Under this constant deviator stress (aa - ar), the instability may develop due the arrest of secondary consolidation or any other perturbation which causes the pore pressure to increase. This instability would, otherwise, not occur if the sand remained drained. This type of pre-failure instability has been studied experimentally, but only under undrained conditions, by Lade and co-workers (Lade etai., 1987, 1988, 1993). 120 121 Using the strain path testing technique, Chu et. al. (1993) did carry out a limited examination of the conditions for instability in saturated dense Sydney sand under partially drained condition. However, they did not address the potential for this instability in relation to a defined undrained instability line, and the related region of strain softening deformation. Depending upon whether the initial state before perturbation exists above or below the undrained instability line, how do different factors, especially the strain increment ratio (dejde^) affect the occurrence of instability needs to be investigated. Chu et al. proposed that at any effective confining pressure, the maximum expansion rate obtained from hydro statically consolidated specimens in a conventional drained (da'r = 0) test at that effective confining pressure could be used as the criterion for the occurrence of instability. In reality, the effective confining pressure during partially drained shearing changes (either increase or decrease). Consequently, their criterion ignores the accompanying changes in (deydej)^^ as the effective confining stress changes from its initial value. This is especially important considering that they examined this criterion only for dense samples, in which large changes in (d^Jde^^^ are associated with change in confining pressure level. Chu et al. did not investigate conditions (stress and drainage) which lead to limited liquefaction (temporary instability) type of behavior. Furthermore, their study was limited to behavior in triaxial compression only. Because of the inherent anisotropy in natural sands, it is important to examine liquefaction and instability in sand under partially drained condition along other stress paths, such as triaxial extension. These investigations over a wider range of densities and initial stress states will clearly be most desirable from a practical standpoint. 5.2 Repeatability in Instability Tests Since very small axial and volumetric strains were instrumental in causing instability, the confidence level in their relative measurement becomes of paramount importance. This confidence is assessed through testing for repeatability of results on specimen pairs at identical initial states, in which instability was induced under identical expansive dejde1 rates. There is no concern regarding the reliability of the measured individual strain components. The resolution of the instruments was within 5 to 10% of the measured values (Chapter 3). Fig. 5.1 illustrates results of instability check on two identical specimens. They were first 122 400 n 200 H rt 0 0.0 0.5 H 1.0 s o CO (a) a J D r c = 30% o Start of instability test 1 i | • I f (d e v / d e a ) i =-0.02 \ \ \ \ \ \ cy ' r c = 200 kPa R c = 2.8 \ R V — - — — — S (b) 0 i 4 B a (%) I 8 0 Point of accelerating strain (c) - 0 Point of decelerating strain 1 i 1 1 20 Time, t (min) 40 - i 1 r 200 ( a ' a + a ' r ) / 2 (kPa) rt Fig. 5.1 Repeatability of instability test results 123 hydro statically consolidated to o're = 200 kPa, and then sheared along the dc'r = 0 path into the zone of undrained strain softening deformation to the targeted R,. = 2.8. At this point, the stress path was switched from drained consolidation to partially drained shear at dejde1 = -0.02. This caused the pore pressure to rise at constant cell pressure op and within a few minutes axial strains started accelerating, marking the onset of instability. The instability, however, was only of the temporary type, because the deformations ceased to increase after a few minutes. Excellent repeatability in the test results may be noted. Both the stress-strain as well as volumetric strain-axial strain relations virtually overlap. This is true not only during the consolidation phase of the test, but also during the partially drained instability. The repeatability is indeed remarkable considering the small magnitude of volumetric strains involved, lending credibility to the test data. 5.3 In Compression - Loose Fraser River Sand 5.3.1 Initial States Inside the Region of Undrained Strain Softening Deformation The zone of undrained strain softening deformation in triaxial compression for the loose (Drc« 3 0%) Fraser River sand, delineated experimentally is shown in Fig. 4.5. As pointed out earlier, the undrained strain softening region is bounded by the line of peaks of strain softening responses (CSR line), and the PT line. The angle of friction at CSR is about 26° and at the PT state at approximately 32° (corresponds to c\lc'3 « 3.4). The initial states of the sand prior to the stress increment which falls in the region of strain softening deformation would be susceptible to spontaneous strain softening (liquefaction). Initial states both inside and below the strain softening region develop a knee at the PT state, which marks the termination of strain softening. Undrained straining beyond the PT state is associated with volume expansion tendencies, and hence decrease in pore pressure. This causes the effective stress path to climb up along the line of maximum obliquity. The angle of maximum obliquity is also unique at 37° in undrained shear, independent of the initial state of sand prior to undrained shear (Vaid and Chern, 1985). 124 Typical results of an instability check on the loosest deposited sand at an initial state o'rc = 200 kPa and R,. = 3.1 are shown in Fig. (5.2) (Test LII). A constant expansive volumetric strain increment ratio, (deyde^ j of-0.20 was imposed. The sand may be noted to develop instability, not at the initial stress state, but soon after, and at very small axial strain levels (of the order of 0.1%), with little change in shear stress. The acceleration in the deformation rate that follows the onset of instability (Fig. 5.2b), leads to runaway strains. As such, this constitutes a case of liquefaction triggered, not undrained but with partial drainage, which is associated with an expansive volumetric strain of as little as <0.02%. This amounts to a void ratio increase of merely 0.0004, and hence the triggering of this instability can hardly be attributed to any physical loosening of the sand, as one might intuitively believe. Extremely small drainage into the sand element can thus trigger instability. The instability did not initiate until the mobilized effective stress ratio R increased close to the value at the PT state of undrained deformation. The runaway strains may be noted to be accompanied by increase in pore pressures, despite some attenuation in the deviator stress. This attenuation in deviator stress occurred partly as a result of an increase in the cross-sectional area of the specimen due to compressional straining, and partly due to the inability of the pneumatic loading system (which supplied the initial static shear stress), to maintain pressure constant in the loading piston, when deformation occurred too fast. As noted earlier, undrained deformation beyond the PT state is always associated with dilative (volume expansion) tendencies, and hence, decrease in pore pressure. The increase in pore pressures observed instead, is, clearly a consequence of the continuously imposed partially drained condition that forces an expansive volumetric strain rate, (deydeji of-0.20. This rate is more expansive than the maximum expansive value of-0.10 along da'r = 0 paths (which correspond to no pore pressure generation) if imposed at any o'r, R state prior to the triggering of the runaway instability (see Fig. 4.12). Instability under identical (de^ de,); of -0.20 may be noted to develop at a lower initial R,. = 2.8 (test LI2), much in the same manner as at the higher R,. = 3.1 already discussed (Fig. 5.2). It, however, initiated when the mobilized R value increased to somewhat less than the R of the undrained PT state. The reasons for this smaller R at the instant of instability when conditions are partially drained, are explained in a later section. But, the causes of this instability are identical to those at the higher 125 Fig. 5.2 Partially drained Instability when initial states are inside region of undrained strain softening deformation 126 initial R, value. Fig. 5.2 also shows the effect of the degree of partial drainage, i.e. magnitude of (dejds^, on the potential for instability at otherwise identical initial state (R, = 2.8). At a much smaller expansive (deydejj of -0.02, the sand suffered only temporary instability (Test LI3). At the initiation of instability, the strains developed at an accelerating rate, but decelerated after the accumulation of a finite strain. The temporary instability develops as a consequence of the imposed dsydej more expansive than along the da'r=0 paths at all a'r, R levels until the trigger of instability. The unstable behavior, however, ceased at the a'r, R state at which the imposed dejdej^ became less expansive than that along the da'r=0 path at that state (Fig. 4.12). This type of instability under partial drainage is similar to limited liquefaction under static undrained loading. That the maximum expansion rate in the conventional drained tests (do'r=0) initiated from R,. = 1 depends on both the initial density and to a smaller extent the effective confining stress was illustrated in Fig. 4.14. Similar argument was presented supporting that it also applied at other nonhydrostatic initial R,. states (Fig. 4.12). In sand with initial state o'rc=200 kPa, R,. = 2.8 under (deydCj); = -0.02 the unstable behavior ceased after a finite runaway strain. Such a behavior also occurred at the identical R<. = 2.8 and (dejde^ = -0.02 initially, but higher effective confining pressure of 330 kPa. The deceleration in strain commenced after the accumulation of larger strain compared to the state with lower a'rc level (see Fig. 5.2b). In general, over the same period of time, larger axial strain will develop at the instant of strain acceleration at higher initial confining pressure. This can be better understood by looking at the drained da'r=0 behavior of sand (Figs. 4.10 to 4.13) which demonstrates that at a given R, the sand becomes more contractive with increasing effective confining pressure level. The influence of the stress ratio level on instability within the undrained zone of strain softening deformation is shown in Fig. 5.3. Identical dsydEj = -0.02 were imposed at R,. levels of 2.8 and 3.1, respectively at constant confining stress o're = 200 kPa. When partially drained conditions were imposed, pore pressures increased, and the effective stress paths moved horizontally towards the left. At both initial states, there was a limit to the development of axial strain, and pore pressure 127 Fig. 5.3 Partially drained instability from initial states inside the region of undrained strain softening deformation (Effect of an increase in R inside the region) 128 generation gradually reached a maximum value, without the stress state reaching the line of maximum obliquity (Fig. 5.3). At the higher stress ratio of 3.1 the axial strain accelerated (initially) at a much earlier time than at the lower stress ratio of 2.8. Close examination of Fig. 5.3 indicate that the specimen exposed to partially drained conditions at the lower stress ratio generated larger pore pressure. This is due to the fact that with initial states corresponding to the higher stress ratios, more volume change had occurred during the consolidation phase of the test, and therefore tendency for volumetric compression (or pore pressure development) was reduced for the subsequent partially drained conditions that would cause instability. The data presented in this section have demonstrated that if the initial state of sand lies inside the region of undrained strain softening deformation, there exists a potential for the sand responding contractively under undrained static loading, and to be unstable under partially drained conditions at constant shear stress. This does not necessarily imply that an immediate liquefaction (known as spontaneous liquefaction) or instability can occur when the conditions are merely switched from drained to undrained, without inducing any stress perturbation, i.e. an increased cd in static loading and an increase in pore pressure at constant cd for instability. Whether either phenomenon occurs or not depends on the volumetric response of the sand along the fully drained da'r=0 path at the ambient stress state. If this volumetric response is not expansive, spontaneous liquefaction will occur on undrained loading. This was indeed the case considered by Lade and co-workers (e.g. Lade et al., 1987). They demonstrated that loose sand with initial state within the region of undrained strain softening deformation manifested instability on merely becoming undrained, i.e. (d&Jde^ = 0. However, loose Fraser River sand even under a small expansive (dejde^ = -0.02 developed only temporary instability (Figs. 5.1 to 5.3). As pointed out earlier, at (dejde^ less expansive than even the maximum expansion rate of the sand at the ambient o'r along da'r=0 path, only temporary instability (or limited liquefaction) is possible. Only for (dejde^i more expansive than the maximum expansion rate at all ambient o'r, R levels is the full strain softening response expected. Similar arguments apply for the occurrence of instability. For loose Fraser River sand at a'rc = 200 kPa, the maximum expansion rate along da'r=0 is -0.10. Thus for imposed de^ d^ less expansive than -0.10 (which includes the undrained mode), no spontaneous liquefaction (or runaway instability) would 129 be expected. This is precisely what was found for undrained and (deydsj); = -0.02. As soon as the imposed de^ /de, = -0.20, which was more expansive than the maximum expansion rate along do'r=0 path at all ambient o'„ R states, runaway instability was observed. This also implies that in Lade et al.'s tests, the volumetric response along dc'=0 was contractive in all cases for axial strains of up to 20%. From the above discussion, it is apparent that the condition for instability does not simply require that the initial state should be inside the region of undrained strain softening deformation. Additional conditions for the occurrence of instability are postulated. If the rate of expansion imposed on the soil element exceeds the rate of expansion along the do'=0 path at the ambient o'r, R states, only then will the effective confining pressure decrease and the sand element become unstable (Equation 4.5). Thus, a rate of volume change (positive, zero, or negative) imposed on the sand could cause it to become unstable depending on the dsydej along da'r=0 paths at o'ro equal to the ambient a'r. 5.3.2 Ageing Effect Inside the Region of Undrained Strain softening Deformation The length of aging time after consolidation was found to have a significant effect on the shear response of Fraser River sand (Section 3.8). Instability checks on two identical specimens inside the zone of strain softening deformation one aged for 2 (LII) and the other for 15 minutes (LI6), following consolidation to the initial state (o'rc = 200 kPa and R,. = 3.1) were made, and the results are shown in Fig. 5.4. Strain increment ratios of-0.20 were imposed on both specimens. As already pointed out in Chapter 3, all instability tests presented in this chapter were performed after an aging of 2 minutes at the end of consolidation. This was needed to ensure consistency in test results for comparison purposes. Since consolidation was carried out as a continuous loading process, some waiting period was deemed necessary after reaching the target initial state to ensure completion of drainage. The results in Fig. 5.4 show that in specimen LI6 which was aged for 15 minutes, the instability did not occur until after about 12 minutes, in contrast to less than 2 minutes for the 2 minutes aged specimen. Thus, the effect of aging is to delay the initiation of instability, although the magnitude of axial strain at the instant of instability (point of accelerating strain) in both specimens was very small. 130 ( a ' a + a ' r ) /2 (kPa) 3 (ds y /de a ) i = -0 .20 (b) CT'rc = 200 kPa 30% 7 •4 " " 1 i 1 , . / l 0 4 8 12 16 Time, t (min) Fig. 5.4 Instability tests inside the region of undrained strain softening deformation (Effect of ageing time at identical stress states) 131 The foregoing discussion shows that with long aging periods, the initiation of instability could get delayed. This is especially important in natural slope stability problems where most slopes are aged at the current stress state for a long period of time. If an instability triggering phenomenon happens to occur at this slope (i.e. a liquified layer causes the pore pressure to increase in the adjoining non-liquefied layer), then the time for the initiation of instability will also be longer. On the other hand, for man-made embankment or soil structures which are aged for much shorter times compared to those for natural slopes, a minor disturbance that causes a redistribution of pore pressure, could trigger instability much sooner than it would occur in natural slopes. The effect of aging is to allow stable behavior to persist even inside the region of undrained strain softening deformation, if the perturbation event is not very long lived. 5.3.3 Initial States Outside the Region of Undrained Strain Softening Deformation Results of instability checks on initial states corresponding to R,. = 2.0, and three levels of effective confining stresses c'^ are presented in Fig. 5.5. Partially drained paths imposed corresponded to (deyds^ j = -0.40 in each case. It may be noted that instability does not develop until the mobilized effective stress ratio approaches the line of peaks corresponding to (deydsj); = -0.40 (see Fig. 4.29). Runaway strains are seen to occur at each initial confining stress level. Since the deJdSi imposed was more expansive than the maximum value ~ -0.10, along da'r = 0 paths (see Fig. 4.14) at all mobilized a'r, R states, positive pore pressures continued to develop even beyond the undrained PT state. This, in fact, is the cause of the runaway strains observed, as was the case for partially drained response for initial states inside the region of undrained strain softening deformation. Again, at each initial confining stress, the stress changes leading to instability occurred with little change in initial shear stress and void ratio (void ratio increase of less than 0.0004), and at very small axial strains of the order of 0.1%. At the instant when the axial strain started accelerating at this imposed strain increment ratio, i.e. (deydej; = -0.40, the mobilized effective stress ratios were found to be slightly smaller than the CSR of the undrained response. These differences are attributed to the imposed small, but finite volumetric strain increments rendering the sand partially drained. In general, as the imposed strain increment ratio became more expansive, the mobilized effective stress ratio at the initiation of 132 ( o ' a + a ' r ) /2 (kPa) Time, t (min) Fig. 5.5 Partially Instability compression for states outside the undrained region of strain softening deformation 133 instability decreased in relation to the undrained CSR, and thus the cutoff boundary "ab" of the region of strain softening deformation moved down towards the origin of the stress space. As shown in Fig. 4.29, this was confirmed independently in the behavior reported earlier for partially drained shear under several constant strain ratios, imposed at a given initial state. It is important to note that for the initial states corresponding to c'rc =100 and 200 kPa, for which the shear stresses xMpa - or) are less than at the cutoff level "ab", instability did not develop under completely undrained conditions (cross and square symbols respectively in Fig. 5.5). Thus initial states construed as stable under undrained deformations may transform into unstable states, if partially drained conditions prevail. Initial states with !/2(aa - or) levels greater than that corresponding to the cutoff "ab" would, however, be unstable under partially drained as well as undrained conditions. 5.3.4 Initial States Inside the Region of Undrained Strain Softening Deformation Region But Below the Cut-off "ab " As pointed out earlier, initial states below the cutoff "ab", but still bounded by the CSR and PT lines, if extended to the origin (i.e. within the region Oab; see Fig. 4.5) could respond in a strain softening manner, under completely undrained condition followed by strain hardening after crossing the PT state (limited liquefaction type response). For instance, the initial state corresponding to o're = 100 kPa and R,. = 2.8 within region Oab, does experience strain softening (limited liquefaction response; see Fig. 4.5). However, such initial states could transform into a flow liquefaction type of deformation if the partially drained condition imposed a dzJ&Ey more expansive than that along the dc'r=0 stress increment path at any previous o"'n R state. Experimental data to support that such a transformation does occur, is presented in Fig. 5.6, which shows that for the initial state c'^100 kPa and Rc=2.8, the temporary instability under undrained assumption indeed changed to unstable behavior within a few minutes, after expansive volumetric strains corresponding to (de^ de^  = -0.40 were imposed. Fig. 5.6 also shows a comparison of the development of axial strain with time under undrained and 134 Time, t (min) Fig. 5.6 Partially instability tests for states inside the region Oab 135 partially drained states. Whereas, instability commences in both cases at very small strains, deceleration of strains in the former case has already commenced at about 1% axial strain - on the way to eventually stable behavior. 5.3.5 Initial States Outside the Undrained Strain softening Deformation Region Below the Cut-off "ab" Results of instability assessments under partially drained loading starting at initial states corresponding to a much lower R,. value of 1.5 at two levels of a're, 200 and 400 kPa are illustrated in Fig. 5.7. Partially drained paths imposed corresponded to (deydei); = -0.40. The sand may be seen to stay stable in both cases, and little strains that develop occur at essentially constant shear stress. The axial strains developed appear to level off with time. Thus, instability will not develop unless pore pressure increases further with continued partial drainage. But, if the imposed de^ de! at any o'r, R level became more expansive than that along do'r=0 stress increment, the sand will demonstrate strain softening behavior eventually, if pore pressures continue increasing. 5 . 4 Instability Under Partially Drained Conditions in Compression - Denser Deposition State The potential for instability under partially drained conditions was also examined for medium-dense Fraser River sand (Drc = 55%). At this initial deposition density the sand is stable under undrained loading for values of c'^ higher than the range investigated in this study; and the maximum shear stress corresponding to the cut-off "ab" is in excess of about 400 kPa (Thomas, 1992). Typical results from two partially drained tests corresponding to (dejde^ = -0.20 and -0.60 on identical specimens at initial states a'rc =200 kPa and Rc=3.1 are shown in Fig. 5.8. Specimen DI1 with (deydeJi = -0.60 is seen to suffer runaway, but DI2 with (dejde^ = -0.20 manifests only temporary instability. As was the case with initially loosest states, the relative magnitudes of volumetric expansion rates imposed and those along do'r=0 paths at the ambient a'r, R states determine the type of instability - runaway or temporary. The maximum expansion rate, (dejde^^, of the sand along do'r=0 path is about -0.30 at =50% (Fig. 4.14). Thus the imposed dejdej^ less than -0.30 could render the sand runaway unstable. Only temporary instability would be experienced if this rate is 200 400 (a ' a + a ' r ) / 2 (kPa) 0.2 0.1 0.0 o. a' 3 c = 200 kPa; (d 8 v / e l ) j = - 0.40 ' (b) V = 400 kPa; = - 0.40 o Start of instability test — i 1 i i 0 400 800 1200 Time, t (min) Partially instability tests for states outside the region undrained strain softening deformation with stress ratio of 1.5 137 (de v / ds T ) j ; R c g 200 - | CN <T3 b o -0.6 ; 3.1 (DI1) A -0.2 ; 3.1 (DI2) X -0.6 ; 2.0 (DI3) D r c = 58% a ' r c = 200 kPa (a' a + a ' r ) / 2 (kPa) 400 CO 2 H l H 0 -0 O Start of instability test 5^1 Point of accelerating strain Point of decelerating strain 0.60 ; 3.1 = -0 .2 ,3 .1 14 - OS > 2 0 hrs w — - » v 20 40 Time, t (min) Fig. 5.8 Partially drained instability tests on denser Fraser River sand - states inside the region of loose undrained strain softening deformation 138 greater than -0.30 along da'=0 path at any ambient a'n R states along the effective stress path to which the sand gets subjected to. Instability will cease when the imposed dejde1 become less expansive than that along the da'r=0 path. Fig. 5.8 also shows partially drained behavior at identical o'rc=200 kPa but lower R<=2.0. The imposed deJdej^ = -0.60, which triggered instability commencing from the higher initial R,. of 2.8, now does not render the sand unstable. In fact little strain develops, and the stress path soon becomes stationary. Thus, for a given (deyde!);, the potential for instability at a given a're is very much dependent on the initial level of R,.. The reason why the denser sand at R,. = 2.0 with imposed dejde1 = -0.60 is stable are: (i) the remoteness of the CSR line from the initial state, which requires larger pore pressure increase to reach it; and (ii) the initial V2(aro3) is much smaller than at the cutoff "ab", which has moved the shaded region of undrained strain softening deformation higher up in stress space for this initial denser than for the loosest state. 5.5 Instability Under Partially Drained Conditions in Extension The zones of undrained strain softening deformation in extension for the Fraser River sand with relative densities at the end of consolidation (Drc) of 30% and 55%, determined by conventional undrained tests under constant rate of strain loading are shown in Figs. 4.7 and 4.8. It may be noted that for the loosest deposited sand, this zone extends right down to the origin, but, for the medium-dense, a horizontal cutoff "cd" exists, which was apparent from the undrained triaxial extension test at o'rc=100 kPa. Three instability checks at Drc of 25% were carried out. The specimens were first isotropically consolidated at their respective effective confining pressures a'rc of 100, 200, and 400 kPa. They were then consolidated in drained extension (a'r = constant) to a stress ratio (a'r / o'J of 1.5 outside the region of undrained strain softening deformation. A partially drained instability check with an imposed strain increment ratio dejdex = deJdeT of-1.33 was now sought. The results of these tests are shown in Fig. 5.9. They show that the stress states at which the strain starts accelerating lie approximately on the undrained CSR line, established from constant rate of strain triaxial extension tests. However, as indicated in Figs. 4.29 and 4.32 for compression and extension modes, this line 139 cd CN -100 H cd -200 H © -.<>^._.r._^._._..._. CT'rc 400 kPa 200 kPa 100 kPa CSR PT ( d 8 v / de i ) i = - 133 —I— 200 (a\ + a'T)/2 (kPa) 400 0.0 cd CO -0.4 H -0.8 i f <!> D r c = 3 0 % o 0 Start of instability test Point of accelering strain 400 Time, t (min) 800 Fig. 5.9 Partially drained instability tests in extension - initial states outside the undrained extension strain softening deformation region) 140 of peaks must be pertinent to the expansive strain increment ratio of -1.33 only. The line of peaks at expansive deJdEly however, does not seem to differ significantly from the undrained line of peaks (CSR) in extension. As was demonstrated for the compression mode, the sand in extension became unstable, because the imposed strain increment ratio was more expansive than the maximum expansion rate in drained da'r = 0 shear at the values of a'r = o're along the stress path traversed (Fig. 4.15). Further instability checks in extension were made on sand inside the region of undrained strain softening deformation. The initial state corresponds to o'r / a'a of 2.1 and effective confining stress c'rc of 200 kPa. As shown by the results in Fig. 5.10, for an imposed expansive dsjde1 of -0.50 (diamond symbol), the sand becomes unstable in less than one minute. Instability also occurs for the imposed expansive deJde,^ of -0.35 and even (dEy/de^p^ = +0.18. It may be noted in Fig; 5.10 that as the strain increment ratio becomes less expansive, and ultimately even compressive, the initiation of instability get delayed, but only by a few minutes (3 for dE^dEj = +0.18). This indirectly confirms that the sand is much more vulnerable to suffering instability in extension than in the compression mode. Under a compressive dE^de, of+0.18 instability does occur in extension, while in compression at the same o'rc, R<. initial states (mirror image in stress space), only expansive values of imposed deydEj would induce instability. The results shown in Fig. 5.10 may also imply that the maximum expansion rate measured in constant cell pressure tests in extension may not be credible. The assumed rate at o'rc = 200 kPa is approximately -0.07 (Fig. 4.15) and does not occur until an axial strain in excess of 10%. This value of deyde! would imply that only expansive dE^dEj could cause instability. However, the results presented herein and together with those under constant rate of strain, presented earlier, show that compressive values of deyde! would also lead to instability. At a strain level of 10% in extension, stress and strain nonuniformity during drained shear in extension may mask the real expansion rate in the zone of deformation (Bishop, 1971). Thus the observed global maximum expansion rate may not be correct. Furthermore, in a conventional drained extension test with constant cell pressure, the mean normal effective stress decreases during shear as opposed to increase during drained compression. This would induce volumetric expansions over and above those due to shear, thus Fig. 5.10 Partially drained instability tests in extension mode on medium-dense sand - initial states outside the undrained strain softening deformation 142 masking out the volumetric compression if the stress path involved loading instead of unloading extension. It is then likely that the true peak volume change rate could even be positive (compression). This was in fact shown earlier in constant rate of strain undrained and partially drained shear with positive (compression) imposed strain increment ratio, and strain softening (liquefaction type) did occur. The potential for instability under positive values of de^d^ would thus be a logical conclusion. Instability checks were now made on medium-dense sand at an initial stress state identical to that in Fig. 5.10 for the loose sand. The imposed dejdei^ was expansive at -0.50. Comparison of the behavior of loose and medium-dense sand is shown in Fig. 5.11 (diamond and star symbols represent loose and medium-dense states, respectively). It noted that after 1 hour from the commencement of instability check, the denser sand develops very small strain. It was therefore assumed that it could remain stable indefinitely. Since the initial state of this medium-dense specimen is outside the region of undrained strain softening deformation specific to this density, sufficient positive pore pressure did not generate to develop a mobilized R equal to that at the line of peaks pertinent to the expansive dejdel of -0.50. However, for the initial state at identical confining stress for this medium-dense sand, inside the region of undrained strain softening deformation (a\ I o'3 = 2.8), instability developed within few minutes (triangle symbol in Fig. 5.11). 5.6 Comparison between Instability and Strain Softening As pointed out in Chapter 2, the occurrence of instability is associated with increase in pore water pressure and thus a reduction in effective stresses at essentially constant deviator stress and total minor principal stress. This causes a decrease in effective confining stress, which takes the stress path into the region of undrained strain softening deformation. Therefore, increase in pore water pressure is the necessary condition for the occurrence of instability. One may argue that the increase in pore pressure as a necessary condition for instability is also true for the occurrence of strain softening. Evidence in support of this argument is now presented. Certain features do, in fact, distinguish between these two types of occurrences, but the basic mechanism remains the same (i.e. the increase in pore pressure) that leads to the catastrophic event. -o.4 -j— 1 1 1 r 0 4 8 Time, t (min) Fig. 5.11 Partially drained instability tests in extension mode (comparison of loose and medium-dense) 144 Possible differences between instability and strain softening can be examined by comparing the results from instability under constant shear stress and strain softening in constant rate of strain shear. It has been shown experimentally that at initial states outside the region of undrained strain softening deformation, instability under partially drained conditions does not trigger until after an effective stress ratio equal to that at the line of peaks (pertinent to the partially drained (dejde^ is mobilized. Runaway strains then manifest if the imposed expansion rates exceed the maximum expansion rate in conventional drained (do' =0) tests, at all a'r, R states encountered along the partially drained strain path. Comparison between instability and strain softening in compression is shown in Fig. 5.12. Two identical specimens at the loose state were first consolidated to a stress ratio of 2.8 along da'r=0 path. At this initial state, a partially drained instability check (indicated by "I" in the Fig. 5.12) was carried out. The specimen indicated by letter "S", was sheared under constant rate of strain. In both, expansive dejdsy of-0.20 was imposed. Both specimens ultimately reached the final effective stress state at the line of undrained maximum obliquity. In the specimen under strain controlled loading (test "S"), full strain softening (flow type liquefaction) occurred, while the test specimen subjected to constant shear stress loading (test "I"), manifested runaway instability. The axial strain development during the instability test (test "I") is shown in 5.12(a), while the stress-strain behavior for test "S" is shown in Fig. 5.12(b). It is also interesting to highlight the effect of aging in making comparisons between instability and strain softening. If the aging period is long the occurrence of instability would be delayed. This was shown in Fig. 5.4. However, in order to appreciate the effect of aging more vividly, the results of two strain controlled compression tests at identical initial state are also shown in Fig. 5.12(b&c). The response of the specimen with the longer aging period is illustrated by the dashed line. With the exception of a small initial increase in ad, a very similar phenomenon of full strain softening (flow liquefaction type) is observed in this test. This essentially confirms that the initial hardening is indeed equivalent to the delay in the occurrence of instability observed in constant shear stress tests. Comparison of instability and strain softening responses for another pair of parallel tests, now at 0.8 0.4 H 0.0 1^ 1 Onset of Instability Constant shear stress test test T ' i * Time (min) ^ 400 -4 \ \ \ \ Constant rate of strain tests \ N. \ N. ___Test "R" with resting period R c = 2.8 Test "S" with no resting period Dr c = 28% (b) B a (%) 12 200 H O Instability test "I" Test "S" -A— Test "R" MO. TT CSR y-- "a - b ( c ) ^ : CJ ' R C = 200 kPa (de y / de a ) i = -0.20 1 1 200 (a' a + a , r ) /2 (kPa) 400 Comparison of strain softening (S) and instability (I) at R,. = 2.8 (Effect of resting period = delay in instability) 146 lower initial stress ratio of 2.0, is shown in Fig. 5.13. Although there is a slight difference in the initial stress ratio (R,. = 2.0 for tests "I" and R,. = 2.1 for tests "S"), the assumption of identical initial state can be made with little error. Since the imposed dejde1 is -0.40, which is more expansive than the maximum rate of expansion in do'r = 0 shear at o'rc = 200 and 400 kPa, full strain softening and/or complete instability occurred. This not only further verifies the findings presented before regarding the criterion for the flow liquefaction response (based on the criterion of maximum expansion rate), but also supports the contention of similarity between strain softening and instability. The similarity between strain softening and instability in stress states which do not ultimately reach zero effective stress is explored in Fig. 5.14. The initial stress ratio was R,. = 2.1 and the imposed deydEj was zero, (dejde^p^ - 0. In the instability check the sand stayed stable even after more than 3 0 hours. Overall a temporary instability was observed. Similar behavior is manifested in strain controlled loading, in which limited liquefaction response occurred. Fig. 5.15 explores comparison of strain softening in constant rate of strain shear and instability at an initial stress ratio of 1.5 under an imposed strain increment ratio of -0.40. This initial state is too far from the region of undrained strain softening deformation. At effective confining stresses of200 and 400 kPa, despite a waiting time of 20 hours, no temporary instability developed. If the imposed partially drained conditions could be held indefinitely, then after large elapsed time from the commencement of the instability check, unstable response may still develop. The tests were terminated after 20 hours, and it may be concluded that the sand was stable, considering the axial strain accumulated over this time was merely 0.05%. However, this may not be the case, since in the parallel test under constant rate of strain, from point A to B at the 200 kPa, an excess pore pressure was 85 kPa, but under constant shear stress instability check, from point A to C, only 45 kPa was generated, despite 20 hours waiting. If sufficient waiting time was available for the generation of pore pressure, the sand would ultimately have suffered instability. Thus in generalization of the equivalence between instability and strain softening, the conditions for the generation of pore pressure plays a very important factor. 147 CO l H 0 Point of accelerating strain O cr' r c = 200kPa(Il) < > A a'TC = 400 kPa (12) 11 (a) L \ 12 ccS 3. 400 H ed 100 „ . , . . 200 Time (min) S ^ S2 (a ' r c = 400 kPa) 300 (b) SI (200 kPa) - i r 2 4 8 a ( % ) 400 CN 200 H ( d e v / de a ) i = -0.40 Dr c = 30% R c = 2.1 —I —i 1 200 400 ( o ' a + a ' r ) / 2 (kPa) 600 (c) Fig. 5.13 Comparison of strain softening (S) and instability (I) at R,. = 2.1 at two effective confining pressures 148 149 Fig. 5.15 Comparison of strain softening (S) and instability (I) at R<. = 1.5 150 The results shown in Fig. 5.15 also reinforce the findings presented earlier in this chapter in which the reduction in o'r in the two tests should be considered. For example, in Fig. 5.15, at point B the effective confining stress has reduced to approximately 115 kPa, so that at each moment of the test the effective confining stress is decreasing and therefore the maximum expansion rate in da'r=0 shear pertinent to this o'r should be considered for valid comparisons. 5.6.1 Further Comments on Strain Softening and Instability As explained in the Chapter 2, the Hill's stability postulate requires that the product of stress and strain increments be positive (during shearing). If this is negative, the condition for instability exists. This is exactly what has been suggested for the occurrence of strain softening (e.g. Valanis, 1985). Lade et al. (1988) have, however, shown that despite a negative product of stress and strain increments, stable behavior could occur, which would invalidate Hill's stability postulate. However, in the opinion of the writer, due to the following reasons, the conclusions made by Lade et al. do not necessary imply the violation of Hill's postulate. In the experiments reported by Lade et al., the sand was consolidated along dc'r=0 path to the desired stress ratio level, below the stress ratio at failure (maximum obliquity). It was sheared drained further (total and effective stress changes were equal) along a stress path in such a way that a negative product of stress and strain increments occurred. From this, Lade et al. concluded that since the stable behavior was observed despite the product of stress and strain increments being negative, Hill's postulate was violated. It is very important to emphasize that if the total and effective stress increment were equal and pore pressure did not increase, then Hill's postulate need not be satisfied when comparing situations where the stress increment causes pore pressure increase. Experiments in which physical instability occurred with generation of pore pressure were not performed by Lade et al., and this apparently led them to conclude that Hill's postulate can be violated without qualification as to the generation of pore pressure. In a different experimental study Chu (1991) also concluded that the strain softening and instability are two different phenomena. The foregoing discussion was presented in order to assess if Hill's postulate can be considered as the criterion for the instability. In the opinion of the writer, whether postulates such as those by Hill's are violated or not does not make any difference in the concept of comparison of instability 151 and strain softening (or the initiation of liquefaction). This is because as long as there is a fully drained loading condition (total and effective stress increments are equal), the states below the drained failure line in stress space are always stable, whether Valanis criterion (strain softening criterion) is met or Hill's postulate. Stress paths in space which encompass a large section of the stress space could still violate Hill's postulate and no instability would occur. But any departure from this type of drainage condition (i.e. when total and effective stress increments are equal), strain softening and instability are similar in occurrence. It is noted that the onset of instability is also the stress state where the initiation of strain softening starts. Therefore, generalization of Hill's postulate is of concern only when fully drained stress paths are included (i.e. when a comparison between fully drained and partially drained response is made). That seems to be a comparison of two different categories of loading regarding the assessment for real (physical) instability. In summary, when stress paths other than fully drained (undrained condition included) are considered, the conditions for the occurrence of strain softening and instability are similar. The above discussion suggests that if strain softening can occur in sand, instability will also occur under the parallel conditions, and vice versa. Therefore, it can be concluded that the instability is indeed related to strain softening, and the two may be a different exhibition of the same mechanism, but under different loading conditions. However, there might be one confusion with the definition of instability. If strain softening occurs and Hill's stability postulate is violated, this may imply that the sand should be unstable. Nevertheless, there is some evidence presented by Lade et al. (1987, 1988) that stable behavior is noted under completely drained (ESP= TSP) conditions. It was argued by Lade et al. that the existence of initial state within the region of undrained strain softening deformation does not necessarily mean that the sand has the potential for instability. The other necessary condition for the occurrence of instability is the generation of excess pore pressure. Chu (1991) reported results of parallel tests under partially drained conditions in which strain softening and instability were compared. While presenting a pair of parallel tests at an initial stress ratio of 1.8 on dense Sydney sand he concluded that"... since the instability test result was negative (i.e. no instability developed) and the strain controlled loading show strain softening response, the necessary condition for instability (but not sufficient) is the occurrence of strain softening, and the 152 instability is the sufficient condition (but not necessary) for strain softening...". The results presented by Chu do not give details of the duration over which instability check was made, and whether in time, a temporary instability was demonstrated. However, as shown'in Fig. 5.15 for Fraser River sand, if the instability check is considered after only a short period of time (as in Fig. 5.5 in comparison with those samples which manifested instability), then stable behavior may be implied. But if the instability check is monitored for a longer period (as it is the case in Fig. 5.14), then the comparison of parallel tests can be assured. From this, it can be argued the condition for the occurrence of instability and strain softening is the same with no necessary or sufficient additional requirements for each phenomenon. The comments presented in above paragraph explain the overall behavior at any state in stress space. However, it should be noted that if the instability check is sought over a short period of time only, and no instability occurs, then at constant rate of strain shear, only the conditions at that state (or the stress increment vector) should be considered. For example, in Fig. 5.14, it can be seen that in the constant rate of strain test, the sand initially strain hardened (point R to M) which implied it is stable. In the parallel instability test, the amount of pore pressure generated between points R and M is shown with diamond symbols, which causes no runaway strain (Fig. 5.14a) implying stable behavior. However, when the overall behavior is considered, both tests show similar response: the constant rate of strain shear exhibits a slightly limited liquefaction type of response, while the constant instability stress test shows a temporary instability response. Therefore, in comparing the instability and strain softening behavior, it is important to consider 'both short-term and long-term time comparison" for the sake of generalization. 5.7 Summary The results of partially drained instability tests in compression and extension presented in this chapter can be summarized as follows: • The location of the undrained and partially drained strain softening deformation regions established already through the constant rate of strain tests was confirmed by instability tests under constant shear stress. 153 It was demonstrated that the condition for instability does not simply imply the existence of initial state inside the region of undrained strain softening deformation. Additional condition are postulated. If the rate of expansion experienced by the soil element exceeds the rate of expansion along da'r=0 path at the ambient a'n R states, only then the effective confining pressure would decrease and the sand element would become unstable. Thus, a rate of volume change (positive, zero, or negative) imposed on the sand could cause it to become unstable depending on the dejdej^ along da'r=0 paths at a'rc equal to the ambient o'r. The sand is much more vulnerable to suffering instability in extension than in the compression mode. Under a given compressive dejdel instability may occur in extension, while in compression at the same c'K, R<. initial states (mirror image in stress space), only expansive values of imposed de^ /de! would induce instability. Evidence in support of the extension mode being more compressible and hence less expansive than compression was verified throughout the instability tests in extension. It was therefore concluded that although the maximum expansion rates dejdey in da'r=0 shear, for the loose Fraser River sand appeared larger in extension than in compression at a given o'rc, the higher values in extension could be attributed mainly due to the development of nonuniformities in the extension loading modes. The occurrence of instability is associated with increase in pore water pressure at essentially constant total stresses. The increase in pore water pressure is the necessary condition for the occurrence of instability and strain softening. Certain features do, in fact, distinguish these two types of occurrences, but the basic mechanism remains the same (i.e. the increase in pore pressure) that leads to the catastrophic event. Possible differences between instability and strain softening were examined by comparing the results from instability under constant shear stress and strain softening in constant rate of strain shear. Comparison between instability and strain softening in compression showed that specimen under constant rate of strain loading full strain softening (flow type liquefaction) 154 occurred, while the test specimen subjected to constant shear stress loading manifested runaway instability. Similar parallel tests for partial strain softening / temporary instability and strain hardening / stable behavior were observed. The effect of ageing was shown to delay the initiation of instability. The effect of ageing is to allow stable behavior to persist even inside the region of undrained strain softening deformation, if the perturbation event is not very long lived. The drained creep was found to shift the point of instability closer to the effective stress failure line. The effect of ageing in making comparisons between instability and strain softening for identical states inside the region of undrained strain softening deformation revealed that if the ageing period is long, the occurrence of instability would be delayed. The initial hardening is indeed equivalent to the delay in the occurrence of instability observed in constant shear stress tests. The Hill's stability postulate requires that the product of stress and strain increments are positive (during shearing). If this is negative, the condition for instability exists. This is exactly what has been suggested for the occurrence of strain softening (e.g. Valanis, 1985). In comparison of strain softening and instability, it is very important to emphasize that if the total and effective stress increment were equal and pore pressure did not increase, only then Hill's postulate need not to be satisfied when compared to situations where stress increment cause pore pressure generation. But any departure from this type of drainage condition (i.e. when total and effective stress increments are equal), strain softening and instability are similar in occurrence. It is noted that the onset of strain softening is also the stress state where the initiation of strain softening starts. Therefore, generalization of Hill's postulate is of concern only when fully drained stress paths are included (i.e. when a comparison between fully drained and partially drained response is made). That seems to be a comparison of two different categories of loading regarding the assessment for real (physical) instability which has been reported by Lade et al. (1987) and Chu et al. (1993). When the stress paths are other than fully drained (undrained condition included) are considered, the conditions for the occurrence of strain softening and instability are similar. CHAPTER 6 STRAIN INCREMENT, STRESS AND STRESS INCREMENT DIRECTIONS 6.1 Introduction The objective of this chapter is to examine the association of strain increment directions to effective stress and stress increment directions when the sand is subjected arbitrary stress probes at a given stress state. This is of relevance in constitutive modeling of soil, which are often idealized as elasto-plastic. In such idealization, slip component of strains have been treated as plastic, and any recovery of strain on unloading as the elastic component of the total deformation. However experimental studies in the literature do not appear to support this when strains are separated into elastic and plastic components by a conventional load/unload sequence. It has been argued that the observed recoverable deformations are not entirely elastic. They also include the reverse slip deformations that occur in random grain assemblies (El-Sohby, 1964;Rowe, 1962; Home, 1965; Nova and Wood, 1978). Therefore, no attempt is made in the following to split total strains into recoverable and irrecoverable components. The association of only total strain increment direction with stress and stress increment directions will be examined. 6.2 Stress and Strain Parameters The stress and strain variables utilized in this chapter are identical to those in elasto-plastic modeling of clays such as the Cam-clay theory (Schofield and Wroth, 1968). The examination of the 156 157 association of strain increment direction to stress and stress increment directions will be limited to compression loading only in which R and (cj - a3) are increasing. However, similar arguments could be extended to extension paths. The stress/strain parameters used are: stress parameters: Deviator stress Mean effective stress Effective stress ratio Stress direction Stress increment direction q = °i - 0-3 p' = 1/ 3 (a\ + 2o'3) = Vb (o'a + 2a'r) in triaxial compression R = o' 1/a' 3:n = q/p' 0 = tan1 (n) a = tan_1 (dq /dp') strain parameters: Incremental deviator shear strain Incremental Volumetric strain so that the work input is: ds8 = % (de! - de3) dey = dei + 2de3 dW = p' dev+qde9 = a'i dex + 2 a'3 de3 Strain increment direction p = tan1 (des / d^ ) These parameters are schematically illustrated in Fig. 6.1. The angles are measured counter clockwise from the horizontal p' and ^  -axes, respectively. Since the principal stress direction in the triaxial compression test are fixed along the principal axes of material anisotropy, there would be a coincidence of the principal axes of stress and strain and the principal axes of strain increment and stress increment. It is convenient to examine a, P, 9 relationship by aligning along the p' and des along the q axes respectively, and then plot strain increments d^ and dea as strain increment vectors at the stress points under consideration. The chosen format for plotting stress-strain data has the dual advantage of providing means for 158 Fig. 6.1 Schematic representation of stress, stress increment and strain increment at a point 159 presenting important arguments graphically, and at the same time employing quantities which are readily incorporated into analytical development. For example, the angle p between the strain increment vector and the hydrostatic axis is a reflection of the rate of volume change with shear strain (Fig. 6. lb). The strain increment vector that represents no volume change corresponds to P equals 90 degrees. It is greater than 90 degrees for volumetric expansion and less than 90 degrees for contraction. The angle 0 between the stress vector and the p' axis is a measure of the soil's mobilized strength as shown in Fig. 6.1(a). 6.3 Loading and Unloading In stress space, six zones of loading and unloading can be identified. These are specified in relation to incremental changes in p', q and 0 during shear, and are depicted in Fig. 6.2. The stress paths along which q and n are monotonically increasing are considered loading paths. Thus, in Fig. 6.2, a values between 0 and 180° (i.e. 0 > a > 180°: zones © and ®), which encompass both increasing and decreasing p' (or o'r) are considered loading increments. The a of 72° corresponds to the stress increment along the conventional drained do'r = 0 path. P = 90° for the undrained stress path, dejdes - 0. In the effective stress space a linear effective stress path results in a nonlinear strain path (such as da'r = 0 shear). Conversely, a linear strain path will give rise to nonlinear effective stress path. The division between loading and unloading paths can also be made in terms of a'r (effective radial stress), instead of p' as the normal stress variable (see Fig. 6.3). 6.4 a, P, 6 Relationships The association of P with 0 and a at a stress point along the loading path is examined for two scenarios (1) identical stress/strain history prior to the current stress state. The results presented in Chapter 4 provide the data base for such examinations. (2) Effect of different stress/strain history prior to reaching the current state. In addition to the data presented in Chapter 4, additional tests were also carried out for the examination of this issue. These are presented and analysed in this chapter. 160 Zone drj dp' dq a = 0 0 to 0 a a = 0 to 90 ° a = 90 ° to 180 o e 4p) 5p) 4^ a= 1 8 0 ° t o ( 1 8 0 ° + 0) a 0 a = ( 1 8 0 ° + 9) to 2 7 0 ° 4 a P a : Stress increment direction 9 : Stress direction a = 270° to 360 ° P' i / a Fig. 6.2 Zones of loading and unloading in p'-q stress space in terms of p' 161 Zone dri d a dq ' / ' / \ a ® a = 0 0 to 6 a 3c P a = G to 72 a = 72° to 180° © a= 180° to (180° + 9) L L. a = (180° + 9) to (180+72)° / a k L a : Stress increment direction 9 : Stress direction a = (180+72)° to 360° i / . / /IS. e Fig. 6.3 Zones of loading and unloading in p'-q stress space in terms of o'r 162 6.4.I a, fi, 6 Relationships for States with Identical Past Stress/Strain History Fig. 6.4 shows the initial stress states considered (designated by numbers ©,©,...) at which these associations were examined. Each initial state was arrived at by following fully drained consolidation path consisting of the hydrostatic, R,. = 1.0 (T|c = 0) path until the desired confining stress was reached. The path was then switched to the conventional do'r = 0 type, until the targeted r| or R was reached. The initial state "a" shown in Fig. 6.5 (state © in Fig. 6.4) with c'rc = 200 kPa and ^  = 2.8 (nc = 1.126) is considered first. For this state 0 = 48.4°. A series of identical specimens of loose Fraser River sand were subjected to a variety of stress increments along different directions simulated by strain paths with dev/d8a = constant. The results are shown in Fig. 6.5. The stress and strain increment directions at "a" are drawn by solid and dotted line segments respectively. The a, p line segment pairs associated with each other are identified by identical symbol labels. The relationships of P to a and 0 for data in Fig. 6.5 are plotted in Fig. 6.6. It appears that P and a during unloading paths (a<0) are approximately equal. It can be shown rigorously that a = P for a linear elastic isotropic material if a = 0° or 90° (stress increments corresponding to hydrostatic and constant mean normal stress directions), or if v = 0 (Poisson's ratio) for the elastic material. In the loading region (0 < a < 180°), however, p deviates more and more from a, as a increases towards 180°. Nevertheless, over most of the loading domain, i.e. for a between 0 and 180°, p may be considered approximately independent of a. This can then be viewed as p being associated with 0 only, as assumed in modeling soils as elasto-plastic solids. Intuitively it would appear to be due to the high level of initial stress ratio (i.e. R,. = 2.8), at which the total deformations could be sensibly viewed as entirely non-recoverable, and hence plastic in the conventional sense. For a in excess of 180° (unloading increments associated with decrease in q but r| still increasing) p loses its association with 0, and suffers large changes for very small change in a. Experimental studies on clays by Lewin and Burland (1970) have also demonstrated that within the compression unloading region (decrease in q or r|) the strain increments are not of the irrecoverable plastic type. 163 0 200 400 600 800 p' (kPa) Fig. 6.4 Initial states a'r, R considered for examining the a, p\ 9 relationships 164 • s co +2.0 +1.0 +0.4 +0.3 +0.2 +0.1 © o -0.2 -0.4 © 1 > CO w r> C N _ ON o m ON O ON rn" m ND ND* •*' 00 © ' ON © ' o ON O NO C N —- <o '"H (CTC-'—N ON ON ON 00 •* V> I—1 ON 00 ND o ON rs rs CN rn m ND ON w i ON 00 ON rn 8 o rs A X * o + o o 166 6.4.1.1 Influence of Stress Ratio Level The association of P with 9 and a at a lower stress ratio of R,. = 2.1 (nc = 0.805 and 9 = 38.8°) but identical a'rc = 200 kPa is shown in Fig. 6.7. For a < 0 (unloading), the strain increment direction is now aligned essentially in the same direction as the stress increment. This seems to be due to the fact that at lower stress ratios, and for the sector of compression unloading, irrecoverable strains may not form a dominant component of the total strain increment compared to those at higher stress ratios. But, for stress increment directions confined within the loading sector (9 < a < 180°), p is more associated with a and not with 9. The region of a over which P could still be regarded as approximately associated with 9 is now much narrower than at R,. = 2.8. The association of P with 0 and a at identical effective confining pressure of 200 kPa but at still lower effective stress ratio 1.5 (r|c = 0.425 and 0 = 23.0°) is shown in Fig. 6.8. Even though the amount of data is limited for a values less than 0 (stress increments associated with decrease in n, i.e., unloading), stress and strain increments directions again coincide. Even for a values within the region of loading compression (0 < a < 180°: zones © and <D in Fig. 6.2), p is still associated with a and not with 0. The association of strain increment direction with stress increment directions at the hydrostatic stress state (a'rc = 200 kPa, nc = 0.0, or 0 = 0°) is shown in Fig. 6.9. The entire range of a values now represent loading paths. Over this region of loading compression (0° < a < 180°: zones © and ® in Fig. 6.2), p is still associated with a. It appears P lages behind a until a = 90° when it equals a. For a >90°, p again starts deviating for a, but not by a significant amount. There are no stress increment directions at R=1.0 that correspond to unloading stress paths. In order to further generalize the findings about the effect of stress ratio on the association of direction of strain increments with direction of stress increments, a limited number of tests at the same confining pressure of200 kPa (c'rc = 200 kPa) but higher stress ratio R,. of 3.4 (nc = 1.332, or 0 = 53.1°) were performed. The results from these tests together with those at other stress ratios already presented are summarized in Fig. 6.10(a). The stress ratio at failure under do'3=0 shear is about 3.7 (r| = 1.421, or 0 = 54.9°). Thus, the ratio 3.4 can be regarded as very close to the drained 167 Fig. 6.7 a, 0, 9 relationships at R,. = 2.1, a're = 200 kPa 168 Fig. 6.8 a, p, 0 relationships at R, = 1.5, a'rc = 200 kPa 169 a ( ° ) Fig. 6.9 a, p, 0 relationships at R,. = 1.0, o're = 200 kPa 170 0 100 0 100 200 a(degree) Fig. 6.10 Summary of a, P, 9 relationships at a'rc = 200 kPa 171 failure along path da'3=0. It is interesting to note that the strain increment direction at this high stress ratio is almost uniquely associated with stress direction 9 only, and is essentially independent of the stress increment direction. For example a change in stress increment direction of over 130° a mere 6° change in strain increment direction occurs. Similar experimental findings have been reported by Tatsuoka and Ishihara (1974) and Anandrajah et al. (1995) who showed a unique direction of plastic strain increment direction as the stress ratio gets closer to the failure line. The results in Fig. 6.10(a) are now presented in Fig. 6.10(b) for the sector which represents compression loading only. It may be argued that as the stress ratio R increases towards the failure value the association of strain increment direction with stress direction 9 only gets closer. In elasto-plastic constitutive modeling of soils, a unique direction of plastic strain increment has indeed been assumed at higher stress ratios. The results shown in Fig. 6.10(b) do tend to support this assumption, but only at higher stress ratios, closer to the failure value. From the test results in Fig. 6.10, it is clear that there is no unique association of strain increment direction with stress direction at lower stress ratios. Instead, the strain increment direction is closely associated with stress increment direction. Angles a and P are, however, not equal for the loading paths, p tends to deviate more and more from a as a increases from 9 towards the limiting loading stress increment direction close to 180°. For n unloading paths, a < 9, the strain increment directions, however, are approximately coincident with stress increment directions. Regardless of the R,. level, all initial states with a'3c = 200 kPa considered for the examination of the association of P with 9 and a were approached by the stress increment directed at a = 72°, with the associated strain increment direction that ensued along these da'r = 0 loading paths. The only exception being the initial hydrostatic stress state, which was approached along the R = 1.0 path. Subsequent stress increment probes at the initial state involved rotation of the direction of stress increment only with stress direction held constant at 9 = 72°, except for continued loading along a = 72°, for which there was no rotation. Fig. 6.11 illustrates the stress and the associated strain increment directions at all initial states considered at c'rc = 200 kPa. Some interesting observations can be made regarding the role of the amount of rotation of the stress increment directions from its 172 Fig. 6.11 Stress and strain increment vectors for different stress ratios at o' = 200 kPa 173 approach value of 72° on p, a, 9 relationships. Only loading stress increment probes are considered for R,. =1.0, 1.5 and 2.1. For the higher R,. =2.8 some unloading probes involving to a > 180° are also included. For a greater than 90° (loading zone (D in Fig. 6.2) that involves decreasing p', the strain increment direction lags behind the stress increment direction regardless of the amount of rotation in a. At a given R, the amount by which P lags behind a increases with the magnitude of rotation in a. For small rotation in a the difference a - p appears to stay essentially constant regardless of the R level. Larger rotation in a from the approaching 72°, on the other hand, tends to reduce (a - P) as R decreases. At lower R levels of 1.0 and 1.5, (a - P) tends to decrease with increase in the magnitude of rotation in a. At the higher R level of 2.1, however, a - P tends to decrease with increase in the amount of rotation in a. According to elasto-plasticity theory, such as that by Hill (1950), if the plastic strain increment direction rotates with the stress increment direction, then the use of a single plastic potential in constitutive formulation may not be reasonable. The results shown in Fig. 6.11 imply similar conclusions. 6.4.1.2 Influence of Effective Confining Stress The association of the direction of strain increment with a along a constant effective stress ratio path is explored in this section. The change in p along such a path would be attributable to change in the level of confining stress only while 9 is held constant. Beginning with a stress ratio R,. of 1.5 (r\a = 0.425 and 9 = 23.0°), it may be noted from Fig. 6.12 that P for a given a is essentially independent of the level of confining stress between 50 to 400 kPa. This implies that the inclination of the strain increment vector to the line of constant R = 1.5 is constant, and thus the stress direction 9 and strain increment directions P relationship is not affected by the level of confining stress. At the same time, the confining pressure does not influence the relationship between the stress increment direction a and the strain increment direction p. This is in agreement with the findings of Poorooshasb et al. (1966) who reported that the inclination of the plastic strain increment vector to the lines of constant R is constant. However, the stress ratio along which this is applicable here is smaller than those used by Poorooshasb et al. It is once again emphasized that there is no relationship between the stress direction and the strain increment direction in Fig. 6.12. Strain increment direction, however is associated with stress increment directions only, and for the r) unloading paths a and p are approximately equal. This conforms to the finding of Roscoe et al. (1963) who demonstrated that Rc=1.5 - a' r c = 50 kPa 200 - n c = 0.425 = 200 kPa 9 = 23.0° - e - = 400 kPa • 'a=p / dil [, dp'| / / / / / / 100 -/ / / / jrfs 0 = 23.2° / / / s? \ ' Z / 0 - / Dr c = 30% i ' — • 1 • r-0 100 200 a, 0 (degrees) Fig. 6.12 Effect of confining pressure on a, P, 0 relationships at R,. = 1.5 175 for unloading strains are predominantly recoverable, a « p. In Fig. 6.13, the results similar to those in Fig. 6.12 are shown at a higher initial stress ratio of 2.1 (r|c = 0.805 and 9 = 38.8°). At this stress ratio, the confining pressure level does not seem to influence the a and P relationship for r| unloading paths (a < 9), and furthermore a and P are essentially equal regardless of the level of confining pressure. For a values between 0 and 160° (loading sector), the confining pressure does somewhat influence the unique association of strain increment direction with the stress and stress increment direction. For a given a in this region (9 < a < 160°), P changes with confining pressure as shown in Fig. 6.14(a) for a few selected values of a. Within this region, a fixed strain increment direction may be approximated over different ranges of stress increment directions, depending upon the confining pressure. For example, P = 65° is associated with a 35° wider range in a, as the effective confining stress increases from 50 to 400 kPa (see also Fig. 6.13). Figs. 6.14(b) and (c) show alternate presentation of the above results for fixed stress increment directions a = 100° and 72°. It is clear that for a given stress increment direction, the strain increment direction does depend somewhat on the level of confining stress, the effect being more pronounced at lower levels of o'rc. The association between the stress, stress increment and strain increment directions is now examined at a high stress ratio R, of 2.8 (nc= 1.125 and 9 = 48:4°) (Fig. 6.15). Again for a values smaller than 9 (unloading sector), a and p are approximately equal, and this equality holds regardless of the level of confining stress. Such a relationship does not, however, hold for a greater than 9 (loading paths). The confining pressure then does influences the a, p relationship more than it does at lower stress ratios. However, for a values greater than 180° (unloading in q) which represents spontaneous strain softening response, the confining pressure again has not much influence on a and P relationship. For a values between the limits 0 < a < 180° (loading paths), the strain increment direction is associated with the stress direction of every confining pressure level. A closer association of P to 0 alone seems to improve as the confining pressure decreases. The test results presented so far indicate that at higher stress ratios, the directions of strain increment are approximately associated with stress direction, typical of plastic behavior. From Fig. 6.15, it is 176 Dr c = 30% / / 200 R c = 2.1 / / T( c = 0.805 / / 9 = 38.8 ° / ' oc= B / / dr,|,dp'| < -/ / / / / / 100 -/ / / / yy -0 = 38.8 0 / / CT1 r c - 50 kPa « ) ' = 200 kPa 0 -/ / = 400 kPa 1 0 100 i 200 a , 0 ( ° ) Fig. 6.13 Effect of confining pressure on a, p, 9 relationships at R,. = 2.1 177 80 H CQ. 40 (a) a= 140° = 100° -o a = 71 .6° — G ri = 0.805 9 = 3 8 . 8 ° 0 -1 200 rj'rc(kPa) 400 400 H ^4 200 H a = 100.0 0 R = 2.1 "7 s f s p = 62.8 ° s p = 65 .2° / ^ > • " 0 a' r c = 50 kPa I ' o = 200 kPa , 7 3 = 7 5 . 0 ° o = 400 kPa 400 H PL, ° " 200 A a = 71.6 (c) p = 5 6 . 6 ° „ "R = 2.1 p = 5 8 . 2 ° p = 6 6 . 7 ° 200 1— 4 0 0 p' (kPa) 600 800 Fig. 6.14 Strain increment directions along constant stress ratio of 2.1 100 200 a , 9 (degrees) Fig. 6.15 Effect of confining pressure on a, P, 9 relationships at R,. = 2.8 179 apparent that for a values within 0 < a < 180° (loading path), the effect of confining pressure is more pronounced compared to that at lower stress ratios. For the high effective stress ratio of 2.8 and three fixed a values, the variation in the strain increment direction with a'3c shown in Fig. 6.16. For each a, it can be seen from Figs. 6.16(b) and (c) that with change in confining pressure, no unique strain increment direction exists for the given stress and stress increment direction. 6.4.2 a, j3, 6 Relationships for States with Different Stress History The influence of prior stress/strain history on a, P, 0 relationships is investigated by stress probes at a given effective stress state that has been arrived at by different prior stress/strain histories. As shown in the previous section (Figs. 6.12, 6.13, and 6.15), the effect a'3c level on the relationship of P to a and 0 was found to be small, and virtually negligible at lower levels of stress ratio R. Therefore it would suffice to examine how p is related to a at fixed levels of R, regardless of the level of a'3c. The association of P to a as noted in Fig. 6.12 is reproduced in Fig. 6.17 at R,. = 1.5 (hollow symbols). This relationship conforms to the reference prior stress/strain history until the current state - confined of hydrostatic loading to a specified o'3c, followed by a switch to dc'r = 0 path until R,. = 1.5. A number of states R = 1.5 were thus arrived at by shearing at constant deyde.,. These are shown by inset diagrams (b, c, d, and e) in Fig. 6.17. For stress probes applied these states, the association of P to a is shown in Fig. 6.17a by solid data points. It may be noted that regardless of the prior stress/strain histories at a given effective stress state has little effect of the relationship of Pto a. Comparative data relating P to a as to the effect of prior stress/strain history similar to that in Fig. 6.17 is now shown at higher R levels of 2.1 and 2.8 in Figs. 6.18 and 6.19. Approximate independent of the relationship between P and a from the prior history at a current stress point may again be noted, although some influence seems to appear as the level of R increases. The above finding on Fraser River sand is to some extent in agreement with those reported on clay (LeLeivre and Wang, 1970) in which no influence of previous stress history was reported on the 180 100 C/3 CD CD t. £J) CD T3 eo. 80 H 60 800 400 H 0 800 (a) a= 140 0 . R c = 2.8 a = 100° TI = 1.126 a = 71.6 0 0 = 48.4° 0 1 200 i 1 400 a' r c (kPa) a = 100.0 0 , ' p = 70.5° (b) -R = 2.8 • / / / / / / p = 75.0° / / r c = 50 kPa / 9' o = 200 kPa , V P = 84.0° o = 400 kPa OH 400 H a = 7 1 . 6 ° j3 ' • (3 = 67.0° (c) H = 2.c , p = 72.8 ° ^ = 83.5° 0 + T T 400 , / l n s 800 p1 (kPa) Fig. 6.16 Strain increment direction along constant stress ratio of 2.8 <L> CO. o H R c = 1 5 a = P / ( a ) n c = 0.425 6 = 23 .0° / ' # E P Dr c = 30% T^L c r ' r c = 50 kPa A = 200 kPa Q = 400 kPa 100 a(degrees) 200 200 H 200 Hollow: Reference Solid: Arbitrary p' (kPa) p' (kPa) 200 200 4 200 p' (kPa) p' (kPa) 6.17 a, P, 0 relationships for states with different stress history at R,. = 1.5 182 Fig. 6.18 a, P, 8 relationships for states with different stress history atRc = 2.1 183 Fig. 6.19 a, P, 0 relationships for states with different stress history at 1^  = 2.8 184 direction of plastic strain increment vectors. However, in this investigation, this effect was examined at different mobilized effective stress ratio levels. 6.4.3 Ageing (Rest Period) and its Related Memory Effect For the test results presented in this chapter, no rest period was allowed under the consolidation stresses of the test prior to application of incremental stress probes. For example, for the tests shown in the Fig. 6.18(c)i (redrawn in Fig. 6.20a), at point Y, no rest period was allowed in either specimens, and it was seen that the resulting response was similar. Furthermore, no appreciable difference was noted between their response as to the influence of prior stress/strain history does not have any effect on the behavior. Another specimen was consolidated along the path OCDY (Fig. 6.20b) commencing from its end of reconstitution hydrostatic stress state of a'3 ~ 20 kPa, the sample was first brought to a stress ratio of 2.0 under do'3 = 0 (path OC). It was then anisotropically consolidated under a constant stress ratio of 2.0 (path CD) up to point D. At this point a rest period of 20 minutes was allowed prior to shearing undrained to point Y and beyond (solid line). The specimen with consolidation path OY (redrawn from Fig. 6.20a) was sheared undrained from state Y without any rest period. It can be seen that initially the stress increment vectors differ from each other by as much as 40 degrees; even though at higher strains the two responses of the specimens matches closely. This behavior can be looked at from the following point of view. By comparing paths (2) inFig. 6.20(a) and (3) in Fig. 6.20(b), the segment DY is common to both specimens. This shows that in path (2) where the specimen was allowed a rest period at point "A" and no resting period at point Y, it does not retain any memory of this resting period at A by the time it reaches state D. In contrast, test (3) where the specimen was allowed a rest period at point D, has a strong memory of its rest period. It is also observed that effect of ageing (resting period) is more pronounced at higher levels of initial R. This is shown in Fig. 6.21 where a series of specimens at identical a'3c but different stress ratios of 1.0, 1.5, 2.0, and 2.8, were sheared under imposed strain increment ratio of -0.2 (or p value of 100.6 degrees). The rest period allowed at each stress ratio at the time of stress probes was almost 2.5 hours. It can be seen that as the stress ratio increases, the initial hardening which is aligned on the drained (do'r = 0) path deviates from the drained path at higher values of R (or larger AR from Fig. 6.20 Ageing and memory effect 186 (a\ + 2a'T)/3 (kPa) Fig. 6.21 Influence of initial stress ratio on ageing effect 187 the starting point indicated by solid circles) than those at lower stress ratios. As the results shown in Fig. 6.22 indicate, the initial hardening is independent of the imposed strain increment ratio if a resting period was allowed (the time which the resting period is allowed in this figure ranges from 20 minutes, triangle symbol, to 6 hours, star symbol). Although the range of imposed strain increment ratios is not wide and only covers imposed strain increment ratios in the range of-0.2 (P = 100.6°) and +0.1 (P = 84.1°), however by comparing the initial stress paths shown in Fig. 6.22 with those stress paths from similar-range of the imposed strain increment ratios (i.e. between -0.2 and +0.1) without any resting period (Fig. 6.5), this conclusion in which the initial hardening is independent of the imposed strain increment ratio seems to be logic. 6.5 Summary • At lower stress ratios (closer to hydrostatic axis), there is no unique association of strain increment direction with the stress direction. Instead, the strain increment direction is closely associated with the stress increment direction. For rj unloading paths, the strain increment directions are approximately coincident with the stress increment directions. This implies that at smaller stress ratios, significant part of the strains are recoverable. • At higher stress ratios (close to failure line), the strain increment direction is almost uniquely associated with stress direction only, and is essentially independent of the stress increment direction. This association is applicable for loading increments (R and q increasing). For unloading, the strain increment direction is still associated with the stress increment direction. In general, as the stress ratio increases towards the failure value, the association of strain increment direction with stress direction gets closer. In elasto-plastic constitutive modeling of soils, a unique direction of plastic strain increment has indeed been assumed at higher stress ratios. The results do tend to support this assumption, but only at higher stress ratios, closer to failure value. • As the stress ratio increases, the rotation of stress increment direction causes the strain increment direction to rotates more. This may imply that the use of a single plastic potential Fig. 6.22 Similar initial stiffness for a range of imposed strain increment ratios when the ageing effect dominates 189 function in elasto-plastic constitutive formulations is not reasonable. At low stress ratio levels, a, P relationship is essentially independent of the level of confining stress. This implies that the inclination of the strain increment vector to the line of constant R is constant, and thus the association of stress direction 0 and strain increment directions P is not affected by the level of confining stress. This is in agreement with the findings of Poorooshasb et al. (1966). At lower stress ratio levels, the previous stress/strain history does not appear to have a significant effect on the association of p to 0 and a. However, as the stress ratio increases, there does appear a small but discernible influence of the effect of previous history on a, p association especially for a ^ 180° (q unloading paths only). These findings on Fraser River sand are to some extent in agreement with those on clay (LeLeivre and Wang, 1970). Ageing at the end of isotropic or anisotropic consolidation to the desired c'r, R state prior to imposition of stress/strain probes in different directions shows that (at lower strain levels) the direction of stress increment vector is influenced n a major way for a given strain increment direction. At higher strain levels, however, this ageing effect tends to disappear, implying that the specimen which was aged at the hydrostaic axis and not at the point of o'n R state prior to probing, appears to have lost memory of its ageing under hydrostatic stresses by the time it reaches the target o'r, R state. In contrast the specimens which were aged at the target o'r, R state prior to probing have a strong memory of their ageing period. At higher stress ratio, the ageing effect is more pronounced. For the same ageing period at each starting stress ratio, the initial stiffness increases with the initial stress ratio. The initial hardening due to ageing seems to be independent of the imposed strain increment ratio. For a change of up to 20° in p, no significant change in a was observed. For similar change in P for specimens with no ageing, a 30° change in a was observed. CHAPTER 7 SUMMARY AND CONCLUSIONS A fundamental investigation of the susceptibility of saturated sands to strain softening (liquefaction) and instability under partially drained conditions has been carried out. The main objectives of the investigation focused on the influence of small volumetric deformations on the sand's potential for liquefaction and instability, when compared to that under the commonly assumed undrained mode. A triaxial testing device that enables loading under strain path rather than the stress path control was designed to induce controlled volumetric deformations in the soil, for simulating field loading conditions representative of different degrees of drainage. Conscious efforts were made to either minimize or properly account for the errors due to factors such as, end restraint, bedding errors and volume change due to membrane penetration effects, specific to the ambient density. The measurement resolutions of the sensing transducers and the data acquisition system were considerably enhanced, and the test procedures refined in order to obtain consistent and repeatable results with a high degree of confidence. Based on the results of a comprehensive series of undrained, conventional drained and partially drained tests on Fraser River sand, reconstituted by water pluviation, the following conclusions may be drawn: • The small deformation response of sand at a given state was dramatically influenced by the length of the time of ageing under the current state of stress. Consistency in laboratory test 190 191 data for valid comparisons can be achieved only by careful attention to the control in time of ageing under the ambient state of stress. The effect of ageing, however, tends to diminish as the strain level gets larger. Contrary to the assumption in the Rowe's stress dilatancy theory, which implies volumetric deformation rates, deydEj dependent only onR, a definite dependence of dev/dE1 on confining pressure, in addition, was found to exist in conventional drained shear. This dependence on the confining stress is central to relating the sand's potential for liquefaction and instability in partially drained shear, based on its volume change behavior in conventional drained and effective stress changes in undrained shear. At a given confining stress and R, the volumetric deformation rates are more contractive before and less dilative after the point of maximum contraction in triaxial extension than in compression, despite the fact that the mean normal effective stress decreases rather than increases. This characteristic is typical of water deposited saturated sands and hydraulic fill sands, and is embedded in their anisotropic response to loading. The zone of undrained contractive deformation in the effective stress space for the Fraser River sand was constrained between the two constant effective stress ratio lines - the line of peak states and the phase transformation line, and was bounded by a horizontal cutoff at a finite value of the shear stress as noted for other sands. Both, the location of the line of peaks as well as the horizontal cutoff depended on the mode of deformation, compression or extension. Partially drained shear under constant expansive defdej^ caused a depression in both the line of peaks and the horizontal cutoff, in both deformation modes. The more expansive the volumetric strain rates, the larger were the resulting depressions. The deformation response of sand under general nonlinear effective stress paths traversed under constant or nonlinear deydsj rates was intimately linked, in addition to the state of ambient effective stress, to the direction of the effective stress increment.. It was, however, relatively independent of the stress history experienced by the soil prior to arriving at the current stress state. 192 Criteria for the development of liquefaction (strain softening) and instability under partially drained states were established based on the behavior of sand under undrained and conventional fully drained shear. These criteria were closely linked to the history of c' r , R traversed by the stress state in the sand element under the dej^ history experienced for the specific partially drained scenario simulated. For initial states within the zone of undrained contractive deformations, the instability under partially drained state did not occur spontaneously as would be implied on undrained loading. It developed instead by an increase in pore pressure at essentially constant shear stress. Similarity and differences between strain softening, typical of liquefaction response, and instability in sand were established. If an increase in pore pressure occurs, strain softening and instability are essentially similar phenomena. It is, however, not true if the deformation occurs under fully drained case with no excess pore pressure generation. In such cases stress increments that satisfy the criteria for instability (Hill's) do not result in unstable behavior. The soil remains completely stable, if fully drained inside the line of maximum obliquity corresponding to its drained friction angle. The effect of ageing in making comparisons between instability and strain softening for identical states inside the region of undrained strain softening deformation revealed that if the ageing period is long the occurrence of instability would be delayed. Ageing allows stable behavior to persist even inside the region of undrained strain softening deformation, if the perturbation event is not very long lived. The drained creep was found to shift the point of instability closer to the effective stress failure line. The initial hardening is indeed equivalent to the delay in the occurrence of instability observed in constant shear stress tests. It was found that at lower stress ratios (closer to hydrostatic axis) and for loading paths, there is no unique association of strain increment direction (p) with the stress direction (0). Instead, the strain increment direction (P) is closely associated with the stress increment direction (a). At higher stress ratios (close to failure line), the strain increment direction p 193 is almost uniquely associated with stress direction 9 only, and is essentially independent of the stress increment direction a. In general, as the stress ratio increases towards the failure value, the association of strain increment direction P with stress direction 9 only gets closer. In elasto-plastic constitutive modeling of soils, a unique direction of plastic strain increment has indeed been assumed at higher stress ratios. The results do tend to support this assumption, but only at higher stress ratios, closer to failure value. For unloading paths, the strain increment directions are approximately coincident with the stress increment directions at low and high stress ratios implying that significant part of the strains may be recoverable. Also, regardless of the level of stress ratio, the confining pressure level does not seem to influence the a and P relationship for unloading paths. As the stress ratio increases, the rotation of stress increment direction causes the strain increment direction to rotates more. This may imply that the use of a single plastic potential function in elasto-plastic constitutive formulations seems not to be reasonable. At low stress ratio levels, a, P relationship is essentially independent of the level of confining stress. This implies that the inclination of the strain increment vector to the line of constant R is constant, and thus the association of stress direction 9 and strain increment directions P is not affected by the level of confining stress. This is in agreement with the findings of Poorooshasb et al. (1966). The previous stress/strain history showed no significant effect on the association of p, 9 and a. As the stress ratio increases, the previous stress/strain history showed more influence on a, P association (although still not that significant) . This is especially true for a > 180° implying the importance of previous history on strain softening initial states. These findings on Fraser River sand are to some extent in agreement with those on clay (LeLeivre and Wang, 1970). The specimens which were aged at the hydrostatic axis and not at the point of the target o'r, 194 R state prior to probing, appears to have lost memory of their ageing at hydrostatic axis by the time it reaches the target o'r, R state. At the same time, specimens which were aged at the target o'r, R state prior to probing have a strong memory of their ageing period. At higher stress ratios, the ageing effect is more pronounced. For the same ageing period at each starting stress ratio, the initial stiffness increases with the initial stress ratio. The initial hardening due to ageing seems to be independent of the imposed strain increment ratio. For a change of 20° in p, no significant change in a was observed. For similar change in p for specimens with no ageing, a 30° change in a was observed. 7.1 Recommendations for further Studies The studies presented in this thesis are largely limited to axisymmetric triaxial testing system. It would be interesting to perform multi-axial strain path tests to investigate the strain softening and instability behavior of granular soils. In this case the control strain increment ratio could be in the form of deyde,. The unstable behavior of granular soils needs to be studied under three-dimensional loading conditions. It has been established that pre-failure strain softening is similar to pre-failure instability. This relationship between instability and strain softening under axisymmetric conditions may be generalized into three-dimensional loading conditions. The current formulated plasticity theories have been based on the flow rule which still relate the plastic strain increment to the stress or strain state only. The plastic flow of granular soil is generally influenced by both the stress state as well as the stress increment direction. Therefore, it is interesting to separate elastic and plastic deformations in the laboratory and confirm the incremental non-linear elasto-plastic modeling. REFERENCES Anandrajah, A., Sobhan, K. and Kuganenthris, N. (1995). " Incremental stress-strain behavior of granular soil". ASCE Journal of Geotechnical Engineering, 121 (1), 57-68. Andersen, A. and Bjerrum, L. (1968). "Slides in subagueous slopes in loose sand and silt". Norwegian Geotechnical Institute Publication 81. Atkinson, J.H. and Evans, J.S. (1985). "Discussion on: The measurement of soil stiffness in the triaxial apparatus by Jardine, R.J., Symes, N.J. and Burland, J.B.". Geotechnique, 35 (3), 378-382. Baldi, G. and Nova, R. (1984). "Membrane penetration effects in triaxial testing". ASCE Journal of Geotechnical Engineering, 110 (3), 403-419. Barden, L. and Khayatt, A.J. (1966). "Incremental strain rate ratios and strength of sand in the triaxial test". Geotechnique, 16 (4), 338-357. Biot, M. A. (1941). "General theory of three-dimensional consolidation". J. Appld. Physics, 12,155-164 Bishop, A.W. (1971). "Shear strength parameters for undisturbed and remolded soil specimens". Roscoe Memorial Symposium, Cambridge University, pp. 3-58. Bishop, A.W. and Green, G.E. (1965). "The influence of end restraint on the compression strength of a cohesionless soil". Geotechnique, 15 (3), 243-266. Bishop, A.W. and Hill, R. (1951). "A theory of the plastic distortion of a polycrystalline aggregate under combined stresses". The Philosophical Magazine, 42 (327), 414-427. Bjerrum, L. (1971)."Subaqueous slope failures in Norwegian fiord". NGI Publication 88. Bjerrum, L., Krindgstad, S., and Kummeneje, O. (1961). "The shear strength of a fine sand". NGI Publication 45. 195 196 Campanella, R.G. and Vaid, Y.P. (1972). "A Simple K0-triaxial cell ". Canadian Geotechnical Journal, 9 (3), 249-260. Casagrande, A.(1975). "Liquefaction and cyclic deformation of sands: A critical review". Proceedings of 5th Pan American Conference on Soil Mechanics and Foundation Engineering, Buenos Aires, Vol. 5, pp. 79-133. Castro, G. (1969). "Liquefaction of sands". Ph.D. thesis, Harvard University, Cambridge, MA. Castro, G. (1975). "Liquefaction and cyclic mobility of saturated sands". ASCE Journal of Geotechnical Engineering Division, 101 (6), 551-569. Chan, C.K. (1975). "Low-friction seal system". ASCE Journal ofGeotechnical Engineering Division, 101 (9), 991-995. Chern, J.C. (1985). "Undrained response of saturated sands with emphasis on liquefaction and cyclic mobility". Ph.D. thesis, The University of British Columbia, Vancouver, Canada. Chu, J. (1991). "Strain softening behavior of granular soils under strain path testsing". Ph.D. Thesis, University of New South Wales, Australia. Chu, J., Lo, S-C.R. and Lee, I.K. (1992). "Strain softening behavior of granular soil in strain path testing". ASCE Journal Geotechnical Engineering, 118 (2), 191-208. Chu, J., Lo, S-C.R. and Lee, I.K. (1993). "Instability of granular soils under strain path testing". ASCE Journal Geotechnical Engineering, 119 (5), 874-892. Dobry, R., Vasquez-Herrera, A., Mohamad, R. and Vucetic, M. (1985). "Liquefaction flow failure of silty sand by torsional cyclic tests". Proceedings, Advances in the Art of Testing Soils under Cyclic Conditions, V. Khosla (ed.), ASCE Convention, Detroit, MI, pp. 29-50. El-Sohby, M.A. (1964). "The behavior of particulate materials under stress". Ph.D. Thesis, University of Manchester, U.K. Frydman, S.,Zeitlen, J.G., and Alpan, I. (1973). "The membrane effectsin triaxial testing of granular soils". Journal of Testing and Evaluation, 1 (1), 37-41. Garrison, R.E., Luternuer, J.L., Gill, E.V., MacDonal, R.D. and Murray, J.W. (1969). "Early diagenetic cementation of recent sands, Fraser River Delta, British Columbia". Seimentology, 12, 27-46. Henkel, D.J. (1960). "The shear strength of saturated remolded clay". ASCE Research Conference on Shear Strength of Cohesive Soils, Boulder, CO, pp. 533-554. Hettler, A. and Vardoulakis, I. (1984). "Behavior of dry sand tested in a large triaxial apparatus". 197 Geotechnique, 34 (2), 183-197. Hill, R. (1950). The Mathematical Theory of Plasticity. Oxford University Press, New York, N. Y. Hill, R. (1959). "Some basic principles in the mechanics of solids without a natural time". J. Mech. Phys. Solids, 7, p209. Home, M.R. (1965). "The behavior of an assembly of rotund, rigid, cohesionless, particles". Proceedings, Royal Society, London, Series A, Parti, 286, 62-78; Part II, 286, 79-97; Part III, 310, 21-34. Ishihara, K., Tatsuoka, F., and Yasuda, S. (1975). "Undrained deformation and liquefaction of sand under cyclic stresses". Soils and Foundations, 15 (1), 29-44. Jardine, R.J., Symes, M.J. and Burland, J.B. (1984). "The measurement of soil stiffness in triaxial apparatus". Geotechnique, 30 (3), 323-340. Kramer, S.L. (1989). "Uncertainty in steady-state liquefaction evaluation procedures". ASCE Journal of Geotechnical Engineering, 115 (10), 1402-1419. Kramer, S.L. and Seed, H.B. (1988). "Initiation of soil liquefaction under static loading conditions". ASCE Journal of Geotechnical Engineering, 114 (4), 412-430. Kuerbis, R.H. and Vaid, Y.P. (1988). "Sand sample preparation-the slurry deposition method". Soils and Foundations, 28 (4), 107-118. Kuerbis, R.H. and Vaid, Y.P. (1990). "Corrections for membrane strength in the triaxial test". Geotechnical Testing Journal, 13 (4), 361-39. Lade, P.V. (1982). "Localization effects in tiraxial tests on sand". Proceedings, IUTAM Symposium on Deformation and Failure of Ganular Materials, P.A. Vermeer and H.J. Luger (eds.), Balkema, Rotterdam, pp. 461-471. Lade, P.V. (1989). "Experimental observations of stability, instability, and shear planes in granular materials". Ingenenieur-archiv, Springer-Verlag, 59 (2), 114-123. Lade, P.V. (1992). "Static instability and liquefaction of loose fine sandy slopes". ASCE Journal of Goetechincal Engineering, 118 (1), 51-57. Lade, P.V. and Pradel, D. (1990). "Instability and plastic flow of soils, I: Experimental observations". ASCE Journal of Engineering Mechanics, 116 (11), 2532-2550. Lade, P.V., Bopp, P.A., and Peters, J.F. (1993). "Instability of dilating sand". Mechanics of Materials, 16, 249-264. 198 Lade, P.V., Nelson, R.B., and Ito, Y.M. (1987). "Nonassociated flow and stability of granular materials". ASCE Journal of Engineering Mechanics, 113 (9), 1302-1318. Lade, P. V., Nelson, R.B., and Ito, Y.M. (1988). "Instability of granular materials with nonassociated flow". ASCE Journal of Engineering Mechanics, 114 (12), 2173-2191. Le Leivre, B. and Wang, B. (1970). "Discussion on stress-probe experiments on saturated normally consolidated clay". Geotechnique, 20 (3), 461-463. Lewin, P.I. and Burland, J.B. (1970). "Stress-probe experiments on saturated normally consolidated clays". Geotechnique, 20 (1), 38-56. Liu, H. and Qiao, T. (1984). "Liquefaction potential of saturated sand deposits underlying foundation of structure". Proceedings, 8th World Conference Earthquake Engineering, Vol. Ill, San Francisco, CA, Vol. 3, pp. 199-206. Lo, S-C. R. (1996). Private Communication. Loung, M.P. (1980)."Stress-strain aspects of cohesionless soils under cyclic and transient loading". Proceedings, International Symposium on Soils under Cyclic and Transient Loading, Swansea, U.K., January 1980. Mejia, C , Vaid, Y.P., and Negussey, D. (1986)."Time dependent behavior of sand". Proc. of 1st Int. Conf. on Rheology and Soil Mechanics, Coventry, U.K., Keedwell (ed.), pp. 312-326. Menzies, B.K. (1987). "A computer controlled hydraulic testing system". Proceedings, Symposium on Advanced Triaxial Testing, ASTM STP 977, Philadelphia, PA, pp. 82-94. Negussey, D. (1984). "An experimental study of the small strain respone of sand". Ph.D. Thesis, The University of British Columbia, Vancouver, Canada. Negussey, D. and Vaid, Y.P. (1990). "Stress dilatancy of sand at small stress ratio states". Soils and Foundations, 30 (1), 155-166. Negussey, D., Wijewickreme, W.K.D., and Vaid, Y.P. (1988). "Constant-volume friction angle of granular materials". Canadian Geotechnical Journal, 25 (1), 50-55. Newland, P.L. and Allely, B.H. (1959)."Volume changes during undrained triaxial tests on saturated dilatent granular materials". Geotechnique, 9 (4), 174-182. Nicholson, P.G., Seed, R.B. and Anwar, H.A. (1993a). "Elimination of membrane compliance in undrained triaxial testing - measurement and evaluation". Canadian Geotechnical Journal, 30 (5), 727-738. 199 Nicholson, P.G., Seed, R.B. and Anwar, H.A. (1993b). "Elimination of membrane compliance in -mitigation by injection compensation". Canadian Geotechnical Journal, 30 (5), 739-746. Nova, R. and Wood, D.M. (1978). "An experimental program to define the yield function for sand". Soils and Foundations, 18 (4), 77-85. Oda, M. (1972). "Initial fabrics and their relations to mechanical properties of granular material". Soils and Foundations, 12 (1), 17-36. Poorooshasb, H.B., Holubec, I., and Sherbourne, A.N. (1966). "Yielding and flow of sand in triaxial compression: Part I". Canadian Geotechnical Journal, 3 (4), 179-190. Poulos, S.J. (1981). "The steady state of deformation". ASCE Journal of Geotechnical Engineering, 107 (5), 553-562. Raju, V.S. and Sadasivian, S.K. (1974)."Membrane penetration in triaxial tests on sand". ASCE Journal of the Geotechnical Engineering Division, 100 (4), 482-489. Raju, V.S. and Venkataramana, K. (1980). "Undrained triaxial tests to assess liquefaction potential of saturated sands-effect of membrane penetration". Proceedings, International Symposium Soils Under Cyclic and Transient Loading, G.N. Pand and O.C. Zienkiewics, (eds.), Vol. 2, pp. 483-494. Ramana, K.V. and Raju, V.S. (1981). "Constant-volume triaxial tests to study the effects of membrane penetration". Geotechnical Testing Journal, 4 (3), 117-122. Roscoe, K.H., Schofield, A.N., and Thurairajah, A. (1963). "An evaluation of test data for selecting a yield criterion for soils". Laboratory Shear Testing of Soils, ASTM STP 361, American Society for Testing and Materials, West Conshohocken, PA, pp. 111-128. Rowe, P.W. (1962). "The stress-dilatancy relation for static equilibrium of an assembly of particles in contact". Proceedings of the Royal Society of London, Series A, 269, pp. 500-527. Rowe, P.W. (1971). "Theoretical meaning and observed values of deformation parameters for soil". Proceedings of the Roscoe Memorial Symposium on Stress-Strain Behavior of Soils, Cambridge University, pp. 143-196. Rowe, P.W., and Barden, L. (1964). "The importance of free ends in the triaxial test". ASCE Journal of the Soil Mechanics and Foundations Division, 90 (1), 1-27. Sarsby, R.W., Kalteziotis, N., and Haddad, E.H. (1980). "Bedding error in triaxial tests on granular meida". Geotechnique, 30 (3), 302-309. Sarsby, R.W., Kalteziotis, N., and Haddad, E.H. (1982). "Comparison of'free-ends' during triaxial 200 testing". ASCE Journal of Geotechnical Engineering, 108 (1), 83-107. Sayao, A.S.F. (1989). "Behavior of sands under general stress paths in the hollow cylinder torsional device". Ph.D. Thesis, University of British Columbia, Vancouver, Canada. Schofield, A.N. and Wroth, P.C. (1968). Critical State Soil Mechanics. McGraw Hill, London, U.K. Seed, H.B. (1979). "Soil liquefaction and cyclic mobility evaluation for level ground during earthquakes". ASCE Journal of Geotechnical Engineering, 105 (2), 201-255. Seed, H.B. (1987). "Design problems in soils liquefaction". ASCE Journal of Geotechnical Engineering Division, 113 (8), 827-845. Seed, H.B., Singh, S., Chan, C.K. and Vilela, T.F. (1982). "Consideration in undisturbed sampling of sands". ASCE Journal of Geotechnical Engineering, 108 (2), 265-283. Seed, R.B., Anwar, H.A., and Nicholson, P.G. (1989)."Elimination of membrane compliance effects in undrained testing". Proc, 12th Int. Conf. SMFE, Vol. 1, A. A. Balkema, Rotterdam, The Netherlands, pp. 111-114 Sivathayalan, S. and Vaid, Y.P. (1998). "Truly undrained response of granular soils with no membrane penetration effects", Canadian Geotechnical Journal, 35 (5), 730-739. Sladen, J.A. and Handford, G. (1987). "A potential systematic error in laboratory testing of very loose sands". Canadian Geotechnical Journal, 24 (3), 462-466. Sladen, J. A., D'Hollander, R.D., and Krahn, J.(1985). "The liquefaction of sands, a collapse surface approach". Canadian Geotechnical Journal, 22 (4), 564-578. Tatsuoka, F. (1981). "A simple method for automatic measurement of volume change in laboratory test". Soils and Foundations, 21 (3), 104-106. Tatsuoka, F. and Ishihara, K. (1974). "Yielding of sand in triaxial compression". Soils and Foundations, 14 (3), 63-76. Tatsuoka, F.,Molenkamp, F., Torii, T. and Hino, T. (1984). "Behavior of lubrication layers of platens in element tests". Soils and Foundation, 24 (1), 13-128. Tatsuoka,F., SantuccideMagistris,F.,Hayano,K.,Momoya, Y. andKoseki, J. (1998). "Somenew aspects of time effects on the stress-strain behavior of stiff geomaterials", Keynote Lecture for 2nd International Conference on Hard Soils Soft Rocks, Evanelista and Picarelli (eds.), Napoli, Balkema, Vol. 2. Taylor, D.W. (1948). Fundamental of Soil Mechanics. John Wiley & Sons. 201 Terzaghi, K. (1956). "Varieties of submarine slope failures". Proceedings, 8th Texas Conference on Soil Mechanics and Foundation Engineering, University of Texas, Austing, TX, p41. Thomas, J. (1992). "Static, Cyclic and post liquefaction undrained behavior of Fraser River sand". M.A.Sc. Thesis, University of British Columbia, Vancouver, Canada. Ueng, T-S., Tzou, Y-M. and Lee, C-J. (1988). "The effect of end restraint on volume change and particle breakage of sands in triaxial tests". Advanced Triaxial Testing of Soil and Rock, ASTM STP 977, R.T. Donaghe, R.C. Chaney and M.L. Silver (eds.), ASTM, Philadelphia, 1988, pp. 679-691. Vaid, Y.P. (1983). "Discussion on cyclic undrained stress-strain behavior of dense sands by torsional simple shear test". Soil and Foundations, 23 (2), 172-173. Vaid, Y.P. and Campanella, R.G. (1974). "Triaxial and plane strain behavior of natural clay". ASCE Journal of the Geotechnical Engineering, 100 (3), 207-224. Vaid, Y.P. and Chern, J.C. (1983). "Mechanism of deformation during undrained loading of saturated sands". International Journal Soil Dynamics and Earthquake Engineering, 2 (3), 171-177. Vaid, Y.P. and Chern, J.C. (1985). "Cyclic and monotonic undrained response of saturated sands" Proceedings, Advances in the Art of Testing Soils under Cyclic Conditions, V. Khosla (ed.), ASCE Convention, Detroit, MI, pp. 120-147. Vaid, Y.P. and Eliadorani, A. (1998). "Instability and liquefaction of granular soils under undrained and partially drained states". Canadian Geotechnical Journal, 35 (6), 1053-1062. Vaid, Y.P., and Negussey, D. (1984). "A critical assessment of membrane penetration in the triaxial test". Canadian Geotechnical Journal, 20 (4), 827-832. Vaid, Y.P., and Negussey, D. (1988). "Preparation of reconstituted sand specimens". In Advanced Triaxial Testing of Soils and Rock. American Society of Testing and Materials, Special Technical Publication No. 977, pp. 119-131. Vaid, Y.P. and Sivathayalan, S. (1996). "Errors in estimates of void ratio of laboratory sand specimens". Canadian Geotechnical Journal, 33 (6), 1017-1020. Vaid, Y.P. and Thomas, J. (1995). "Liquefaction and post-liquefaction behavior of sand". ASCE Journal of Geotechnical Engineering, 121 (2), 163-173. Vaid, Y.P., Chung, E.K.F., and Kuerbis, R.H. (1989). "Preshearing and undrained response of sand". Soils and Foundations, 29 (4), 49-61. 202 Vaid, Y.P., Chung, E.K.F., and Kuerbis, R.H. (1990). "Stress path and steady state". Canadian Geotechnical Journal, 2 7 (1), 1-7. Vaid, Y.P., Eliadorani, A., Sivathayalan, S., and Uthayakumar, M. (1999). "Laboratory characterization of stress-strain behavior of soils by stress/strain path loading". ASTM, Geotechnical Testing Journal, under review. Vaid, Y.P., Sivathayalan, S., Eliadorani, A., and Uthayakumar, M. (1996). "Sand characterizing for static and dynamic liquefaction", Report: CANLEX Laboratory Testing at UBC. Valanis, K.C. (1985). "On the uniqueness of solution of the initial value problem in strain softening materials". Journal Applied Mechanics, 52 , 649-653. Vardoulakis, I.G. and Graf, B. (1985). "Calibration of constitutive models for granular materials using data from biaxial experiments". Geotechnique, 3 5 (3), 299-317. Whitman, R.V. (1985). "On liquefaction". Proceedings, 11th International Conference on Soil Mechanics and Foundation Engineering, Vol. 4, pp. 1923-1926. Yoshimi, Y., Tokimatsu, K., Kaneko, O. andMakihara, Y. (1984). "Undrained cyclic shear strength of a dense Niigata sand". Soils and Foundations, 2 4 (4), 131-145. Yoshimi, Y., Tokimatsu, K. andHosaka, Y. (1989). "Evaluation of liquefaction resistance of clean sands based on high quality undisturbed samples". Soils and Foundations, 2 9 (1), 93-104. APPENDIX A MEMBRANE PENETRATION CORRECTION CURVES As discussed in Chapter 3, a rubber membrane covers the surface of the specimen. During an undrained test, partially drained condition, or drained test under effective confining stress changes, a change in effective confining pressure causes the membrane to change its original position (either indentation or rebound). Thus, both the volume change and pore water pressure measurements can be affected by membrane penetration. For example, an undrained test conducted by closing the drainage line cannot really obtain a constant volume situation due to the membrane penetration. Such a deviation from the undrained state can make the conventional undrained test for assessing the liquefaction potential of sands highly erroneous and on the unsafe side (Raju and Venkataramana, 1980; Nicholson et al., 1993b). The control quality of a partially drained condition will also be affected if the volume change cannot be measured accurately. As the effective confining stress applied to a sample changes, the total sample volume contained within the sample membrane is changed. The change in sample volume (AVX) is actually the sum of two volume-change components. The total measured volume change can then be taken as the sum of these components as: A ^ = e * A + e v , A (A-l),(3.1)viz the first component is due to the variation in the amount of membrane penetration and second component is the true or skeleton volume change of the soil sample. In Eq. B-1, the em is the compliance-induced volume change per unit area of the membrane (AJ, e^ , is the true sample volumetric strain and V T is the overall sample volume. The different methods developed to evaluate 203 204 the magnitude of volumetric membrane compliance to differentiate between the two volume change component (em and e^ ). Newland and Allely (1957) first suggested that volume changes caused by membrane penetration could be determined as the difference between the total volumetric strain and three times the axial strain assuming that the soil behaved isotropically under isotropic loading. Several investigators questioned the validity of this assumption since soil does not necessarily behave isotropically under isotropic loading (Roscoe et al., 1963; Vaid and Negussey, 1984). Vaid and Negussey (1984) examined the fundamental assumptions involved in the assessment of sample volume changes as a result of membrane compliance in triaxial tests on granular soils. They concluded that the necessary assumptions render invalid methods that (i) use dummy rod inclusions (Roscoe et al., 1963; Raju and Sadasivian, 1974) or (ii) assume isotropic sample behavior. Frydman et al. (1973) by using hollow cylindrical samples and varying the sample volume and surface area (and the ratio between them) plotted the total volumetric strain against the ratio between membrane surface area and initial sample volume (AJVT). From the plot, the slope was equal to the unit volumetric membrane penetration (e^ which is defined as the volumetric membrane compliance per unit membrane area. By plotting em versus log of effective confining pressure, a linear relationship with slope m is obtained. The slope of the relationship is referred to as the normalized penetration (m). This method yields more reliable estimates of volumetric compliance than previous techniques. However, hollow samples are extremely difficult to prepare and duplicate with the precise sample density required. Thus, simpler technique is desirable. In this study, the membrane penetration corrections were applied in accordance with the method proposed by Vaid and Negussey (1984). Significant advantage of this method is that it does not require specimens of different sizes. In this method, the membrane penetration correction are determined by conducting an isotropic consolidation load and unload test on a single soil sample. The deviation of triaxial test soil specimen unloading strain from isotropy with respect to axial strain is shown to be due to membrane penetration effects in the radial direction of loading. Therefore, during isotropic unloading, S =3 8 vu au 205 (A-2) which by substituting it in Eq. (B-l) yields to: E V = e m A J V T + 3eau (A-3) by letting Vj/A,,, equal D/4 where D is the diameter of the sample, the unit membrane penetration can be determined as: It is now well established that the relationship between unit membrane penetration (membrane-induced volume change) and the effective confining stress becomes essentially linear when plotted on a semi-logarithmic scale. The slope between unit membrane penetration and log o'3 will hereafter be referred to as the normalized penetration (m). m will thus be formally defined as em per log cycle change in o'3: m = eJlog(a'e/o£ (A-5) by substituting in Eq. (B-l), the true skeleton volume change can be computed as: eVj, = eVr-4mlog(oi/oi)/D<, (A-6) A - l Factors Influencing Membrane Penetration in more Detail The membrane compliance may have a significant influence on pore-pressure accumulation depending on factors such as effective confining pressure, mean grain size, relative density, and membrane thickness (Ramana and Raju, 1982; Baldi and Nova, 1984; Nicholson et al., 1993a). Despite the considerable studies devoted to this subject, there exists little quantitative knowledge of the effect of relative density on the behavior of granular soils. Interestingly, Frydman et al. (1973) concluded that the relative density of the soil is not an important factor and found little difference 206 between membrane penetration on uniform glass microspheres having relative densities of 30% and 70%. The effect of relative density can be best studied based on the normalized unit membrane penetration (m). An effort was made to investigate this aspect in more detail, mainly, for evaluation of the test results presented in the this thesis. However, to some extent the effect of mean grain size and grain shape are also presented. The influence of membrane thickness was not considered since most of the commercially available membranes used are approximately of the same thickness. All the previous investigators observed the importance of grain size on membrane penetration. The grain size distribution for the three sands; namely, Fraser River sand, Ottawa sand, and Silica sand were fairly uniform, as shown in Fig. A-1. The results obtained in terms of unit membrane penetration versus logarithm of effective confining pressure, for Fraser River sand with initial relative density (at 20 kPa) of D,,. = 21% (loose), 32% (medium-loose), 55% (medium-dense), and 82% (dense) were given in Fig. 3.6(a). The increase in relative density causes a lower normalized unit membrane penetration (m; i.e. slope of em vs. log o'3). Similar results were also observed for Ottawa sand with the difference that after a relative density of 63%, the "m" value stays the same for a density of 83% (Fig. A-2a). At the same time, as Fraser River sand, for Silica sand (Fig. A-2b) the value of "m" decreases with relative density. The summary of the results for these three sands in terms of "m" value and relative density was given in Fig. 3.6(b). From above results, the following observations can be made: • As expected, the increase in grain size leads to an increase in membrane penetration (comparison between Fraser River and Silica sand; D 5 0 of 0.9 mm compared to 0.3 mm). By an increase in the grain size, the effect of membrane correction is less significant for the loosest deposited specimens than the denser specimens; although the loosest specimens still need more correction to be applied. Another words, the effect of mean grain size for specimens with higher relative densities becomes more pronounced. For loosest specimens, the ratio of "m" value between coarsest Silica sand and finest Fraser River sand is 3.5, but 207 208 10 100 1000 c ' r ( k P a ) Fig. A-2 Membrane penetration correction curves (a) Ottawa sand (b) Silica sand 209 for the densest this ratio becomes 7.3. For the rounded Ottawa sand, there seems to be a threshold relative density in which the normalized unit membrane penetration does not decrease with density. For rounded Ottawa sand which is likely to be less brittle (or softer) against load and no crushability of particles at the edges, the normalized membrane penetration does not decrease with relative density at a relative density of approximately 63%. Beyond this density, the same "m" value is obtained. However, in the case of friable Fraser River sand and the Silica sand with much higher brittleness, due to crushability of particles at the edges more packing can be achieved and therefore the "m" values decrease with relative density. Comparison of the results for Fraser River sand and Ottawa sand shows the importance of particle shape. The D 5 0 of these two sands are close enough that (0.4 mm for Ottawa sand compared ot 0.35 mm for Fraser River sand) the effect of particle shape could be observed. Two differences may be noticed; first, the rounded Ottawa sand has higher normalized unit membrane penetration (eg 0.005 for loosest Ottawa sand compared to 0.0033 for Fraser River sand or 52% higher value for Ottawa sand). Secondly, the effect of relative density is more pronounced for Fraser River sand than the Ottawa sand. For Fraser River sand, for almost the same range difference in density, the "m" value at loosest state is 2.2 times bigger than the value for densest state, but for Ottawa sand, the "m" value at loosest state is 0.7 times bigger than that for densest state. Previous studies have shown that the effect of relative density is less significant compared to the mean grain size (Frydman et al., 1973; Nicholson et al., 1993a). The results obtained for coarser Silica sand, with mean grain size D 5 0 = 0.9 mm , and finer Ottawa and Fraser River sands, with mean grain size D 5 0 = 0.35 mm, indicate the importance of relative density as shown in Figs. A-2 and 3.6(b). The effect of relative density appears to play a significant role in membrane penetration. It cannot be concluded that for finer sands the effect of relative density may be regarded as less important in comparison with coarser sands. Comparison of "m" values for two rounded samples of Ottawa and Silica sands may show 210 that for finer Ottawa sand, the effect of relative density is not less significant. Indeed, the opposite may be seen here; for the difference in "m" values of the loosest Ottawa specimen is 0.74 times bigger than that for densest specimen while this difference is 0.55 times for coarser Silica sand. Therefore, due to the relative importance of relative density, its effect should be taken into account for a more accurate evaluation. It has been suggested that the maximum adjustment to "m" values that would have to be made for extreme cases of high- or low- density samples should be no more than 10% (Seed et al, 1989). However, the results presented in this section do not confirm these conclusions. Taking into consideration the variability of grain size and grain size distribution of granular soils in nature, it appears unrealistic, if not impossible, to try to develop a procedure that could model membrane penetration based only on mean diameter D J 0 or any other single parameter. As shown in Fig. 3.6(b), the variation of unit membrane penetration does not vary linearly with the relative density for all sands. However, this could be the case for coarser Silica sand. Therefore, in performing the tests at different densities, the interpolation of the "m" values from two other densities may not be correct. APPENDIX B DEPENDENCE OF d^/d^ ON o' r AND R FOR OTTAWA SAND 0 200 400 a ' r c (kPa) Fig. B-l Relationship between dejde^, R, and da'r=0 during da'r=0 shear (loose Ottawa sand) 211 APPENDIX C DRAINED aVr=0 EXTENSION RESPONSE OF FRASER RIVER SAND Fig. C-l Drained do'r=0 extension response of loose Fraser River sand 212 213 Fig. C-2 Drained dc'=0 extension response of Fraser River sand at fixed confining pressure (Effect of relative density) APPENDIX D PARTIALLY DRAINED TEST RESULTS AT <s'„ = 50,100, 400 kPa Fig. D-1 Partially drained response at R,. = 1.0 ; o'rc = 400 kPa 214 g. D-2 Partially drained response at R,. = 1.5 ; o'rc = 400 kPa 216 Fig. D-3 Partially drained response at IL. = 2.1 ; o'rc = 400 kPa Fig. D-4 Partially drained response at R,. = 2.8 ; o'rc = 400 kPa 218 0 2 4 6 0 200 400 ( a ' a + o ' r ) / 2 (kPa) Fig. D-5 Partially drained response at R,. = 1.0 ; a'^ = 100 kPa 219 (a , a + a , r ) / 2 (kPa) Fig. D-6 Partially drained response atRc=1.5;a'rc = 50 kPa 220 

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