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An experimental study of the small strain response of sand Negussey, Dawit 1984

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AN EXPERIMENTAL STUDY OF THE SMALL STRAIN RESPONSE OF SAND by DAWIT NEGUSSEY B.Sc, Southern I l l i n o i s University, 1971 M.Sc.C.E., University of Minnesota, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF-THE REQUIREMENTS 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 December 1984 © Dawit Negussey, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for s c h o l a r l y purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or pub l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering  The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date December 28, 1984 ABSTRACT Fundamental behaviour of Ottawa sand, i n the s t r a i n range of 1 x 10~ 2 to 1 x 10 - 5, i s investigated by dir e c t measurement of deformations i n a load controlled conventional t r i a x i a l system. Experiments are aimed at examining common concepts and previous experimental j u s t i f i c a t i o n s for incremental e l a s t i c , e l a s t o - p l a s t i c and particulate frameworks for characterizing sand behaviour. From fundamental in t e r p r e t a t i o n of test data, an alt e r n a t i v e s t r e s s - s t r a i n relationship i s proposed for proportional loading with r e l a t i v e density represented as a separate parameter. Maximum Young's moduli evaluated from resonant column tests are found to be approximately equal to i n i t i a l unloading moduli from conventional t r i a x i a l tests and i n i t i a l moduli from v i r g i n loading and subsequent reloadings are much l e s s . I n i t i a l unloading moduli are r e l a t i v e l y unaffected by the cycle of loading and deviator stress l e v e l from which unloading i s i n i t i a t e d . The value of Young's modulus at a stress state Is not unique but depends on the stress path and s t r a i n h i s t o r y . Nonrecovered s t r a i n d i r e c t i o n s , at small s t r a i n , depend on stress d i r e c t i o n as opposed to the generally accepted dependence on stress state at large s t r a i n . Proportional loading paths are uniquely related to li n e a r s t r a i n increment directions and maintain p a r a l l e l mean normal stress equipotentials i n s t r a i n space. Energy density increments i n two proportional loading paths having i d e n t i c a l mean normal stress h i s t o r i e s remain proportional. P a r a l l e l nonproportional loadings result i n a unique s t r a i n increment d i r e c t i o n , r e l a t i v e l y i i i independent of hydrostatic stress l e v e l , with l i n e a r stress r a t i o equipotentials i n s t r a i n space. In small s t r a i n response, shear str a i n s r e s u l t mainly from shear stress increments and not from changes i n stress r a t i o . Shear volume response i s contractant f o r both an increasing and decreasing shear stress increment, whereas the sense of shear s t r a i n increment alternates with the sense of shear stress increment. When the sense of s t r a i n state i s opposite to the sense of an applied stress increment, the r e s u l t i n g s t r e s s - s t r a i n response i s s o f t e r than when both are of the same sense. Strain paths for compression side shear loading are i d e n t i c a l to paths of extension side shear unloading and vice versa. More shear and volumetric strains develop on extension side shear loadings and the r a t i o of volumetric to shear s t r a i n i s also higher as opposed to comparable compression side shear loadings. At higher stress r a t i o states, decreasing mean normal stress at constant shear and increasing stress r a t i o conditions; extension side volumetric s t r a i n responses are associated with contraction following i n i t i a l swelling and prior to d i l a t i o n . This contraction phase i s not present on the compression side. Y.P. Vaid Thesis Supervisor iv TABLE OF CONTENTS Page ABSTRACT i i LIST OF FIGURES v i i i INDEX OF NOTATIONS xv ACKNOWLEDGEMENTS x v i i I - INTRODUCTION 1 II - STRESS-STRAIN RESPONSE OF SAND 5 2.1 Observed Behaviour 5 2.2 Idealization of Behaviour at Either Elastic or Plastic 8 2.3 Particulate Considerations 10 2.4 Decomposition of Strain 13 2.5 Trends in the Study of Sand Behaviour 16 III - EXPERIMENTAL CONSIDERATIONS 19 3.1 Test Equipment 19 3.1.1 Limitations and Improvement of the Triaxial Test 20 3.1.1.1 Ram Friction 22 3.1.1.2 Membrane Penetration 22 3.1.1.3 End Restraint 23 3.1.1.4 .Bedding Error 26 3.1.2 Stress Path Control 28 3.1.2.1 Description of the System 29 3.1.2.2 Underlying Principles 32 3.1.3 Measurement Devices 34 3.1.3.1 Load and Deformation Monitoring .... 34 V 3.1.3.2 Standard References and Data Acquisition 35 3.2 Material Tested and Sample Size 37 3.2.1 Material Tested 37 3.2.2 Sample Size 37 3.3 Testing Procedure 39 3.3.1 Sample Preparation 39 3.3.1.1 Adopted Method 41 3.3.2 Sample Set-Up 43 3.3.3 Repeatability of Test Results 44 IV - EXAMINATION OF PREVAILING FRAMEWORKS FOR DESCRIBING SMALL STRAIN RESPONSE 50 4.1 Incremental Elastic Representation 50 4.1.1 Moduli in Uniaxial Loading 51 4.1.1.1 I n i t i a l Moduli, E± 51 4.1.1.2 Relationships Between E. and E ... 54 r i max 4.1.1.3 Comparison of E m a x , E^, E^ u and E ^ . . 67 4.1.2 The Influence of Stress Paths 71 4.1.2.1 Stress-Strain Relations in Different Paths 72 4.1.2.2 The Influence of Stress Paths on Incremental Moduli 72 4.1.3 Additional Remarks 83 4.2 Particulate Considerations 85 4.2.1 Stress Dllatancy in Conventional Triaxial Paths 86 v i 4.2.1.1 The Influence of Stress Level 86 4.2.1.2 The Influence of Density 90 4.2.2 Stress Dilatancy in Different Stress Paths ... 95 4.2.3 Further Remarks 101 4.3 Strain Separation for Elasto Plastic Representation.. 103 4.4 Concepts Based on Normalized Work '109 4.5 Concluding Remarks 115 V - BEHAVIOUR OF SAND IN PROPORTIONAL LOADING 117 5.1 Behaviour at One Relative Density 118 5.1.1 Experimental Observations 118 5.1.2 Proportional Stress-Strain Relationships 123 5.2 Extensions to Other Relative Densities 129 5.2.1 Experimental Observations 131 5.2.2 Proportional Stress-Strain Relationships 143 5.2.3 Required Parameters 149 VI - NON-PROPORTIONAL STRESS PATHS 150 6.1 Total Loading Paths 150 6.1.1 Parallel Loading Paths 151 6.1.1.1 Effect of Consolidation Stress on Strain Paths 151 . 6.1.1.2 Effect of Density on Strain Paths ... 158 6.1.1.3 Stress Ratio Equipotentials 163 6.1.1.4 Quantitative Relationships Between Parallel Paths 173 6.1.2 Quantitative Relationships Between Stress and Strain Directions 186 v i i 6.2 Other Non-Proportional Paths 192 6.2.1 Constant Shear Stress Paths 192 6.2.2 Constant p' Paths 201 VII - SUMMARY AND CONCLUSIONS 209 REFERENCES 213 v i i i LIST OF FIGURES 3.1 Membrane Penetration Per Unit Surface Area with Changing Effective Confining Pressure. 3.2 Details of Axial Deformation Measurement. 3.3 Schematic of the Pneumatic Stress Path Analog System. 3.4 Grain Size Distribution Curve for Ottawa Sand. 3.5 Repeatability of Test Results In Hydrostatic Compression. 3.6 Repeatability of Test Results in Shear Loading. 3.7 Comparisons of Response to Shear at Three Levels of Mean Normal Stress. 3.8 The Influence of Relative Density on Strain Response to Hydrostatic Compression. 4.1 Results of Conventional Triaxial Tests at Different Confining Pressures. 4.2a Comparative Interpretation of a Triaxial Test Result on the Basis of Data at Small and Large Strain. 4.2b Transformed Plot of Triaxial Test Results in the Small Strain Range. 4.3 Variation of Static Moduli with Strain Level and Confining Pressure. 4.4 Young's Moduli From Resonant Column Testing in Longitudinal Mode. 4.5 Young's Moduli with Strain and Confining Stress Level as Determined from Resonant Column Tests. 4.6 Change of Dynamic Moduli with Strain Level and Confining Pressure. ix 4.7a Comparison of E and E, as Determined From Resonant Column max i and Triaxial Tests. 4.7b Comparison of Dynamic and Static Moduli at 1 x 10 - l + Axial Strain Level with Confining Pressure. 4.8 Alternative Comparison of ^ m a x and E^ at Different Confining Pressures. 4.9 Loading and Unloading Response in a Conventional Triaxial Path. 4.10 A Comparison Between E and Various I n i t i a l Moduli, E . max i 4.11 Stress Paths Investigated. 4.12 Stress Strain Response From Tests Performed Along the Various Stress Paths. 4.13 Different Stress Paths and Common Points of Intersection. 4.14a Path Dependence of Moduli Evaluated at Common Stress Points: 6a' Conventional Triaxial and -=—7- = 2 Paths. 4.14b Path Dependence of Moduli Evaluated at Common Stress Points: 6 a' Conventional Triaxial and -?—r " 4 Paths. 4.14c Path Dependence of Moduli Evaluated at Common Stress Points: Conventional Triaxial and Constant Mean Normal Stress Paths. 4.15 Comparison of Moduli Evaluated at Common Stress Points Following Different Stress Paths. 4.16 Strain Paths for Conventional Triaxial Stress Paths. 4.17 Stress Dilatancy Plot of Conventional Triaxial Test Results. 4.18 Strain Paths for Conventional Triaxial Stress Paths in Reloading. X 4.19 Stress Dllatancy Plot of Conventional Triaxial Test Results During Reloading. 4.20 Strain Paths for Conventional Triaxial Tests on Loose Sand. 4.21 Stress Dilatancy Plot of Conventional Triaxial Test Results on Loose Sand. 4.22a Strain Paths for Stress Paths of Incremental Stress Ratio of 2. 4.22b Strain Paths for an Incremental Stress Ratio of 4 Path. 4.22c Strain Paths for Constant Mean Normal Stress Paths. 4.23 Relative Location of Different Stress Paths on a Stress Dilatancy Plot. 4.24 Strain Paths for Conditions of Hydrostatic Loadings and Unloadings. 4.25 Strain Paths for Conventional Triaxial Loadings and Unloadings. 4.26 Comparison of Stress and Non-Recovered Strain Direction at the Same Stress Point. 4.27 The Influence of Mean Normal Stress on Normalized Work. 4.28 The Influence of Relative Density on Normalized Work. 5.1 Strain Paths for Proportional Loading and p' Equipotentials. 5.2 Relationship Between Strain Increment Ratio and Stress Ratio. 5.3 Geometric Features of Strain Paths and p1 Equipotentials. 5.4a Comparison of Measured and Predicted Axial Strain in Proportional Loading Paths. 5.4b Comparison of Measured and Predicted Volumetric Strain in Proportional Loading Paths. 5.5 Variation of Incremental Energy Density Ratio with Stress Ratio. x i 5.6 Strain Paths for Proportional Loading and p' Equipotentials: (a) Df* = 30% and (b) D r = 70%. 5.7 The Influence of Density on Strain Paths from Proportional Loading. 5.8 Relationships Between Strain Increment Ratio and Stress Ratio at Different Relative Densities. 5.9 Relationships Between Strain Increment Ratio and Relative <> Density for Different Stress Ratios. 5.10 Contractant Constant R State Surface for Ottawa Sand. 5.11 Variation of Incremental Energy Density Ratio with Stress Ratio: (a) D = 30% and (b) D = 70%. r r 5.12 Relationships Between Incremental Energy Density Ratio with Stress Ratio. 5.13 Relationships Between Ratios of Energy Density Increment and Relative Density. 5.14 Comparison of Measured and Predicted Volumetric Strains for Hydrostatic Loading at Various Relative Densities. 5.15 Comparison of Measured and Predicted Strains for Proportional Loading in Extension Mode and at Various Relative Densities: (a) Axial Strain and (b) Volumetric Strain. 5.16 Comparison of Measured and Predicted Strains for Proportional Loading in Compression Mode and at Various Relative Densities: (a) Axial Strain and (b) Volumetric Strain. 6.1a Strain Paths for Conventional Triaxial Stress Paths from Different Consolidation States. 6.1b Strain Paths for Constant Mean Normal Stress Paths from Different Consolidation States. x i i 6.1c Strain Path for a Stress Path of Constant Incremental Stress Ratio of 4. 6.1d Strain Paths for Stress Paths of Constant Incremental Stress Ratio of 2. 6.2a The Influence of Relative Density on I n i t i a l Strain Paths for Conventional Triaxial Compression. 6.2b Strain Paths.for p' = 450 kPa Tests at Different Relative Densities. 6.2c Strain Paths for R = 1.67 Stress Paths at Different Relative Densities. 6.2d Strain Paths for Hydrostatic Compression at Different Relative Densities. 6.3a Conventional Triaxial Strain Paths and Stress Ratio Potentials in Strain Space. 6.3b Stress Ratio Potentials in Strain Space for Constant Mean Normal Stress Paths. 6.3c Stress Ratio Potentials in Strain Space for Constant Incremental Stress Ratio of 2 Stress Paths. 6.4a Energy Density with Stress Ratio in Conventional Triaxial Paths. 6.4b Energy Density with Stress Ratio in Constant Mean Normal Stress Paths. 6.4c Energy Density with Stress Ratio in Constant Incremental Stress Ratio of 2 Stress Paths. 6.5a Normalized Energy Density with Stress Ratio for Conventional Triaxial Stress Paths. 6.5b Normalized Energy Density with Stress Ratio for Constant Mean Normal Stress Paths. x i i i 6.5c Normalized Energy Density with Stress Ratio for Constant Incremental Stress Ratio of 2 Stress Paths. 6.6 Average Normalized Energy Density with Confining Stress Ratio for A l l Stress Paths. 6.7 Comparison of Experiment and Prediction in Conventional Triaxial Paths. 6.8a Incremental Stress and Strain Ratio Relationships at 50 Percent Relative Density. 6.8b Incremental Stress and Strain Ratio Relationships at 30 Percent Relative Density. 6.9 Trends and Features in Incremental Strain and Stress Ratio Relationships. 6.10a Shear and Volumetric Strain States in a Constant Shear Stress Path for Decreasing Stress Ratio and Increasing Mean Normal Stress Conditions in Extension Mode. 6.10b Shear and Volumetric Strain States in Constant Shear Stress Path with Decreasing Stress Ratio and Increasing Mean Normal Stress in Compression Mode. 6.11a Shear and Volumetric Strain States in Constant Shear Stress Paths with Increasing Stress Ratio and Decreasing Mean Normal Stress in Extension Mode. 6.11b Shear and Volumetric Strain States in Constant Shear Stress Paths with Increasing Stress Ratio and Decreasing Mean Normal Stress in Compression Mode. 6.12 Volume Contraction in a Constant Shear Stress Path Under Increasing Stress Ratio and Decreasing Mean Normal Stress in Extension Mode. x i v 6.13a Strain Paths for Extension Side Shear Unloading and Compression Side Shear Loading at Constant Mean Normal Stress. 6.13b Strain Paths for Compression Side Shear Unloading and Extension Side Shear Loading at Constant Mean Normal Stress. 6.14a Relationships Between Shear Stress and Shear Strain for Compression Side Unloading and Extension Side Loading of Shear Stress at Constant Mean Normal Stress. 6.14b Relationships Between Shear Stress and Shear Strain for Extension Side Unloading and Compression Side Loading of Shear Stress at Constant Mean Normal Stress. XV INDEX OF NOTATIONS Bfc tangent bulk modulus 6e D dilatancy = 1 - — a d energy density r a t i o per cycle of consolidation stress r a t i o D^ r e l a t i v e density state 6 increment E Young's Modulus K± i n i t i a l E i n loading E i r i n i t i a l E i n reloading E i u i n i t i a l E i n unloading E maximum E max E f c tangent E e a x i a l s t r a i n a e r a d i a l s t r a i n r e shear s t r a i n = e - e s a r e volumetric s t r a i n = e + 2e v a r 9 d i r e c t i o n of s t r a i n vector K upper bound of energy increment r a t i o modulus number K inverse stress r a t i o for zero l a t e r a l deformation o K lower bound of energy increment r a t i o m slope of the relationship between e /a, and e i n small s t r a i n a d a response u Poisson's r a t i o XVI n modulus exponent, represents an for a confining stress equal to atmospheric pressure and divided by P n stress r a t i o = ^, n stress r a t i o at c r i t i c a l state and zero rate of volume change cv Pfl atmospheric pressure o-^  + 2a^ p* e f f e c t i v e mean normal stress = ^ q shear stress = °\ ~ a2 Q energy density state a' R stress r a t i o = —,— a' r 6a' r incremental stress r a t i o = — r oa r S slope of the relationship between W and e s s a^ major e f f e c t i v e p r i n c i p a l stress a^ minor e f f e c t i v e p r i n c i p a l stress a' a x i a l e f f e c t i v e stress a a, deviator stress = a - a d a r a'r r a d i a l e f f e c t i v e stress (j) f r i c t i o n angle <J>^  a material c h a r a c t e r i s t i c i n t e r p a r t i c l e f r i c t i o n angle $ f r i c t i o n angle at c r i t i c a l state and zero volume change W work done i n shear s i|> gradient of stress equipotentials In s t r a i n space. x v l i ACKNOWLEDGEMENTS I am greatly indebted to my supervisor, Professor Y.P. Vaid, f o r hi s guidance, encouragement and enthusiastic i n t e r e s t throughout t h i s research. I would also l i k e to thank Professors P.M. Byrne, R.G. Campanella and W.D.L. Finn for t h e i r advice and counsel. My colleagues J.C. Chem, E. Cheung, A. Sayao, M. Zergoun and e s p e c i a l l y , P. L u i , shared a common active i n t e r e s t i n s o i l mechanics. I thank them a l l for the i r h e l p f u l discussions and constructive c r i t i c i s m s . The task of equipment development was made easier by the advice and assistance of Mr. F. Zurkirchen". Ms. S.N. Krunic typed the manuscript with care and the figures were s k i l l f u l l y drafted by Mrs. M. Sayao. I am grateful for the i r patience and hard work. Support and assistance provided by the Univ e r s i t y of B r i t i s h Columbia, the Natural Science and Engineering Research Council of Canada and Golder Geotechnical Consultants i s acknowledged with deep appreciation. A very s p e c i a l thanks go to my wife, Atsede, and my son, Brook; whose encouragement, optimism and support sustained me through stress and s t r a i n . V 1 CHAPTER I - INTRODUCTION The engineering of sand i s often associated with boundary value problems that require a r a t i o n a l s t r e s s - s t r a i n law for s a t i s f a c t o r y s o l u t i o n . Sand possesses a r e l a t i v e l y high permeability and i n s i g n i f i c a n t creep deformation properties. E s s e n t i a l features of the s t r e s s - s t r a i n behaviour of sand are considered to be contained within a time independent domain. A number of co n s t i t u t i v e models f or sand have been proposed. A l l are founded upon c e r t a i n assumptions and hypotheses; with guidance and motivation engendered from experimental observations. For several decades, research i n t e r e s t has focused heavily on the strength behaviour of sand. Because of uncertainties Inherent i n s o i l exploration and assessment of strength properties, foundations are usually designed with generous safety factors against f a i l u r e . Except perhaps along l o c a l i z e d stress concentrations, working stress l e v e l s are much below f a i l u r e . For both loose and dense sand subjected to conventional t r i a x i a l stress paths, a s i g n i f i c a n t proportion of peak strength i s mobilized at small s t r a i n . Working stresses under drained conditions generally induce st r a i n s that are less than 1 percent. S i m i l a r l y , tendencies toward volume contraction and the corresponding development of p o s i t i v e pore pressure during undrained loading are phenomena that occur mainly at small s t r a i n . On account of possible errors and experimental l i m i t a t i o n s , small deformations associated with strains below 1 x 10 - 5 are d i f f i c u l t to measure d i r e c t l y . Behaviour at such s t r a i n l e v e l s i s relevant for dynamic response and i s assessed i n d i r e c t l y through resonance and wave 2 propagation considerations assuming e l a s t i c behaviour. With c a r e f u l equipment design; small s t r a i n response, between 1 x 10"5 and 1 x 10~ 2, can be investigated through d i r e c t measurement of associated deformations. In previous studies of drained response by d i r e c t measurement of deformations, in t e r e s t was mainly concerned with s t r a i n levels i n excess of 1 x 10~3. Shear induced volume change i s a d i s t i n c t c h a r a c t e r i s t i c of pa r t i c u l a t e materials l i k e sand. Such volume change has often been regarded to be synonymous with dilatancy, or shear volume expansion, i n many previous considerations. It i s , however, widely known that shear induced volume change can either be contractant or d i l a t a n t . For stress r a t i o s approaching f a i l u r e and associated large s t r a i n l e v e l s , d i l a t a n t effects generally predominate. Whereas, for stress r a t i o s normally associated with working stress l e v e l s and thus small s t r a i n , contractant shear volume change assumes greater importance. Over a l l , contractancy would appear to be of much more fundamental in t e r e s t i n the study of sand behaviour. In a broader context, generation of positive pore pressure i n undrained shearing i s a consequence of contractancy. Current understanding of the contractant behaviour of sand i s very l i m i t e d . Experimental observations relevant to dilatancy may not be applicable to situations of contractant response. Recoverable deformations become n e g l i g i b l e i n the d i l a t a n t region, whereas, both recoverable and nonrecoverable s t r a i n components become s i g n i f i c a n t i n the region of contractancy. E l a s t i c - p l a s t i c separation of strains i s usually deemed fe a s i b l e through a load unload procedure. However, 3 the experimental evidence i n support of such separation does not appear to be convincing. The inevitable existence of i n e l a s t i c s t r a i n s as a consequence of reverse s l i p during unloading can not be denied. This research focuses on a fundamental experimental investigation of the behaviour of sand at small s t r a i n . Some common frameworks and experimental j u s t i f i c a t i o n s for describing the behaviour of sand as incrementally e l a s t i c , e l a s t o - p l a s t i c and from p a r t i c u l a t e considerations are c r i t i c a l l y examined as to th e i r v a l i d i t y i n the region of small s t r a i n s . Through exercise of systematic and precision oriented experimentation, new observations and consistent trends i n small s t r a i n response of sand are presented. A fundamental in t e r p r e t a t i o n of test data and an al t e r n a t i v e d e s c r i p t i o n of s t r e s s - s t r a i n behaviour i s attempted. Sand behaviour i s examined predominantly at a r e l a t i v e density of 50 percent i n simple stress paths. The study i s concerned with drained deformation response. Behavioural e f f e c t s of stress parameters and r e l a t i v e density state were examined maintaining wide i n t e r v a l s i n both stress and r e l a t i v e density such that the ef f e c t of the variables can be established with certai n t y . Test results are presented and compared i n conventional s t r e s s - s t r a i n representation i n order to examine common concepts i n small s t r a i n region. As well, correspondence between states of stress and s t r a i n are investigated i n s t r a i n rather than stress space. Pursuit of these objectives required, as a prerequisite, r e l i a b l e data from c a r e f u l l y performed experiments. Small s t r a i n observations cannot be r e l i a b l y extracted from test setups assembled for the study of large s t r a i n response. A measurement range and s e n s i t i v i t y suitable for large s t r a i n study are generally too coarse and i n s e n s i t i v e for r e l i a b l e observation of small s t r a i n phenomena. Experimental procedures and improvements adopted to enhance large s t r a i n i n v e s t i g a t i o n may not be useful or even be counterproductive for small s t r a i n observations. As a r e s u l t , test equipment and procedures are c r i t i c a l l y evaluated and overhauled with the e x p l i c i t i n t e n t i o n of optimizing accuracy and f u l f i l l m e n t of necessary assumptions with regards to small s t r a i n study. This necessitated Improvement i n test equipment i n order to f a c i l i t a t e precise loading control, including the a b i l i t y to follow a r b i t r a r y stress paths; and c r i t i c a l evaluation of available methods, and development of a l t e r n a t i v e r a t i o n a l procedures, for membrane compliance correction. 5 CHAPTER II - STRESS-STRAIN RESPONSE OF SAND U n t i l the recent past, the performance of sand was predominantly evaluated s o l e l y i n terms of f a i l u r e . In most cases, design was based on l i m i t equilibrium analyses and without regard to associated deformations. However, rapid introduction of computing f a c i l i t i e s made possible the use of numerical solutions to many deformation problems i n geotechnical engineering. The desire for implementing this newly acquired c a p a b i l i t y to p r a c t i c a l use has imposed a sudden demand for some form of c o n s t i t u t i v e input. Over the past several years, a number of researchers have contributed to a fundamental understanding of the s t r e s s - s t r a i n behaviour of sand through experimental study. The studies ushered i n several improvements i n test techniques and equipment.including the use of a variety of test apparatus. However, for the most part, in t e r e s t has focused on large s t r a i n response (generally i n excess of 1%) and that too under a limited number of stress paths, a l l leading to f a i l u r e . The study of small s t r a i n phenomena has been very l i m i t e d . 2.1 Observed Behaviour I n i t i a l i n t e r e s t i n examining shear volume change i n sands was motivated by a desire to explain experimentally observed phenomena related to shear strength testing (Taylor, 1948; Bishop, 1950). An energy correction concept was introduced to separate work related to d i l a t i o n or contraction from that due to f r i c t i o n . Closer agreement between strengths of dense and loose sands was obtained following energy correction. Consideration of shear volume change i n the 6 prefailure region did not emerge until the introduction of stress dilatancy concepts several years later (Rowe, 1962). With the objective of describing sand response within a linear elastic framework, earlier experimental investigators attempted to establish the influence of shear and confining stress on moduli and Poisson's ratio (Chen, 1948; Jakobson, 1957; Makhlouf & Stewart, 1965). Although the test equipment and experimental techniques were much less refined, their qualitative observations regarding the strong influence of density or void ratio, shear and confining stress on strain response remain valid. Such observations of nonlinearity and stress level dependence of stress-strain behaviour are features central to empirical representations that are in wide use today. Further Investigations demonstrated that the stress strain behaviour of sand is path dependent and is influenced by stress history (Lade and Duncan, 1976; Lambrechts and Leonards, 1978; Varadarajan et a l , 1983). Upon stress reversal from a stress state near failure, anisotropy induced during shear in one direction has been shown to alter shear deformation characteristics in the opposite direction (Arthur, 1971; Thurairajah, 1973). However, when stress reversal was initiated from a stress state significantly below failure, the response was found to be independent of the loading history in the opposite direction (Tatsuoka and Ishihara, 1974). Apart from the strong influence of density on stress-strain response, structure or s o i l fabric, as determined by predominant grain orientation, has been shown to significantly alter the nature of inherent anisotropy, in otherwise identical specimens (Arthur and 7 Menzies, 1972; Yamada and I sh ihara , 1979; Oda, 1972, 1976). The effect of inherent anisotropy on small s t r a i n response has also been demonstrated to be s i g n i f i c a n t , whereas strength propert ies have been found to be r e l a t i v e l y unaffected. These observations have led to the suggestion that inherent anisotropy becomes erased by s tress induced anisotropy that develops with shear loading (Yamada and Ish ihara , 1979). P luv ia ted sand samples acquire a preferred gra in o r i e n t a t i o n o and as a re su l t have a s t i f f e r response i n the d i r e c t i o n of depos i t ion . The degree of inherent anisotropy, as i n f e r r e d from s t r a i n response i n hydrostat ic compression tes t s , has been shown to diminish with increas ing density (El-Sohby and Andrawes, 1972). However, the inf luence of the d e n s i f i c a t i o n procedure In changing the nature of inherent anisotropy has not been c l a r i f i e d . S tra in paths under proport iona l loading and unloading at low s tress r a t i o l e v e l s have been found to be l i n e a r (El-Sohby, 1969; El-Sohby and Andrawes, 1973) . Changes i n mean normal s tress were i n f e r r e d not to a l t e r Inherent anisotropy so long as the s tress r a t i o remained constant. Rowe (1971) has pointed out that the l i n e a r i t y of loading and unloading s t r a i n paths during proport ional loading of sand also implies l i n e a r i t y for the nonrecovered s t r a i n paths. This behaviour bears c lose resemblance to observed l i n e a r i t y of t o t a l , recovered and nonrecovered s t r a i n paths during proport ional loading of metals (Mehan, 1961; Rees, 1981; 1981a) The strength of sand has been known to be influenced by rate of loading (Whitman, 1957; Home, 1965). However, reported s t r e s s - s t r a i n studies i n v a r i a b l y dwell i n a time independent domain. There Is general ly no expressed concern r e l a t i v e to procedures adopted 8 to suppress or assess the relative influence of transient response (commonly identified with creep) during previous investigations. Yet, due consideration and minimization of transient effects Is essential for quantitative correspondence between test results. 2.2 Idealization of Behaviour as Either Elastic or Plastic In analyses of material response, i t is often convenient to idealize behaviour as either a purely elastic or plastic continuum. The v i a b i l i t y of such idealized representation for sand has been examined in previous experimental investigations, and primarily from simple shear and tr i a x i a l tests (Cole, 1967; Roscoe, 1970; Frydman and Zeitlen, 1969). A simple shear test permits a continuous rotation of principal axes of stress and strain, whereas these axes are fixed in a tr i a x i a l test. Coincidence of principal axes of strain rate and stress increment would characterize elastic behaviour (Jaeger, 1964); whereas, principal axes of strain rate and stress would coincide for a plastic material ( H i l l , 1964). Roscoe (1970), in his Rankine Lecture, reported on important experimental findings from simple shear tests on sand regarding orientations of principal axes of stress, stress increment and strain rate. A coincidence of principal axes of stress and total strain increment was noted under monotonic loading but following attainment of a minimum void ratio. This implies that the stress-strain response of sand when subjected to increasing shear stress and beyond a stage of maximum volumetric contraction resembles ideal plastic behaviour. An i n i t i a l zero rate of volume change 9 invariably coincides with mobilization of shear strength corresponding to a constant volume f r i c t i o n angle of <b for sand and at r e l a t i v e l y cv large shear s t r a i n . Although less convincing, there was also evidence In support of coincidence of p r i n c i p a l axes of stress increment and s t r a i n rate during unloading and reloading, thus suggesting some basis for an e l a s t i c approximation of unloading and reloading response of sand. However, during v i r g i n loading and prior to maximum contraction, coincidence of neither p r i n c i p a l axes of stress nor stress Increment with p r i n c i p a l axes of s t r a i n rate could be inferred with certai n t y . This uncertainty i n part r e f l e c t s the inherent lim i t a t i o n s of the simple shear apparatus r e l a t i v e to determination of stress state and measurement of small deformations (Saada et a l , 1980; 83; Cole, 1967; Budhu, 1984; Arthur et a l , 1980). At the same time, i t also r e f l e c t s the comparable presence of e l a s t i c and p l a s t i c responses i n the early stages of shearing and that behaviour would not be adequately represented as purely e l a s t i c or p l a s t i c . T r i a x i a l test r e s u l t s have been used to demonstrate the usefulness of pseudo-elastic i d e a l i z a t i o n for dense sand (Frydman and Ze i t l e n , 1969). Considerations were r e s t r i c t e d to shear stress states below that corresponding to the material f r i c t i o n angle, and therefore small s t r a i n . Experimental results that suggest an absence of shear volume change were shown and this observation was advanced as j u s t i f i c a t i o n for a purely e l a s t i c i d e a l i z a t i o n of s t r e s s - s t r a i n response. However, contractant shear induced volume changes occur at small stress r a t i o s and a l l l e v e l s of r e l a t i v e density, becoming more pronounced with decreasing density. Consequently, the experimental evidence presented i n support of a pseudo e l a s t i c characterization of sand would appear to be r e a l i s t i c when considering dense states only. With decreasing r e l a t i v e density, an e l a s t i c approximation would become less desirable as both e l a s t i c and p l a s t i c responses assume r e l a t i v e importance. 2.3 P a r t i c u l a t e Considerations On a v i s u a l scale, sand i s a p a r t i c u l a t e material. I t s aggregate behaviour i s intermediate between a f l u i d and a s o l i d with the added shear volume change a peculiar c h a r a c t e r i s t i c . Under i d e a l i z e d conditions of p a r t i c l e geometry and packing, the load deformation c h a r a c t e r i s t i c s of a granular mass can be derived from grain contact considerations. Regular packing of spheres have therefore been studied to gain fundamental understanding of deformation response. At a plane of contact between two spheres within regular packing; torsion, shear and normal stresses can e x i s t . In the absence of torsion and shear; Hertz's contact theory (TImoshenko and Goodier, 1970) predicts a l i n e a r r e l a t i o n s h i p between e l a s t i c volumetric s t r a i n and the 2/3 power of mean normal stress for a set of Isotropic e l a s t i c equal spheres. Consideration of torsion and shear at grain contact introduce i n e l a s t i c i t y (Dereswicz, 1957) and shear volume change of void space. The kind of shear volume change Is determined by density of packing. A simple cubic array, which constitutes a loose packing with a void r a t i o of 0.910, would experience contraction; whereas dense packing, such as i n a face centered cubic array (void r a t i o of 0.351) would d i l a t e . The nature of shear volume change of granular materials can therefore be either contractant or dilatant depending on 11 the state of packing or density. The study of i d e a l i z e d geometry, packing and material brings out e s s e n t i a l deformation response features to focus. It makes possible the q u a l i t a t i v e understanding of sand behaviour within a r e s t r i c t e d framework. However, i t s d i r e c t extension towards describing sand would f a l l short of i t s intended objective (Rowe, 1971). Individual sand p a r t i c l e s are i r r e g u l a r shaped and grain sizes vary. Also, some amount of p a r t i c l e crushing would i n e v i t a b l y occur. S l i p and rearrangement would take place at a l l stress l e v e l s and stress paths. In such i r r e g u l a r assemblies, the d i s t r i b u t i o n of contact stresses for p a r t i c l e s not i n l i m i t i n g equilibrium cannot be determined. S t r e s s - s t r a i n parameters would vary with o r i e n t a t i o n to c r y s t a l structures. Because of these l i m i t a t i o n s , i n t e r p r e t a t i o n of sand behaviour within a p a r t i c u l a t e framework required an emperical approach using al t e r n a t i v e assumptions. Treatment of sand from p a r t i c u l a t e considerations gained new ground with the introduction of stress dilatancy theory (Rowe, 1962). As pointed out previously, a desire to separate stress r a t i o into f r i c t i o n a l and dilatancy components appears to have motivated i n i t i a l i n t e r e s t (Rowe et a l , 196A, Bishop, 1964). Since the stress r a t i o , R=o|/ i s associated with assumed s l i p s t r a i n increment r a t i o , the stress dilatancy equation, R = KD, was suggested to hold r e a l promise of providing a r e a l i s t i c nonassociated flow rule for p a r t i c u l a t e materials (Rowe, 1971; Barden and Khayatt, 1966; Cole, 1967). However, the association between stress r a t i o and d i l a t i o n rate D i s not unique. The difference between upper and lower bounds i n K, . (corresponding to <(> and $ ) for sand i s of the order of 25 percent and cannot be considered small even though proponents of stress dilatancy suggest otherwise (Barden and Khayatt, 1966). Stress dilatancy theory has been considered v a l i d for stress paths of increasing stress r a t i o (Rowe, 1971) and shown to be path independent (Tatsuoka, 1976; Cole, 1967). However, reported experimental v e r i f i c a t i o n s are inv a r i a b l y from large s t r a i n response and along paths that ultimately lead to f a i l u r e . A separate class of stress paths that do not lead to f a i l u r e but are nevertheless paths of increasing stress r a t i o e x i s t (Hardin, 1983). V a l i d i t y of stress-dilatancy along such paths that have an orientation between R = 1/KQ and a hydrostatic stress path has not been demonstrated. A hydrostatic path would not conform to stress-dilatancy because the associated energy r a t i o would not be negative (Barden et a l , 1969). A s i n g u l a r i t y would be objectionable on physical grounds and hence there must exist a neighborhood about the hydrostatic axis within which stress dilatancy would not apply. This same argument can be extended to question the suggested v a l i d i t y of stress-dilatancy i n constant stress r a t i o paths. In an extension to his e a r l i e r work, Rowe (1971) made an alter n a t i v e proposal f o r Incorporating constant stress r a t i o response within a stress dilatancy framework. He proposed a conceptual uncoupling of stress increments into v e r t i c a l and r a d i a l components and Imposed compressional and extensional stress dilatancy relationships to the uncoupled components respectively. For any uncoupled stress increment along a proportional loading path, both uncoupled stress components would not be associated with increasing stress r a t i o directions simultaneously. Thus, the conceptual uncoupling of stress increments and implied v a l i d i t y of stress-dilatancy i n a d i r e c t i o n of decreasing stress r a t i o i s contrary to e a r l i e r assumptions. Furthermore, the quoted expression for the uncoupled r a d i a l stress increment does not appear to be correct, casting further doubt on the merits of Rowe's proposal. Shearing of sand along a f a i l u r e path induces contraction up u n t i l s i g n i f i c a n t strength mobilization. Most p r a c t i c a l deformation problems l i e well below f a i l u r e . Recoverable strains i n this small deformation region are r e l a t i v e l y s i g n i f i c a n t and thus a pseudo p l a s t i c i d e a l i z a t i o n , as would be implied by stress dilatancy, may not be appropriate. Although there has been i n t e r e s t i n using a stress dilatancy relationship as a flow rule i n stress deformation modelling of sand (Nova and Wood, 1979; Bardet, 1983), i t s presumed v a l i d i t y and extrapolation to the region of small s t r a i n s t i l l remains uncertain (Arthur, 1971). Arguments i n favour of a d i f f e r e n t stress dilatancy r e l a t i o n s h i p for small s t r a i n response have been put forward (Nova and Wood, 1979). However, this hypothesis has yet to be v e r i f i e d experimentally. 2.4 Decomposition of Stra i n In general, neither a pseudo e l a s t i c nor p l a s t i c representation has been found to f u l l y characterize the small s t r a i n response of sand. More appeal has therefore been made to simultaneous but separate e l a s t i c - p l a s t i c i d e a l i z a t i o n of behaviour, assuming correspondence between recovered and s l i p deformations to e l a s t i c and p l a s t i c s t r a i n s , respectively. The required decomposition of t o t a l s t r a i n s into e l a s t i c and p l a s t i c components has generally been considered possible through a load unload procedure. Whether such a separation holds true for sand must, however, be demonstrated by comparing presumed s t r a i n components against c h a r a c t e r i s t i c behaviour. A given response would need to s a t i s f y set preconditions to j u s t i f y c l a s s i f i c a t i o n s as either e l a s t i c or p l a s t i c . The possible e l a s t i c character of recoverable deformations was examined by Holubec (1968). He presented experimental results on dense sand that show no hy s t e r e i s i s upon unloading and reloading up to 80 percent of maximum shear strength. Furthermore, he found incremental e l a s t i c properties evaluated at stress points to be independent of stress h i s t o r y . These observations i n and of themselves are necessary but not s u f f i c i e n t to est a b l i s h the e l a s t i c character of recovered deformations. For an e l a s t i c material, integration of incremental s t r e s s - s t r a i n equations must also be path independent but the test results have been shown not to s a t i s f y this condition (Coon and Evans, 1969). Moreover, the incremental equations used by Holubec were shown to require an additional term (Merkele and Merkele, 1969). Furthermore, the reported experimental re s u l t s appear to be erroneous i n that the reloading curves s t i f f e n instead of soften with s t r a i n . C l e a r l y , Holubec's op t i m i s t i c outlook from examination of dense sand would not be as encouraging i f consideration were to focus on loose behaviour. Inspite of i t s shortcomings, Holubec's work i s s t i l l widely referred to j u s t i f y e l a s t i c treatment of recovered deformations. V a l i d i t y of e l a s t i c - p l a s t i c s t r a i n separation can also be examined from consideration of the nonrecovered deformations. The d i r e c t i o n of p l a s t i c s t r a i n increment would be uniquely determined by the state of stress i f the behaviour i s p l a s t i c . Observations consistent with t h i s c r i t e r i a have been reported by Poorooshasb, et a l , 1966; Lade and Duncan, 1976. Existence of a potential function whose gradient determines the i n c l i n a t i o n of the p l a s t i c s t r a i n increment was claimed. The p l a s t i c potential curves for a given void r a t i o were inferred to form a family of geometrically si m i l a r curves closing on the f a i l u r e and hydrostatic l i n e s . A closer examination of the experimental evidence shows that the unique association of stress state and p l a s t i c s t r a i n increment vector has been demonstrated only for stress states i n the proximity of f a i l u r e and hence at large s t r a i n . As such, these findings are re-affirmations of the stress dilatancy theory of Rowe (1962) and evidence from simple shear test results of Cole (1967), both of which were based on t o t a l s t r a i n response. S l i p at p a r t i c l e contacts i s the predominant form of deformation at large s t r a i n . Since recoverable deformations are r e l a t i v e l y small neither t h e i r i n c l u s i o n or omission would have serious implication. However, the s i t u a t i o n at small s t r a i n i s much d i f f e r e n t i n that both recovered and non-recovered deformations assume r e l a t i v e importance. Extrapolation of findings at large stress r a t i o to encompass response at small stress r a t i o i s therefore u n j u s t i f i e d . In a granular mass, s l i p and reverse s l i p would occur during loading and unloading. The i s o l a t i o n of e l a s t i c deformation of sand grains from s l i p occurring at grain contacts i s v i r t u a l l y impossible. Aside from attempts to examine recovered and nonrecovered deformations for symptoms of e l a s t i c or p l a s t i c behaviour, s i g n i f i c a n t e f f o r t has also been directed towards modifying established e l a s t o - p l a s t i c concepts to suit test results on sand. In this regard, i n t e r s e c t i n g y i e l d surfaces and uncoupling of strains into two and three components are features that have been adopted to force f i t s o i l within some form of e l a s t i c - p l a s t i c type of framework. There i s as yet no concensus on a c r i t e r i o n for y i e l d i n g and a nonassociated form of flow rule i s often used. Even a f t e r r a d i c a l a l t e r a t i o n of established material models, i t has been possible to characterize behaviour at only one void r a t i o and mostly dense states. The number of required parameters i s cumbersome, sometimes i n excess of 10, and have to be determined from several tests (Desai et a l , 1981; Bardet, 1983). Generally, models need to be calibrated with reference to selected experimental data and tend to be less e f f e c t i v e when c a l l e d upon to make predictions under stress paths d i f f e r e n t from those considered for c a l i b r a t i o n (NSF/NSERC Workshop, 1980). Undoubtedly, a lo t more remains to be done and the future d i r e c t i o n and refinement of s o i l models w i l l require new experimental findings and a l t e r n a t i v e ideas. 2.5 Trends i n the Study of Sand Behaviour As stated e a r l i e r , the small s t r a i n response of sand i s a r e l a t i v e l y unexplored t e r r i t o r y . Although there i s growing i n t e r e s t i n using the stress dilatancy equation as a flow rule, i t s v a l i d i t y at small s t r a i n i s uncertain. In some e l a s t i c - p l a s t i c approaches, y i e l d i n g has been postulated to start at the onset of loading (Nova and Wood, 1979). This would imply there i s no i n i t i a l region wherein response would be e n t i r e l y e l a s t i c . A concept of normalized work, Introduced by Moroto (1976), has inspired new e l a s t i c - p l a s t i c model proposals for sand (Ghabaussi and Momen, 1984, 1982; Tobita and Yanagisawa, 1980; Varadarajan et a l , 1983). In this concept, the relationship between the work in t e r g a l due to shear, normalized by the corresponding mean normal e f f e c t i v e stress, and shear s t r a i n i s suggested to be unique. The slope of this r e l a t i o n s h i p has been further observed to represent a material parameter independent of mean normal e f f e c t i v e stress and density. The experimental v e r i f i c a t i o n and possible implications of this concept do not, however, appear to have been c r i t i c a l l y examined. In view of the i m p o s s i b i l i t y of separation of e l a s t i c and p l a s t i c strains and the observation that t o t a l strains may not be adequately represented as e l a s t i c , appeal has been made to considering a l l deformations as p l a s t i c (Mroz, 1983; Chang, 1983). This i s contrary to previous suggestion that small s t r a i n response be represented as pseudo e l a s t i c . A r a d i c a l departure from current concepts of s t r e s s - s t r a i n modelling has been proposed by Zytynski, Randolph, Nova and Wroth (1978). They reject the idea of a purely e l a s t i c response at any stage and point out that i t would be impossible to i s o l a t e e l a s t i c and p l a s t i c s t r a i n components. Furthermore, they and subsequently Bardet (1983), have demonstrated inconsistencies i n some current assumptions that lead to extraction of work i n a cycle of loading. Zytynski et a l proposed an al t e r n a t i v e conceptual representation of sand as linked springs and f r i c t i o n a l blocks. This would allow simultaneous development of e l a s t i c and s l i p deformations at a l l times. Although such a mechanical i d e a l i z a t i o n may be useful as a conceptual framework, i t appears u n r e a l i s t i c for representing actual response. Further progress i n t h i s d i r e c t i o n has not been reported to date. In general, developments have advanced progressively from successful p r e s c r i p t i o n of f a i l u r e to reasonable but q u a l i t a t i v e description of large s t r a i n behaviour. Study of small s t r a i n phenomena and development of procedures for quantitative description of sand appear to be the d i r e c t i o n for future progress. CHAPTER III - EXPERIMENTAL CONSIDERATIONS An experimental study of a d i f f i c u l t material such as sand requires consideration of a number of important fa c t o r s . Suitable and r e l i a b l e test equipment would need to be assembled. The test material and an appropriate sample s i z e must be selected. Testing procedures would need to be improved and developed with the aim of performing consistent and repeatable t e s t s . The study of small s t r a i n phenomena requires precise load a p p l i c a t i o n and measurement of deformatons with confidence. Considerations given to these Important elements of experimentation are discussed i n this chapter. 3.1 Test Equipment Laboratory test equipment should f a c i l i t a t e development of homogenous stress and s t r a i n conditions within the-sample, allow adequate control of boundary stresses and permit accurate measurement of boundary displacement. The behaviour of sand i s stress path dependent. A sand specimen i s In general anisotropic and thus the rotation of p r i n c i p a l stresses would influence stress s t r a i n behaviour. Equipment that allows unlimited stress path control as well as permit rotation of p r i n c i p a l axes Is ultimately desired. Some attempt and progress i n t h i s d i r e c t i o n has been reported (Arthur, et a l , 1980). For the moment, however, di f f e r e n t factors that control s o i l behaviour would have to be investigated using d i f f e r e n t test equipments. A useful evaluation and comparison of various test apparatus has been done by (Saada et a l , 1980; 1983; Ladd et a l , 1977; Budhu, 1984). The existence of p o t e n t i a l l y serious d e f i c i e n c i e s , with regards to s p e c i f i c a t i o n and homogeneity of stress 20 and s t r a i n , i n many testing devices i s well known. The conventional t r i a x i a l apparatus has been widely used for study of s o i l behaviour. I t s inherent l i m i t a t i o n s are well recognized. Several modifications to this test have been made in the past and a wealth of background study e x i s t s . In comparison to many other testing devices, a r e s t r i c t e d but much more rigorous s t r e s s - s t r a i n analyses i s possible for the t r i a x i a l t e s t . There i s i n general a broad f a m i l i a r i t y and acceptance of behaviour characterized by conventional t r i a x i a l t e s t s . Consequently, a decision was made to Investigate small s t r a i n phenomena u t i l i z i n g an improved t r i a x i a l t e s t ing system. Examination of the influence of rotation of p r i n c i p a l axis was set aside for study using tors i o n a l hollow cylinder equipment i n a separate forthcoming i n v e s t i g a t i o n . A resonant column device was also used to test a few t r i a x i a l specimen i n order to compare s t a t i c and dynamic moduli at small s t r a i n . 3.1.1 Limitations and Improvement of the T r i a x i a l Test The degree to which the conventional t r i a x i a l test approximates idealize d assumptions r e l a t i v e to homogeneity of stress and s t r a i n within the test specimen has been of long standing i n t e r e s t . A n a l y t i c a l and experimental studies of stress and s t r a i n d i s t r i b u t i o n within test specimen have led to various suggestions for improvement. Internal and external displacement patterns have been studied using a var i e t y of experimental techniques, whereas stress d i s t r i b u t i o n c h a r a c t e r i s t i c s have generally been inferred from displacement observations. External measurement of a x i a l deformations along segments of a t r a i x i a l specimen suggest that the assumption of uniform axial strains is reasonable for tr i a x i a l compression (Roscoe etal, 1963). In addition, radial strains were found to be uniform over the height of the sample for small strains. Large deviations from uniformity were observed in t r i a x i a l extension tests at post peak strain levels. Re-examination of this result (Barden and Khyatt, 1966) showed that nonuniformities in large strain extension tests occur as a result of necking. This effect was significantly reduced through use of lubricated (also called free or frictionless) ends. A study of internal strain distribution that was made using embedded strain gages (Januskevicius and Vey, 1965) also showed uniform strain distributions for specimens with minimum end restraint. X-ray techniques were used by Kirkpatrick and Belshaw (1968) and Kirkpatrick and Younger (1971) to study the nature of internal displacement. Provided end restraint efects are negligible, radial displacements were shown to be proportional to radial position by monitoring the location of embedded lead shots. These results imply equality of radial and tangential strains in tr i a x i a l specimen. Uniform deformations are best achieved in conventional t r i a x i a l specimen with a correct emphasis on necessary improvements. In general, the problem of nonuniform deformation would be less significant as opposed to obstacles to true soil deformation measurement in small strain considerations. In this sense, the real benefit of some previously adopted improvement measures, e.g. the use of free ends, would be open to question. Inherent limitations specific to the conventional tr i a x i a l test and suitable improvement, alternatives that were implemented to promote a reliable measurement of small deformations i n sand are discussed i n the following. 3.1.1.1 Ram F r i c t i o n When the applied a x i a l load i s measured outside the c e l l , the actual load carried by the specimen i s d i f f e r e n t than the registered load due to ram f r i c t i o n . Error due to ram f r i c t o n has been avoided by measuring the load i n t e r n a l l y within the c e l l (Barden and Khayatt, 1966; El-Sohby and Andrawes, 1972; Green, 1969). Continuous a i r bleed bushings can also v i r t u a l l y eliminate resistance due to ram f r i c t i o n (Chan, 1975). The magnitude and signif i c a n c e of ram f r i c t i o n would depend on ram and sample areas as well as equipment c h a r a c t e r i s t i c s . The a x i a l force was measured externally. By using a low ram to sample area r a t i o together with l i n e a r b a l l bushings and a lubricated 0-ring s e a l , ram f r i c t i o n e f f e c t s were made n e g l i g i b l e . 3.1.1.2 Membrane Penetration The c y l i n d r i c a l surface of t r i a x i a l specimen i s covered by a rubber membrane. When stress paths of changing c e l l pressure are followed, measured volume changes are subject to error r e s u l t i n g from membrane penetration (Newland and A l l e l y , 1959). Even though this e f fect has important implications as to int e r p r e t a t i o n of strains i n drained tests, there appears to be a considerable ambiguity regarding correction procedures (Newland and A l l e l y , 1959; Roscoe et a l , 1963; Raju and Sadasivan, 1974; Wu and Chang, 1982). A c r i t i c a l review of current correction methods and development of more r a t i o n a l alternatives carried out i n conjunction with this study have been reported previously (Vaid and Negussey, 1984). Experimental observations have shown that a sand skeleton rebounds i s o t r o p i c a l l y upon hydrostatic unloading (El-Sohby and Andrawes, 1972). Hence i t was possible to determine volume change due to membrane penetration as the difference between t o t a l and s o i l volume changes during i s o t r o p i c unloading. For the sand and membrane used i n t h i s i n v e s t i g a t i o n , the membrane penetration curve shown i n Figure 3.1 was used to correct measured volume changes under conditions of changing e f f e c t i v e confining pressure. 3.1.1.3 End Restraint V a l i d i t y of assumed homogeneity of stress and s t r a i n within t r i a x i a l specimens have been observed to depend on conditions of end r e s t r a i n t . The influence of end r e s t r a i n t on strength has been shown to be minimal when a height to diameter r a t i o of 2 or larger i s used (Taylor, 1948; Bishop and Green, 1965). Furthermore, the effect of end r e s t r a i n t on s t r a i n increment r a t i o does not appear to be severe (Barden and Khayatt, 1966; Rowe, 1971; Green, 1971). Provision of lubricated rubber interface between the s o i l and r i g i d end platens has been adopted to diminish end r e s t r a i n t (Rowe and Barden 1964, Barden and Khayatt, 1966) and thus enable uniform deformations at a reduced height to diameter r a t i o of one. This was found e f f e c t i v e i n aiding development of multiple and hence general f a i l u r e i n sand. However, f r i c t i o n l e s s ends were better approximated with use of a double rather than single membrane (Bishop and Green, 1965). Use of free ends was attempted i n t h i s study, but i n i t i a l r esults were not encouraging. Uniform deformation developed only along the lower h a l f of the specimen. This phenomenon was also reported previously (Rowe and Barden, 1964; Green, 1969). A two stage sample forming process was found e f f e c t i v e i n improving the mode of deformation (Green, 1969). F i r s t , the top half of the specimen was O o Effective confining pressure, KPa Fig. 3.1 Membrane Penetration Per Unit Surface Area with Changing E f f e c t i v e Confining Pressure 25 formed over a dummy block. The mold was then inverted and the block removed to complete sample forming. The method of sample forming adopted i n this research precludes this solution. On the other hand closure of the top drain l i n e was suggested to enhance uniform deformation (Rowe and Barden, 1964). This remedy was not found effective i n this research and also by Green (1969). There appears to be no conclusive explanation for the phenomenon as well as the o apparent remedies. Review of previous studies strongly suggests that the end restraint error contribution to small stra i n response may not be serious. This view i s supported by conventional t r i a x i a l compression test results of free and fixed ends reported by Rowe and Barden (1964) and a similar comparison by Barden and Khayatt (1966) for t r i a x i a l extension tests. -Both studies suggest that the advantages of using free ends become apparent only for large s t r a i n conditions. The strain increment ratios are even much less affected (Barden and Khayatt, 1966; Rowe, 1971). Hydrostatic loading of samples with free ends was found to result i n about 10 percent higher volumetric strains when compared to similar but fixed end conditions by El-Sohby and Andrawes (1972). The results were based on tests made on specimens that have a height to diameter r a t i o of one. Thus, the underestimate of volumetric strains i n a conventional sample having a height to diameter ra t i o of two would be less than 5 percent. In addition, use of smooth polished metal platens with small diameter and centrally located porous stones, the techniques adopted i n the study, would further reduce the margin of error. Considering the degree of scatter evident i n the reported results of El-Sohby and Andrawes (1972); i t 26 appears that small s t r a i n response can be studied without resorting to the use of free ends. This not only s i m p l i f i e d experimental procedure but was also desirable because the associated bedding errors were eliminated as w i l l be described subsequently. 3.1.1.4 Bedding Er r o r Use of lubicated platens have been shown to contribute to serious bedding error i n the measurement of a x i a l deformations (Sarsby et a l , 1980). For the material tested i n t h i s study, the contribution of bedding errors to a x i a l deformation have been indicated to be comparable to membrane penetration i n obscuring r e a l volume change (Sarsby et a l , 1982). C a l i b r a t i o n and development of correction curves have therefore been advocated. Compressibilities i n measuring equipment and bedding between elements of the apparatus were i d e n t i f i e d to be the major components of aggregate bedding error (Daramola and Vaughan, 1982). However, bedding error i s not a function of sample height and i t s r e l a t i v e influence would diminish with use of larger sample heights. Up to 80 percent of measured a x i a l deformation i n samples with a height to diameter r a t i o of one may be due to bedding error i n small s t r a i n observations (Sarsby et a l , 1980). This estimate would be reduced to about 40 percent and would thus s t i l l remain s i g n i f i c a n t for samples with a height to diameter r a t i o of 2. D e t a i l s of equipment set up relevant to a x i a l deformation measurement adopted i n this study are shown i n Figure 3.2. The top cap and the loading ram were permanently attached. Use of small diameter stones not only minimized end r e s t r a i n t but also v i r t u a l l y eliminated movement due to improper seating of porous stones on to end platens. The load c e l l was located above the LVDT bracket such that LOAD CELL EYE LINK RECOIL SPRING I I LVDT BRACKET I I I I BUSHING Q P I CELL TOP PLATE STEEL RAM ALUMINIUM TOP CAP I PO S S POROUS STONE  SMOOTH ANODIZED SURFACE SAND SAMPLE 3.2 Details of A x i a l Deformation Measurement i t s compliance did not influence deformation measurement. T i l t i n g of the top cap was r e s t r i c t e d by the guide bushing. Design improvements and o v e r a l l test considerations were reviewed to optimize characterization of true s o i l behaviour. The influence of end r e s t r a i n t was minimized and lubricated platens were not considered necessary. This had the further advantages of v i r t u a l l y eliminating bedding errors as well as f a c i l i t a t i n g use of improved sample preparation techniques. 3.1.2 Stress Path Control The choice of stress paths that can be followed i n a conventional t r i a x i a l test are very r e s t r i c t e d . Conventionally, the c e l l pressure i s maintained constant. Compressional or extensional deviator stress i s applied under either a stress or s t r a i n controlled condition. Isotropic loading can be effected by simply increasing the c e l l pressure. As well, constant shear paths can be followed by applying the desired deviator stress and manipulation of either c e l l or back pressure. Other stress paths would require simultaneous changes and adjustments i n l a t e r a l and a x i a l s t ress. Independent control of c e l l and deviator stresses would tend to result in uneven stress path that would be d i f f i c u l t to r e p l i c a t e p r e c i s e l y . A stress path analog was developed to f a c i l i t a t e simultaneous control of c e l l and deviator stress such that smooth and repeatable stress paths could be followed. E s s e n t i a l features and underlying p r i n c i p l e s of the system are presented below. Additional d e t a i l s including operation and performance have been described previously (Vaid and Negussey, 1983). 29 3.1.2.1 Description of the System A schematic layout of the stress path device i s shown i n Figure 3.3. The loading system consists e s s e n t i a l l y of an adjustable r a t i o , reversing and volume booster pneumatic relays, three pressure regulators and a double acting loading piston. The signal pressure regulator R 3 controls the c e l l pressure P 3 which i s also the input pressure to the pneumatic relays. The relays ultimately deliver the output pressure to the a i r piston, which applies the desired deviator load to the sample. The signal pressure regulator can be operated either manually or coupled to a variable speed motor i f a constant rate of loading of the sample i s desired. The pneumatic relays transform the signal pressure P 3 to an output pressurure Po i n the following manner: P Q = S P 3 Adjustable Ratio Relay (3.1) PQ = K - P 3 Reversing Relay (3.2) P Q = K + P 3 Volume Booster Relay (3.3) i n which S and K are positive constants. If the reversing relay i s used concurrent with the r a t i o relay, the output pressure obtained w i l l be: P 0 = S(K - P 3 ) (3 .4) Since the output pressure i s r e s t r i c t e d to be positive; K > P 3 . AIR S U P P L Y ARR RR — < e VBR -6) - * — ® © R T ) V J 13 & * V .4 V 2 P = P, - A = A-| -A = Ac AR—<R> LVDT DOUBLE ACTING AIR PISTON L C • - JL DT -A= A t SOIL S A M P L E X P p =U P = P r . A = A c - C E L L ch BASE C L A M P IE 75" i / / / / / / / F i g . 3.3 Schematic of the Pneumatic Stress Path Analog System LEGEND FOR FIGURE 3.3 SCHEMATIC OF STRESS ANALOG SYSTEM SCRIPTS A - AREA ARR - ADJUSTABLE RATIO RELAY LC - LOAD CELL (M) - MOTORIZED OR MANUAL P - PRESSURE R - REGULATOR RR - REVERSING RELAY T - TRANSDUCER V - VALVE VBR - VOLUME BOOSTER RELAY SUB-SCRIPTS BOTTOM PISTON CHAMBER CELL OUTPUT PORE OR BACK PRESSURE RAM ROD SAMPLE TOP PISTON CHAMBER SHUT-OFF VALVE THREE WAY VALVE B -C -0 -P -r -R -S -T -SYMBOLS ® -e -S i m i l a r l y , use of the volume booster i n place of the r a t i o relay results i n P 0 = S(K + P 3) (3.5) Both K and P 3 are positive and a r e s t r i c t i o n on their r e l a t i v e magnitudes i s not necessary. For components used i n t h i s investigation, the r a t i o relay factor, S, can vary between 1/30 and 30. The K factors f o r the reversing and volume booster relays can be between 15 to 1000 KPa, with the upper range being usually limited by available supply l i n e pressure. 3.1.2.2 UNDERLYING PRINCIPLES For the t r i a x i a l test set up shown In Figure 3.3, l e t A± = Upper chamber area of loading piston A 2 = Lower chamber area of loading piston ( t h i s i s les s than Ai by an amount equal to piston rod area) A g = Sample area A r = Sample loading rod area Pj = Pressure i n the upper piston chamber ?2 = Pressure i n the lower piston chamber P 3 = C e l l pressure u = Porewater pressure i n the sample V e r t i c a l equilibrium of the sample then requires v; = p i A i - p 2 A 2 + V A S - V - v (3.6) or a; = ^ r * - Vr* + p 3 ( 1 - ir> - u ( 3 , 7 ) s s s also aj; - P 3 - u (3.8) In Equations (3.7) and (3.8), and are respectively the v e r t i c a l and l a t e r a l e f f e c t i v e stresses i n the t r i a x i a l sample. These stresses would be expressed incrementally i n the following form: A a v = ^ A " 0 " A P 2 ( A ~ ) + A P 3 ( 1 " IT* " AU ( 3 ' 9 ) s s s Ao^ = AP 3 - Au (3.10) Since the porewater pressure i s constant i n drained tests, Au = 0 and ACT' = APo. h 3 The incremental d i r e c t i o n of a desired stress path would be given by: Ao' APi Ai AP2 A 2 A ( - ) — "IT) <3-U> s ACT/ A P 3 ^ A ' ' A P 3 U ' ' ^ A h J s 3 s By holding either of ?2 o r p l constant, increments i n the other can be expressed i n terms of AP3 as A Ao' A j ' J A h s A P 1 = A P 3 t l f ^ + A 1 " 1 W (3-12) 3 4 when = 0; and A Ao' A A P 2 = A P 3 [ - - i (3.13) * h s when A P 1 = 0. The terms within the square brackets can be determined and would remain constant for a lin e a r path. Given i n i t i a l values of ?i , P 2 and P 3 ; o o o equations 3.12 and 3.13 can be converted to expressions i d e n t i c a l to equations 3.4 or 3.5. Thus smooth stress paths can be followed by co n t r o l l i n g only the c e l l pressure P 3 and choosing appropriate constants S and K on the pneumatic r e l a y s . It turns out that the r a t i o relay setting S i s a function of only the stress d i r e c t i o n whereas K for either the volume booster or reversing relays depends both on stress d i r e c t i o n and stress state. 3.1.3 Measurement Devices 3.1.3.1 Load and Deformation Monitoring Transducers were used to measure a x i a l load and monitor pressures together with gauges for v i s u a l inspection. The c e l l and backpressure were monitored by separate transducers. Two additional pressure transducers were required for setting and monitoring the stress path analog. A l l pressure transducers were calibrated with reference to a precision dead weight gage teste r . The load c e l l was calibrated against dead weight. A x i a l deformations were measured with a sensitive transducer suitable for detection of small displacements. The transformer was 35 mounted on to a fixed reference outside the influence of c e l l pressure v a r i a t i o n (Figure 3.2 and 3). The core reacted against but was not attached to a stubby bracket that was r i g i d l y clamped to the loading rod. A plexiglass polished b a l l was f i t t e d to one end of the core. The other end was inserted into the transformer through a centering c o l l a r . Between the b a l l end and the base of the c o l l a r a soft r e c o i l spring was placed. These arrangements and d e t a i l s as shown i n Figure 3.2 were found necessary to minimize t i l t i n g and drag of the core against the transformer wall. The displacement transducer was calibrated against a micrometer and was also checked i n p a r a l l e l with a precision d i a l gauge. Displacements of the order of 1 x 1 0 - 3 mm can be detected by the transducer r e l i a b l y . Volume change was measured by monitoring the height of water column In a 4 mm bore grduated pipette. A standby 2 mm bore pipette was also available for more accurate volume change measurement. This pipette was used primarily In the testing of small 35.6 mm diameter samples for membrane penetration study to r e s u l t i n a comparable accuracy of volumetric s t r a i n measurement between sample s i z e s . A low range (35 KPa) d i f f e r e n t i a l pressure transducer was used to monitor volume change e l e c t r o n i c a l l y by sensing the height of water column within the pipette. Volume changes of 0.01 cc could be detected i n the 4 mm pipette, both v i s u a l l y and by the d i f f e r e n t i a l transducer. 3.1.3.2 Standard References and data A c q u i s i t i o n A l l transducers were excited by a common power supply that was set at 6.00 v o l t s . Supply pressure to the test frame was regulated to a maximum of 750 KPa and below l i n e operating range. The laboratory 36 represented a temperature controlled environment to within ± 1/4°C. These conditions remained standard for a l l t e s t s . The zero load output of the two transducers used with the stress path analogue was balanced to 1.000 mv. A l l other devices were read above zero load output without balancing. Pore and c e l l pressure transducers were referenced to a water l e v e l at sample mid-height. The d i f f e r e n t i a l pressure transducer reference was read with atmospheric common pressure and at zero water l e v e l i n the volume change pipette. The suspended load c e l l output was taken as reference zero load. At the beginning and end of each t e s t , zero references were read three times. An average of these readings was taken as the zero reference for a t e s t . These i n i t i a l and f i n a l reference readings were e s s e n t i a l l y i d e n t i c a l for a l l tests reported. Test data was acquired on a cassette tape. At the beginning of each t e s t , I n i t i a l dimensions and sample i d e n t i f i c a t i o n were entered on to the casette through a desk top computer. Data scanning was then i n i t i a t e d by manual or automatic t r i g g e r i n g at prescribed time i n t e r v a l s . A t o t a l of ten channels were monitored i n each scan. The f i r s t two carried time and predata information. Transducer readings were contained i n the subsequent seven channels whereas the l a s t channel monitored the e x c i t a t i o n voltage. Each current and previous scans were displayed on the video monitor of the data a c q u i s i t i o n system. This feature permitted ready comparison of successive readings and was useful for setting the stress path analog parameters. At the end of the t e s t , the data acquired on cassette was transferred to disk for processing and duplicate storage. 37 3 . 2 Material Tested and Sample Size 3 . 2 . 1 Material Tested A n a t u r a l l y occurring medium s i l i c a sand from Ottawa, I l l i n o i s , commonly known as Ottawa sand, was used i n this study. Its mineral composition i s primarily quartz with a s p e c i f i c gravity of 2 . 6 7 and a material f r i c t i o n angle of about 2 5 ° . Individual p a r t i c l e s are rounded and the average p a r t i c l e s i z e D ^ Q i s 0 . 4 mm. Reference maximum and minimum void r a t i o s used are 0 . 8 2 and 0 . 5 0 , r e s p e c t i v e l y . A l l test samples were formed using a fixed oven dry weight of 6 4 0 gms. No detectable difference i n gradation between a fresh batch and one subjected to several recycling could be observed, as shown i n Figure 3 . 4 . Similar conclusions were reached i n studies at confining pressures of up to 2 5 0 0 KPa by Vaid et a l ( 1 9 8 3 ) . This i s due to quartz being a r e l a t i v e l y hard mineral, and i n d i v i d u a l sand p a r t i c l e s are rounded. Both gradation curves conform to the l i m i t s s p e c i f i e d by ASTM Designation C - 1 0 9 - 6 9 for Ottawa sand and p a r t i c l e crushing was not detectable. 3 . 2 . 2 Sample Size The r e l a t i v e s i g n i f i c a n c e of equipment related experimental error can be minimized by a proper choice of sample s i z e . Aside from adopting a widely accepted height to diameter r a t i o of 2 to 1 , there i s often less emphasis on deliberate choice of sample s i z e . A number of previous studies have been based on 3 8 mm diameter samples. With reference to experimental l i m i t a t i o n s discussed previously, the influence of ram f r i c t i o n and error due to membrane penetration decrease with increasing sample diameter. End r e s t r a i n t effects 38 MEDIUM SAND h 4 20 28 40 48 100 140 200 Diameter ( mm) Fig.3.4 Grain Size D i s t r i b u t i o n Curve for Ottawa Sand larger sample height. Disturbance to the sample during set up would be less severe for a larger diameter sample. Given the equipment c a p a b i l i t i e s discussed previously, a choice of sample size that would promote r e l i a b l e detection of comparable a x i a l and volumetric s t r a i n magnitudes was desired. Taking into account the above considerations a 63.5 mm diameter sample with, an approximately, 2 to 1 height to diameter ra t i o was adopted. As w i l l be described subsequently, the actual sample height varied depending on the r e l a t i v e density of Interest. This i s because the s o i l weight and diameter remained the same and density was controlled by change i n height. Overall, this choice of sample size was found favourable i n reducing the r e l a t i v e significance of equipment related test errors. It also permitted r e l i a b l e detection of a x i a l and volumetric strains at comparable levels and i n the order of 1 x 10" 5. 3.3 Testing Procedure 3.3.1 Sample Preparation At a given instant i n time, a test sample i s assumed to represent a point i n stress space. The sample preparation technique must therefore promote development of homogeniety of density and structure within an individual sample. Experimental investigations i n v a r i a b l y require testing of several samples. A high degree of sample reproduction c a p a b i l i t y i s therefore essential for meaningful correspondence and r e p e a t a b i l i t y of test r e s u l t s . The s t r e s s - s t r a i n response of sand i s sensitive to variations In sample preparation technique. Researchers have i d e n t i f i e d a number of factors to be s i g n i f i c a n t i n sample forming. Method of placement, 4 0 height and rate of pouring have been suggested to be important (Kolbuszewski, 1948; Muira and Toki, 1982; M u l i l i s et a l , 1975; Tatsuoka et a l , 1982). The medium of deposition, whether a i r or water, aff e c t s structure and i n i t i a l packing (Kolbuszewski, 1948 and 1948a). Directi o n of pouring has been shown to determine grain orientation and hence inherent anisotropy (Arthur and Menzies, 1972; Yamada and Ishihara, 1979; Oda, 1972). Placement water content and method of d e n s i f i c a t i o n also control sand behaviour. There i s a broad disagreement regarding the r e l a t i v e importance of some of these fa c t o r s . This may i n part be because evaluations have been based on empirical r e s u l t s and t h e i r degree of importance depends on the method of sample preparation. Of a l l current disagreements, however, the e f f e c t of height of drop i s the most unclear and contradictory. I t s influence on i n i t i a l void r a t i o was found to be s i g n i f i c a n t (Kolbuszewski, 1948 and 1948a; Tatsuoka et a l , 1982; Ishihara and Towhata, 1983) to the extent that desired i n i t i a l densities were controlled by height of pouring. On the other hand, results that have Indicated a minor (Mululis et a l , 1975) to n e g l i g i b l e (Muira and Toki, 1982) influence of height of drop on i n i t i a l density have also been reported. This contentious issue was examined from t h e o r e t i c a l and experimental considerations i n a limited background study (Vaid and Negussey, 1984a). Terminal v e l o c i t y of sand grains would be attained almost instantaneously and within n e g l i g i b l e drop height when plu v i a t i o n takes place i n water. However, for pluviation through a i r , the influence of height of pouring on i n i t i a l density was found to vary from s i g n i f i c a n t to i n s i g n i f i c a n t i n the low range of drop heights. 41 P l u v i a t i o n through water would thus be more e f f e c t i v e i n promoting homogeneity and r e p e a t a b i l i t y of test samples and was therefore used i n th i s study. 3.3.1.1 Adopted Method The basic method of sample preparation adopted l n t h i s study has been i n long use at UBC. Further refinements were introduced to enhance attainment of desired objectives of s i m p l i c i t y and precise r e p l i c a t i o n of structure as well as density. Samples were formed with extreme care and diligence to ensure r e p e a t a b i l i t y and consistency of test r e s u l t s . The adopted method may also be considered as an improved version of that suggested by Chaney and M u l i l i s (1978). A fixed mass of oven dried sand was weighed i n a dry f l a s k . Water was added to near f i l l i n g . The sand water mixture was then boiled for about 20 minutes. After cooling to room temperature, i t was kept under vacuum u n t i l sample forming. Porous stones were also boiled i n water and cooled to room temperature. De-aired water was used i n a l l saturation and drainage l i n e s . Samples of 63.5 mm diameter were formed i n rubber membrances that were stretched and folded over a s p l i t mold. Membranes having 60 mm diameter and 0.3 mm thickness were cut to 190 mm lengths. The membrane was sealed to the base of pedestal and folded over the former to an established l e v e l . It was stretched to the wall of the s p l i t mold by applying a small vacuum suction. The cavity thus formed was f i l l e d with de-aired water. The e f f e c t i v e vacuum suction applied to the boiled sand was released. A special tapered rubber stopper with a glass tube nozzle was f i t t e d to the f l a s k . De-aired water was added u n t i l l overflow. By gentle overturning of the flask preventing flow, 42 the nozzle t i p was allowed to make shallow penetration of the water surface within the former. The f l a s k was then fastened to a movable stand with a fi x e d clamp. Deposition of sand proceeded under g r a v i t a t i o n a l influence and mutual displacement with water. A l l samples were formed by deposition from the top and center of mold. However, when the fl a s k was near empty, the stand was moved l a t e r a l l y to produce a l e v e l surface. The diameter of the mold, membrane thickness and dry weight of sand were constant. Thus, desired r e l a t i v e densities were obtained by c o n t r o l l i n g specimen heights. P r i o r to assembling the sample former, a target height was established. This was done by placing an aluminum dummy sample of known height between the bottom pedestal and top cap. A f l a t head nut was f i t t e d temporarily to the load end of the loading rod which was permanently attached to the top cap. A reference reading was then taken on top of the loading rod using a d i a l gauge that was mounted on a movable stand. Knowing the height of the dummy sample, the d i a l target reading required for a sample of the desired height was calculated. Following deposition and l e v e l l i n g , the f l a s k was removed from the stand and the top cap was put i n place. The d i a l gauge was located on top of the loading rod. The specimen was then densified by high frequency low amplitude v i b r a t i o n induced along the top of c e l l base. This procedure was found e f f e c t i v e i n preventing t i l t i n g of the top cap and uneven settlement. D e n s i f i c a t i o n with top cap i n place and thus e f f e c t i v e confinement was found useful i n promoting development of uniform densities within test samples (Vaid, 1981, 1983). Both top and bottom drainage l i n e s were kept open and change i n height was continually monitored during d e n s i f i c a t i o n . On approaching the target 43 height, d e n s i f i c a t i o n was terminated. From several attempts, points of termination that would lead to a target height to c e l l assembly and desired i n i t i a l r e l a t i v e density were determined. At the appropriate termination reading to target s e t t i n g , the membrane was pulled over the top cap and sealed with an 0 r i n g . Keeping the base drainage open, the top drainage l i n e was plugged. A vacuum of 17 KPa was then applied to the sample along the base drainage l i n e . Thus the sample was given e f f e c t i v e confinement by v i r t u e of the applied vacuum and the mold was dismantled. Height monitoring was subsequently terminated and the t r i a x i a l c e l l was assembled. The base drainage l i n e was shut o f f to maintain confinement and the vacuum l i n e was disconnected. The c e l l was f i l l e d with de-aired water and the sample preparation phase was completed. 3.3.2 Sample Set-up After placing the t r i a x i a l c e l l within the loading frame, i n i t i a l readings of a l l transducers were taken and the c e l l l i n e was connected to the c e l l pressure system. A c e l l pressure of 20 KPa was applied and the confining vacuum was released. The sample drainage l i n e was then connected to the volume change device, a i r free, and the loading ram connected to the loading piston. An intermediate scan of_ a l l transducers was taken. Volume change during sample connection was estimated by assuming isotropy from height difference between the intermediate and i n i t i a l scans. With the sample maintained undrained, saturation of the specimen and membrane leakage were checked. The sample was then brought to an e f f e c t i v e hydrostatic stress state of 50 KPa with a back pressure of 100 KPa. When desired to bring the stress state to a stress r a t i o , R, a deviator stress was applied under drained conditions, while maintaining the e f f e c t i v e confining pressure constant. A reference reading of a l l transducers was made. This stress state was taken as the i n i t i a l stress state for the te s t . A l l of the above steps were carried out with continued recording of associated deformations. Sample dimensions at the i n i t i a l stress state were used i n s t r a i n c a l c u l a t i o n s . Procedures were c a r e f u l l y repeated to enhance sample reproduction and consistency of test r e s u l t s . 3.3.3 Repeatability of Test Results Experimental observations and conclusions derived therefrom depend on consistency and r e p e a t a b i l i t y of te s t s . Repeatability of otherwise consistent test r e s u l t s on sand depend on reproduction of r e l a t i v e density, r e p l i c a t i o n of structure, measurement accuracy and exact duplication of test routine. Through procedures described previously, the weight of sand grains and i n i t i a l sample dimensions can be controlled to enable reproduction of r e l a t i v e density to within one percent of the desired target. In order to re p l i c a t e grain structure as c l o s e l y as possible, the same sample forming technique was followed throughout. One int e r e s t i n g feature that was adopted i n the testing routine was the repeated monitoring of deformations at constant e f f e c t i v e stress and following a p p l i c a t i o n of a stress increment. Although time independence i n the sense of sand compared to clay behaviour i s acceptable when considering a long time frame, time dependent effects may not be Ignored i n a laboratory s e t t i n g . Depending on stress l e v e l and stress paths, various amounts of time 45 dependent deformation was observed to occur at constant e f f e c t i v e stress and as a consequence of s l i p and load shedding. Only time independent readings were associated with corresponding stress increments. Appropriate measurement devices were selected, test equipment was modified and testing techniques were improved i n order to enhance accuracy i n the application of stress and measurement of r e s u l t i n g deformations. The r e p e a t a b i l i t y of test results i s an o v e r a l l manifestation of the above tasks. Some results from repeated testing of i d e n t i c a l samples i n hydrostatic compression are presented i n Figure 3.5. Excellent r e p e a t a b i l i t y may be noted i n mean normal stress against both volumetric and a x i a l s t r a i n p l o t s . Differences i n both a x i a l and volumetric s t r a i n observations at equal mean normal stress show a slow rate of accumulation with s t r a i n magnitude. Individual test results are smooth and well defined and these differences have therefore l i t t l e e f f e c t on description of incremental behaviour. Response of two Id e n t i c a l samples to constant mean normal stress shearing i s shown i n Figure 3.6. The comparison includes only shearing response and represents r e s u l t s a f t e r hydrostatic compression, as i s customary. It may be noted that reproduction within the s t r a i n range of in t e r e s t i s exceptionally good. Behaviour representing d i f f e r e n t densities and stress l e v e l s for constant mean normal stress and hydrostatic compression are shown i n Figures 3.7 and 3.8, respectively. Clear and consistent relationships can be i d e n t i f i e d at small s t r a i n range. The results represent a r e l i a b l e magnification of small s t r a i n phenomena as observed from data obtained by d i r e c t measurement of deformations. 46 47 F i g . 3.6 Repeatability of Test Results i n Shear Loading 48 F i g . 3.7 Comparisons of Response to Shear at Three Levels of Mean Normal Stress 50 CHAPTER IV - EXAMINATION OF PREVAILING FRAMEWORKS FOR DESCRIBING SMALL STRAIN RESPONSE Previous experimental investigators have attempted to describe the stress s t r a i n response of sand within several frameworks such as incrementally e l a s t i c , e l a s t o - p l a s t i c and p a r t i c u l a t e . For the most part, such frameworks have been derived from d e t a i l e d study of behaviour either at very small s t r a i n s , less than 1 x lO"* 4; or with emphasis at large s t r a i n response, above 1 x 10~ 2. Sometimes behaviour at these two extremes has been force connected, such as i n attempts to specify dynamic moduli i n terms of shear strength. Otherwise, each region has been treated separately neglecting the existence of the other. Such a s i m p l i f i c a t i o n meets d i f f i c u l t y , however, when the range of in t e r e s t i s extended from either side to include the small s t r a i n range, 1 x 10 - l + to 1 x 1 0 - 2 . A v a r i e t y of c a r e f u l l y conducted tests on sand covering small s t r a i n response are presented i n this chapter. The usefulness of widely accepted frameworks and s i m i l a r i t y of behaviour to common but fundamental assumptions w i l l be examined. 4.1 Incremental E l a s t i c Representation In a vast majority of current geotechnical analyses, sand at given void r a t i o Is id e a l i z e d as an incrementally l i n e a r e l a s t i c i s o t r o p i c continuum. Appropriate moduli for incremental equations are prescribed, as state variables i n stress s t r a i n space, using suitable a n a l y t i c expressions that are derived from large s t r a i n data. The usefulness of this procedure for describing small s t r a i n moduli under u n i a x i a l loading conditions as well as the influence of stress path on 51 small s t r a i n moduli w i l l be examined. A l l experimental results and conclusions therefrom are based on tests at a common r e l a t i v e density of 50 percent. 4.1.1 Moduli i n Un i a x i a l Loading Small s t r a i n test results from conventional t r i a x i a l compression tests at d i f f e r e n t levels of confining stress (03) are shown i n Figure 4.1. In Incremental e l a s t i c approximations, such s t r e s s - s t r a i n r e s u l t s are most often simulated by hyperbolas. A hyperbola i s described by an equation In which the dependent and independent variables are related by two parameters. The f i r s t parameter i s the ultimate value of the dependent va r i a b l e . In hyperbolic simulation this parameter i s considered to be spec i f i e d by the f a i l u r e c r i t e r i a appropriate for the confining stress i n question. The second parameter i n a hyperbolic equation i s given by the i n i t i a l slope, E^, of the hyperbolic curve. 4.1.1.1 I n i t i a l Moduli, E x I n i t i a l slope and thus modulus (E^) i s a key parameter for a hyperbolic description. I n i t i a l moduli are determined from transformed p l o t s , such as Figure 4.2(a) for the test at a*j = 50 kPa. It may be noted that the data points do not form a straight l i n e f i t . Thus, the s t r e s s - s t r a i n curve i s not pe r f e c t l y hyperbolic. For sand, such an imperfect f i t i s i n general the rule rather than the exception. As a r e s u l t , i n i t i a l moduli so determined are independent of small s t r a i n data. Neither the improved accuracy nor the a v a i l a b i l i t y of ad d i t i o n a l small s t r a i n results would lead to a better d e f i n i t i o n of E . F i g . 4.1 Results of Conventional T r i a x i a l Tests at Different Confining Pressures 54 On considering test results i n Figure 4.1 i n the region of small s t r a i n , 1 x 10 - l + to 1 x 10~ 2 alone; a separate straight l i n e f i t and i n i t i a l moduli can be noted, as shown i n Figure 4.2(b). These re s u l t s suggest that small s t r a i n response can also be approximated by a hyperbolic function which i s separate from that for large s t r a i n . Use of two separate i n i t i a l moduli and thus a dual hyperbolic f i t to the data would appear to r e s u l t i n an improved s t r e s s - s t r a i n r e l a t i o n s h i p . However, whereas the additional parameter has been prescribed by f a i l u r e c r i t e r i a i n large s t r a i n considerations, the corresponding parameter can not be sp e c i f i e d readily for small s t r a i n data. Hence a dual hyperbolic representation would appear to require an a l t e r n a t i v e assumption for the required second parameter. Such an alt e r n a t i v e approach i s considered l a t e r i n Chapter VI. 4.1.1.2 Relationships Between E. and E E ± max There has been some uncertainty about the meaning of E^ and i t s rela t i o n s h i p with E m a x> as derived at extremely small s t r a i n (of the order of 1 x 10~ 6) from resonant column t e s t s . Before proceeding, however, i t would be useful to restate that the E^ determined from large s t r a i n considerations i s a f i c t i t i o u s modulus and has i n general l i t t l e to do with small s t r a i n response. In the following comparison of E to E ; unless spe c i f i e d otherwise, a reference to E pertains max I I to i n i t i a l moduli determined from small, 1 x 10 _ 1 + to 1 x 10 - 2, s t r a i n data alone as shown i n Figure 4.2(b). Changes i n i n i t i a l moduli, E^, with confining pressure derived from Figure 4.2(b) are shown i n Figure 4.3. Secant moduli obtained d i r e c t l y from Figure 4.2(b) and at s t r a i n l e v e l s of 1 x IO"1* and 1 0 - 3 F i g . 4.3 V a r i a t i o n of S t a t i c Moduli with Strain Level and Confining Pressure 57 are also shown. Both and secant moduli at comparable s t r a i n l e v e l s may be noted to increase with confining s t r e s s . There i s however l i t t l e difference between and E at 1 x 10 - 1 + s t r a i n . In order to c l a r i f y questions regarding E m a x and E^, resonant column tests were conducted on t r i a x i a l specimen formed at the same r e l a t i v e density. Tests were made at corresponding confining pressures and a x i a l s t r a i n levels of 1 x 10" 6 to 10"1*. In order to allow ready comparison of secant Young's Moduli without recourse to estimation of Poisson's r a t i o or cross referencing of s t r a i n l e v e l s , only long i t u d i n a l excitations were applied. Thus overlap i n i n i t i a l void r a t i o , confining stress and s t r a i n l e v e l was maintained. Relationships between secant modulus, confining stress and s t r a i n l e v e l as determined from the resonant column test results are shown i n Figure 4.4. Secant moduli at comparable s t r a i n l e v e l s increase with confining s t r e s s . Furthermore, modulus degradation with s t r a i n appears to progress at a r e l a t i v e l y slow but sim i l a r rate_ at a l l confining pressures for s t r a i n levels between 1 x 1 0 - 5 and 10 - l t. Below a s t r a i n of 1 x 10~ 5 the rate of modulus degradation i s observed to increase with confining pressure. However, presentation of the data In Figure 4.4. i n semi-log form, Figure 4.5, shows that moduli can be considered r e l a t i v e l y constant below 1 x 1 0 - 6 s t r a i n and as i s commonly used t h i s form of representation f a s c i l i t a t e s determination of E r max These observations are, i n general, consistent with current understanding, as well as addit i o n a l r e s u l t s by L u i (1984). Values of E and E at 1 x 10 - l + a x i a l s t r a i n from resonant column max tests have been plotted against confining stress i n Figure \.6. Relatively independent of confining stress, the r a t i o of E to E at 500 ^ 400 . . 350 250 300 150 200 v~3 = 50 KPa 100 Dr = 50 % F I P . 4.4 Young's Moduli From Resonant Column Testing i n Longitudinal Mode Fig. 4 . 6 Change of Dynamic Moduli with Strain Level and Confining Pressure 61 1 x 10 -^ s t r a i n from resonant column tests i s about 1.3. Whereas, the corresponding r a t i o of to E at 1 x 10 - 1 + s t r a i n from t r i a x i a l t e s t i n g , Figure 4.3, i s between 1.0 to 1.1. E and E. as well as E °' ° ' ' max i at 1 x 10"1* from both resonant column and t r i a x i a l test results are comapred i n Figure 4.7a and b. E can be observed to be grater than max E^ by a factor of about 2.3 and at a s t r a i n l e v e l of 1 x 10""4* moduli from resonant column are higher than moduli from t r i a x i a l tests by a reduced factor of 1.6. Both E and E, are observed to increase nonlinearly with max i confining s t r e s s , Figure 4.7(a), and i n approximately the same proportion to each other. The re l a t i o n s h i p of E and E, with max i confining stress can also be viewed i n log-log space as i n Figure 4.8, wherein data points corresponding to resonant column and t r i a x i a l r e sults can be observed to f i t cl o s e l y along approximately p a r a l l e l straight l i n e s . This implies that both E and E. may be described by max I power functions of stress that have the same exponent but d i f f e r e n t c o e f f i c i e n t s . The exponents would be given by the slope of moduli and confining stress relationships i n log-log space, as i n Figure 4.8. For both results shown, the slope and thus power exponent i s about 0.47. It may be of Interest to note that this value i s i n the mid range of reported values for various sands and E^ from large s t r a i n considerations by Duncan et a l (1980). It i s also close to the exponent of 0.5 that i s often used to re l a t e moduli and confining stresses i n very small s t r a i n c y c l i c loading considerations (Hardin and Drenevich, 1972; Seed and I d r i s s , 1970). Because their power exponent i s the same, the r a t i o of E and E,, once established, would remain max i unchanged at corresponding confining stresses. These observations may be of some inte r e s t and possible usefulness but do not provide a basis 62 * 100 200 300 400 500 600 <r3' (KPa) Fie;. 4.7a Comparison of E and E . as Determined from Resonant r max x Column and T r i a x i a l Tests 63 Fig. 4.7b Comparison of Dynamic and St a t i c Moduli at 1 x 10 A x i a l S t r a i n Level with Confining Pressure 64 100 50 20 £ UJ 10 -4— X RESONANT COLUMN , E m a x TRIAXIAL TEST, E; Dr = 50 % J L i i I I I J I M i l l 10 20 50 100 (KPa) F i g . 4.8 Alternative Comparison of E and E. at Different ° max l Confining Pressures for i nterpreting discrepancies between E m a x and E^. Observations of s t a t i c and dynamic moduli from higher s t r a i n l e v e l s might lead to further i n s i g h t . Because of l i m i t a t i o n s i n d i r e c t measurement of small deformations, E^ was determined by extrapolation of secant moduli from s t r a i n levels above 1 x 1 0 - 5 to zero crossing. Observations of moduli at s t r a i n l e v e l s of 1 x 10 - 1 + would appear to provide an al t e r n a t i v e and more r e l i a b l e perspective for examining relationships between & m H X and E^. It was previously noted that s t a t i c and dynamic moduli at a s t r a i n l e v e l of 1 x 10 - l + d i f f e r by a factor of about 1.6. The primary cause for this difference appears to be related to stress h i s t o r y . The resonant column sample was subjected to no less than a few thousand cycles of loading and unloading because resonant frequencies for the sand tested and i n the range of confining pressures used were of the order of 250 Hz. Whereas, the t r i a x i a l sample was s t i l l within the f i r s t quarter of i t s f i r s t cycle of loading. Comparison i s therefore being made of i n i t i a l loading with reloading behaviour. The accumulated deformation during i n i t i a l loading would contain a greater proportion of non recoverable deformation than would be the case for reloading. Hence s t a t i c moduli at 1 x 10-1* s t r a i n would be less than corresponding dynamic moduli at the same s t r a i n l e v e l . I f extrapolation of moduli to zero s t r a i n were to be made from moduli that were derived from s t r a i n l e v e l s at and In excess of 1 x IO'1*, the extrapolated moduli at zero s t r a i n from i n i t i a l loadings would .clearly be much less than those obtained from c y c l i c loading. This difference i n intercept moduli i s s i m i l a r to the previously discussed difference i n i n i t i a l moduli for small and large s t r a i n considerations. This i s 66 because, as the strain amplitude approaches zero, cycl ic loading history becomes of no consequence. Hence, the true zero strain intercept moduli for static and cycl ic loading become one and the same; which means E should be equal to E . . The modulus degradation of max n i sand should therefore begin from E for both static as well as cycl ic max loading conditions. Therefore, E^ determined from small strain results must also be considered f ict i t ious as was the E derived from large o i strain data. Their usefulness in prescribing moduli within various segments of strain response is analogous therefore to the conceptual use of apparent cohesion in specifying failure strength for granular materials over a range of confining pressures. As is the case for apparent cohesion, E^ should vary to enable Improved correspondence between stress and strain . From observations so far, three i n i t i a l moduli that characterize broad behaviour classifications can be Identified. The f i r s t would be the i n i t i a l modulus determined by following conventional procedure and with emphasis on two data points corresponding to mobilization of 70 and 95 percent of shear strength (Duncan et a l , 1980). As pointed out previously, this modulus is generally derived from large strain observation at near fai lure and steady state conditions. It would therefore be inclined to be associated with significant s l ip deformations. At the other end of the spectrum, E would represent K ' max K an i n i t i a l modulus for a response region wherein the proportion of recoverable strains would be more significant. Of course, the transformation from justif ied approximation at one extreme to the other is a gradual one. However, because this gradual transformation is d i f f i cu l t to track and actual deformations related to response near E tend to be r e l a t i v e l y very small, i t has i n the past been max customary to u t i l i z e large s t r a i n representation throughout. This procedure has been found to invariably lead to overestimation of small s t r a i n response and has not been e n t i r e l y s a t i s f a c t o r y . Even for limited small s t r a i n results and considerations, the required use of f a i l u r e parameters i n conventional hyperbolic representation has been found objectionable. Alternative suggestions for d i s t o r t i o n of the stress s t r a i n curve and forced f i t t i n g of data have been introduced by Hardin and Drenevich (1972). These additional steps were required to overcome basic inconsistencies i n procedure and are a result of mixing small and large s t r a i n response. It has been shown, Figure 4.2(b) that t r a n s i t i o n a l i n i t i a l moduli representing an approximate response region between 1 x 10 - l + and 1 x 10~ 2 s t r a i n can be i d e n t i f i e d . Both recoverable and non-recoverable deformations would assume r e l a t i v e s i g n i f i c a n c e and this s t r a i n region i s i n general of greater relevance to many deformation problems. These i n i t i a l moduli have further been shown to maintain a r e l a t i v e l y fixed proportion to E ^ values at corresponding confining pressure. However, adoption of these i n i t i a l moduli within current procedure would require further development r e l a t i v e to s p e c i f i c a t i o n of slopes i n transformed plots, as noted previously. 4.1.1.3 Comparison of E , E., E, and E. ; m a x - — i - — i u i r Conventional t r i a x i a l and resonant column tests were made at a common r e l a t i v e density of 50 percent i n order to examine relationships between alternative i n i t i a l Young's Moduli of E , E., E. and E. . max i i u i r 68 Test r e s u l t s from c y c l i c loading i n conventional t r i a x i a l paths and from which E,, E. and E, have been determined are shown i n Figure i i u i r 4.9. These results are presented with E and secant moduli at a max s t r a i n of 1 x 10 - l + that were obtained from resonant column tests as shown i n Figure 4.10. As argued previously, d i r e c t measurement of deformations cannot be considered suitable for r e l i a b l e evaluation of moduli at well below the measurement accuracy of the testing apparatus. At the same time, for a given confining pressure and i n the v i c i n i t y of zero s t r a i n ; E^, i n i t i a l loading modulus; E ^ u « i n i t i a l unloading modulus; and E ^ , i n i t i a l reloading modulus; must a l l converge to E m a x . In r e a l i t y and because of measurement l i m i t a t i o n s ; E., E. and E. , shown i n Figure i ' i u i r 4.9 represent secant moduli at s t r a i n l e v e l s of about 1 x 10-1*. Moduli from resonant column testing are determined from a mean slope of a hysteresis loop and represent a secant modulus. Both E^and E ^ f as well as E at 1 x l O - 4 s t r a i n from resonant column testing are s i g n i f i c a n t l y lower than E and thus r e f l e c t modulus degradation with s t r a i n , max & ' Figure 4.10. The extent of degradation i s more severe for E^ as opposed to E^^. While there i s a r e l a t i v e l y large difference between E^ and for f i r s t reloading, further s t i f f e n i n g on subsequent reloadings was small as E^ r appeared to increase very l i t t l e . As may be noted, E i r Is s l i g h t l y below E at 1 x 10 - 1 + s t r a i n from resonant column re s u l t s and t h i s small difference would appear to be the effect of s t i f f e n i n g due to a large number of repeated loading and unloading. On the other hand, E, i s approximately equal to E and there i s Iu r r J max 69 Fi g . 4.9 Loading and Unloading Response i n a Conventional T r i a x i a l Path 70 500 ~ <? 4 0 0 Q_ UJ (fi 3 O O 3 0 0 2 0 0 100 LEGEND A TRIAXIAL LOADING B TRIAXIAL RELOADING C TRIAXIAL UNLOADING • E ^ , R E S O N A N T COLUMN - — E AT I x 1 0 " 4 STRAIN, RESONANT COLUMN E initial . TRIAXIAL S M A L L STRAIN A , B , C - SEE FIG. 4.9 D r = 5 0 % 100 200 300 4 0 0 (TjtKPa) 500 600 Fig;. 4.10 A Conroarison Between E and Various I n i t i a l Moduli, E. max l therefore very l i t t l e modulus degradation with s t r a i n during unloading. For p r a c t i c a l purposes, both E ^ u and E^ can be considered to be independent of deviator stress l e v e l and cycles of loading and unloading. This implies r e l a t i v e l y unique values of E^, E^ f, E^^ and E would be associated with each confining stress and void r a t i o , max Furthermore, because E i s approximately equal to E ^ , r e l i a b l e estimates of E can be obtained from small s t r a i n s t a t i c test max r e s u l t s . It should be noted, however, that behaviour and relationships so far observed pertain to conventional t r i a x i a l tests and thus do not r e f l e c t the possible influence of stress paths. 4.1.2 The Influence of Stress Paths The stress s t r a i n behaviour of sand has long been recognized to depend on stress paths. In many p r a c t i c a l s i t u a t i o n s , stress paths may d i f f e r from conditions of u n i a x i a l loading. Incremental approaches are usually employed l n numerical analyses to simulate response to a r b i t r a r y stress paths assuming isotropy and moduli to be state va r i a b l e s . These incremental evaluations are made from hyperbolic representations of s t r e s s - s t r a i n data from u n i a x i a l loading. One of the aims of the following experiments was to observe the general character of stress s t r a i n relationships from widely d i f f e r i n g stress paths. Another was the evaluation of incremental moduli at common stress states that were reached by following d i f f e r e n t stress paths. These moduli are then compared to moduli determined from u n i a x i a l tests i n order to assess the r e l a t i v e influence of stress paths and es t a b l i s h o v e r a l l trends. 72 4.1.2.1 Stress-Strain Relations in Different Paths The stress path analog and small deformation measurement capability developed in this research enabled reliable observation of behaviour in a variety of stress paths. Some typical stress paths investigated are shown in Figure 4.11, in which the conventional tri a x i a l path corresponds to label 2. A l l selected stress paths satisfy conditions of increasing or constant stress ratio, mean normal and shear stresses. There was no unloading in terms of any stress variable. As shown in Figure 4.12, these paths result in widely different stress-strain responses. Failure and deformation of sand are strongly influenced by stages in overall shear to mean normal stress ratio. Relationships between deviator stress and axial strain resemble a hyperbola only in paths that ultimately lead to a failure stage. On the other hand, increases in deviator stress in paths having an obliquity, well below failure (test #4 in Figure 4.12) lead to contraction. Moduli increase with strain level and a parabolic rather than hyperbolic approximation may be more suitable. Consequences of assuming moduli to be state variables are examined in the following section. 4.1.2.2 The Influence of Stress Paths on Incremental Moduli Considering conventional t r i a x i a l test results, for a given magnitude of deviator stress different moduli may be specified depending on the confining stress level. Moduli are evaluated at 7 3 F i g . 4.11 Stress Paths Investigated 74 F i g . 4.12 Stress Strain Response From Tests Performed Along the Various Stress Paths 75 stress points, with the i m p l i c i t assumption of path independence. To examine the influence of stress paths on moduli, tests were conducted to common stress states following paths shown i n Figure 4.13. Stress points C, G and F were reached i n two d i f f e r e n t ways by following either a constant confining or mean normal stress path. Point F was also reached following a path i n which the incremental stress r a t i o was held to 4. Successive intersections between t r i a x i a l and a path with an incremental stress r a t i o of 2 are represented as points A, B, C, D and E. These test results were used to evaluate and compare incremental moduli that have been determined along d i f f e r e n t stress paths at common stress point. which homologous stress points have been l a b e l l e d are shown i n Figures 4.14(a),(b) and ( c ) . However, moduli at common stress states determined from deviator s t r e s s - a x i a l s t r a i n space cannot be compared d i r e c t l y . Considering a simple l i n e a r e l a s t i c representation within a small neighbourhood of the in d i v i d u a l stress states; the incremental expression for tangent modulus would be: Corresponding plots of deviator stress against a x i a l s t r a i n on t 2w6o* (4.1) or 6a, 6a' 6e 5c 2u) (4.2) a a 76 ^(KPa) F i g . 4.13 Different Stress Paths and Common Points of Intersection 77 79 F i g . 4.14c Path Dependence of Moduli Evaluated at Common Stress Points: T r i a x i a l and Constant Mean Normal Stress Paths Moduli that would be determined from Figures 4.14a, b and c are then (4.3) a such that 5a' E t = E t + sT* ( 1 ~ 2 y ) ( 4 , 4 ) a for the special case of the conventional t r i a x i a l test, 6a^ = 0 and E f c = E£. In other cases E f c * E^.. For conditions wherein 6a' 6a. = 0; E' = 0 and E = -=—- (1 - 2u) which would lead to the d t t 6e a fa m i l i a r expression for tangent bulk modulus as: 't 3(1 - 2u) Letting r represent the r a t i o of stress increments, as 6a' r = ^—p ; then E' would be expressed i n terras of r, as r 6a' a and such that Et = E t ( ?r i ) ( 4 / 7 ) The parameter r defines the stress path under consideration. Thus, moduli at the same stress point but obtained from considering d i f f e r e n t stress paths and stress increment directions can therefore be compared through the use of Poisson's r a t i o , u. Equivalent Poisson's r a t i o s were found to l i e generally between 0.2 and 0.4. The v a r i a t i o n being a function of the combined e f f e c t s of stress l e v e l and stress path and also r e f l e c t s i n part the imposition of the simplifying assumption of isotropy. An average Poisson's r a t i o of 0.3 was used to determine corresponding moduli for d i f f e r e n t stress paths at the same stress states, aj" and = (aj - a 3 ) . A comparison of re s u l t s , as shown i n Figure 4.15, demonstrates that moduli determined from uni a x i a l loading results d i f f e r from those derived from considering other stress paths. In general, moduli for paths oriented to the l e f t of the conventional t r i a x i a l path (see Figure 4.11) would be overestimated. Consequently, predicted deformations for such paths would be lower than actual. With increasing s t r a i n l e v e l and on approaching f a i l u r e states, the r e l a t i v e difference i n moduli would diminish r a p i d l y . However, at this stage, deformations would be in t o l e r a b l y large and concern would focus on 82 F i g . 4.15 Comparison of Moduli Evaluated at Common Stress Points Following Different Stress Paths f a i l u r e rather than deformation. For stress states at small stress r a t i o where deformation considerations are important, the two moduli may be seen to d i f f e r by a factor as large as 2. On the other hand, equivalent moduli estimated from paths oriented to the right of the t r i a x i a l path are seen to be underestimated, Figure 4.15. Predicted deformations for such paths would thus be higher than actual and there i s a s l i g h t but consistent tendency for the r e l a t i v e difference i n moduli to increase with confining stress. Overall, i n a majority of p r a c t i c a l situations wherein assessment of deformations i s the primary objective, both shear and confining stresses tend to increase simultaneously over most regions. Thus, evaluation of moduli at discrete stress points assuming path independence would appear to lead towards exaggerated prediction of deformations i n a majority of cases. 4.1.3 Additional Remarks Experimental studies of moduli led to incremental techniques for i n t e r r e l a t i n g confining and shear stresses with a x i a l s t r a i n for stress conditions akin to the t r i a x i a l path (Duncan and Chang, 1970). Even under simpler stress conditions of a x i a l symmetry or plane s t r a i n , at le a s t one more s t r a i n component needs to be determined for complete correspondence between states of stress and s t r a i n . Poisson's ratios were found to vary with s t r e s s . Imposition of an average value enabled characterization of behaviour by a two parameter model, i n which dilatancy could not be simulated. Stress dilatancy features were l a t e r incorporated through a third parameter for large stress r a t i o responses (Byrne and Eldridge, 1982). Such further refinement enhanced r e a l i s t i c reproduction of volumetric response. Incremental e l a s t i c representations are currently used i n a vast majority of geotechnical deformation problems. This i s a consequence not only of t h e i r convenience and s i m p l i c i t y but also because t h e i r performance i s often comparable to other constitutive models for sand, at least i n the current state of development (NSF/NSERC Workshop, 1980). Even with further development of more r e l i a b l e comprehensive models, s i t e conditions are i n many cases not defined well enough to j u s t i f y added complexity. It has therefore been suggested that such incremental e l a s t i c procedures w i l l remain relevant for drained deformation problems ( G r i f f i t h s and Smith, 1983). In this regard, further improvements r e l a t i v e to small s t r a i n response and also stress path e f f e c t s would appear to make them more e f f e c t i v e as well as enhance t h e i r current wide appeal. There are various disadvantages inherent i n incremental l i n e a r e l a s t i c approaches. Procedures for evaluating required parameters tend to ignore small s t r a i n response. The need to t i e together small s t r a i n parameters with f a i l u r e c r i t e r i a have been found l i m i t i n g . Although i d e a l i z e d as being i s o t r o p i c , sand i s i n general anisotropic. Thus rotation of p r i n c i p a l axes of stress, which so often occurs i n many r e a l s i t u a t i o n s , would have influence on stress s t r a i n response but cannot be accounted for by th i s form of representation. The stress path dependence of moduli has been shown, using test results and thus the notion that required parameters remain state variables i n stress does not hold. Comprehensive relationships that would overcome these various limitations are ultimately desired. These objectives have been pursued i n the past from both a continuum as well as particulate considerations. 85 4.2 Pa r t i c u l a t e Considerations Study of the gross deformation c h a r a c t e r i s t i c s of sand has also been attempted from fundamental considerations at the part i c u l a t e l e v e l . Because sand grains vary i n s i z e , are i r r e g u l a r l y shaped and have random packing, developments have of necessity remained empirical. The most s i g n i f i c a n t of contributions reported from this l i n e of development has been the stress dilatancy theory of Rowe (1962). For large s t r a i n and along selected stress paths, the stress dilatancy theory has been v e r i f i e d experimentally. However, i t s relevance to small s t r a i n response and along a r b i t r a r y stress paths of increasing stress r a t i o s t i l l remains i n doubt and w i l l be examined i n this section. The stress dilatancy equation, R = KD, proposed by Rowe originates from considerations of s l i d i n g between two p a r t i c l e s . S l i d i n g along a contact plane would be i n i t i a t e d when the corresponding Mohr-Coulomb shear strength i s exceeded. Rowe postulated that the orientation of the plane of s l i d i n g would be so as to minimize the r a t i o of incremental energy input to output along the p r i n c i p a l directions of stress and s t r a i n . This same mechanism was then extended to describe the deformation of a random assembly of ir r e g u l a r p a r t i c l e s i n contact. The p a r t i c l e s were considered to be r i g i d and deformations to be a result of non recoverable s l i p . Reported agreement with experimental r e s u l t s has been a mainstay for the stress dilatancy theory. The bulk of supporting experimental evidence has come from conventional t r i a x i a l t e s t s . With improved test procedures and test equipment, the stress dilatancy equation has been shown to describe sand behaviour for conditions of increasing stress 86 r a t i o s t a r t i n g from a hydrostatic state to peak and ultimate state (Barden and Khayatt, 1966). Confirmations were considered most favourable for dense states and upon reloading for loose conditions. Additional agreement with stress dilatancy was reported from plane s t r a i n and simple shear test results (Barden et a l , 1969a; Cole, 1967). These further v e r i f i c a t i o n s together with previous confirmations have led to the view that stress dilatancy i s path Independent. In a l l reported experimental v e r i f i c a t i o n s , regardless of stress path and stress r a t i o l e v e l , consideration has always been based on t o t a l s t r a i n s . It has been presumed that even upon reloading, s l i d i n g deformation predominates over e l a s t i c deformation of grains. The focus of experimental study i n this section has two parts. The f i r s t deals with a closer examination of stress dilatancy at small stress r a t i o and hence small strains i n conventional t r i a x i a l paths. This issue has been raised recently by Nova and Wood (1979) and Nova (1982) and reservations r e l a t i v e to the form of the stress dilatancy relationship at small stress r a t i o have been expressed on the bases of conceptual arguments. The second objective i s to examine the stress path Independence of the stress dilatancy equation at small s t r a i n . As i n a l l previous investigations, t o t a l strains w i l l be used throughout and no attempt w i l l be made to separate s l i d i n g and non s l i d i n g deformation components. 4.2.1 Stress Dilatancy l n Conventional T r i a x i a l Paths 4.2.1.1 The Influence of Stress Level A plot of volumetric s t r a i n against a x i a l s t r a i n from the results of a series of conventional t r i a x i a l tests at the same r e l a t i v e density 87 of 50 percent but d i f f e r e n t l e v e l s of confining stress are presented i n Figure 4.16. Incremental s t r a i n r a t i o s were obtained as tangent slopes Se i n Figure 4.16 from which dilatancy, D = (1 - -£~)> was determined. a Test data i n Figure 4.16 show that at low s t r a i n l e v e l s , D i s constant and appears to be r e l a t i v e l y independent of confining pressure. Thus i n the stress r a t i o dilatancy p l o t , Figure 4.17, r e s u l t i n g curves would have a common i n i t i a l v e r t i c a l segment. This implies a s t r a i n increment r a t i o c h a r a c t e r i s t i c to the stress path rather than the stress state would p r e v a i l at small s t r a i n . With Increasing s t r a i n l e v e l s , the stress dilatancy curves t r a n s i t i o n to a slope p a r a l l e l to but not coincident with K^, which corresponds to the i n t e r p a r t i c l e f r i c t i o n angle <j>^. Figure 4.17 also shows that the onset of increasing dilatancy, D, i s delayed with increasing confining pressure and that the spacing between subsequent p a r a l l e l l i n e s and the actual l i n e increases s l i g h t l y . There i s thus a discernable but s l i g h t dependence on confining stress i n the relationship betwen R and D beyond the small stress r a t i o response region. It i s generally recognized that at a given void r a t i o d i l a t i o n becomes suppressed with increasing confining stress. Therefore, at l e a s t i n a q u a l i t a t i v e sense, increasing density and confining stress have compensating e f f e c t s . The influence of density has been recognized i n stress dilatancy expressions by extreme settings r e l a t i v e to K and K . However, the possible existence of order on the basis cv y of confining stress and within these extreme l i m i t s of the stress dilatancy expression has never been suggested neither as an assumption or a hypotheses nor has i t been i d e n t i f i e d experimentally. The stress dilatancy theory i n a l l i t s stages of development has remained e n t i r e l y 89 90 oblivious to the possible influence of confining stress. At a given density, better agreement, i n the sense of approaching K^, with the stress dilatancy equation was previously reported for conditions of reloading as opposed to i n i t i a l loading (Barden and Khayatt, 1966). This opinion i s , however, not supported by the small s t r a i n reloading test tesults shown i n Figure 4.18. Again as for v i r g i n loading, dilatancy i n the small stress ratio region i s constant and very close to the value for i n i t i a l loading. Thus i n the stress dilatancy plot, Figure 4.19, the i n i t i a l v e r t i c a l and transition segments are s t i l l i n evidence. Although s t r a i n increment magnitudes associated with loading exceed those during reloading, both share a common str a i n Increment r a t i o . This implies that s t r a i n increment directions are unchanged and thus the stress dilatancy expression would not vary. 4.2.1.2 The Influence of Density A common i n i t i a l s t r a i n path was also followed i n loose sand as i l l u s t r a t e d by test results at a relative density of 30 percent and at two confining stresses, Figure 4.20. In a stress dilatancy plot, Figure 4.21, this again corresponds to an i n i t i a l near v e r t i c a l segment. However, this stress dilatancy relationship i n i t i a t e s to the l e f t of that for medium dense sand, and with increasing str a i n transitions towards p a r a l l e l alignment to K^ and K ^ at low and high confining stresses, respectively. Thus, the influence of confining stress i n a stress dilatancy plot i s similar but more significant for loose as opposed to medium dense sand. Therefore, comparison of loose and dense behaviour i n stress dilatancy terms appears to require q u a l i f i c a t i o n with respect to lev e l of confining stress. The 92 F i g . 4.19 Stress Dilatancy Plot of Conventional T r i a x i a l Test Results During Reloading 0 0.1 0.2 0.3 0.4 AXIAL STRAIN, £ Q (%) Fig. 4.20 Strain Paths for Conventional T r i a x i a l Tests on Loose Sand 94 F i g . 4.21 Stress Dilatancy Plot of Conventional T r i a x i a l Test Results on Loose Sand 95 approximately constant i n i t i a l dilatancy factors increase with r e l a t i v e density. For dense states, relationships would originate below the l i n e , whereas for loose conditions stress dilatancy curves i n i t i a t e above K • If the confining stress i s high, the relationship tends to be p a r a l l e l to K and for a lower confining stress the alignment approaches K^. Hence, with Increasing s t r a i n l e v e l , apparent agreement with stress dilatancy would depend not only on density but also on confining stress l e v e l . 4.2.2 Stress Dilatancy i n D i f f e r e n t Stress Paths The preceding experimental r e s u l t s at small stress' r a t i o and i n conventional t r i a x i a l paths indicated that the stress dilatancy r e l a t i o n s h i p did not continue as a straight l i n e to a state of hydrostatic compression. Constant t o t a l s t r a i n directions that were independent of confining stress but dependent on density were observed for low stress r a t i o states. The hypothesis and subsequent experimental v e r i f i c a t i o n s advanced to j u s t i f y the v a l i d i t y of the stress dilatancy equation for a l l states of increasing stress r a t i o s t a r t i n g from a state of R = 1 could not be supported. Further consideration of diverse stress paths i n which the requirement of monotonically increasing stress r a t i o , as stipulated by stress dilatancy, are s a t i s f i e d have been pursued to provide additional c l a r i f i c a t i o n . Results from constant mean normal stress, constant incremental stress r a t i o as well as conventional t r i a x i a l paths, a l l at an i n i t i a l r e l a t i v e density of 50 percent, have been plotted i n terms of volumetric against a x i a l s t r a i n to determine D for corresponding stress 96 r a t i o s , as i n Figures 4.22a, b and c. It may be noted i n Figure 4.22a that constant incremental stress r a t i o paths of 2 maintain a constant dilatancy factor of -0.5. The dilatancy term, D, would be zero for one dimensional deformation. A negative value of D implies both a x i a l and r a d i a l s t r a i n increments are p o s i t i v e . Hence energy i s input i n both p r i n c i p a l directions whether or not t o t a l or nonrecovered strains are considered. For an incremental stress r a t i o path of 4, the i n i t i a l dilatancy factor i s 0.37 (Figure 4.22b); whereas constant mean normal stress paths are i n i t i a l l y associated with a dilatancy of 0.83 (Figure 4.22c). Conventional t r i a x i a l paths which were considered i n the previous section maintain a dilatancy factor of about 0.42. These observations have been combined i n a stress dilatancy p l o t , Figure 4.23, which shows that a unique i n i t i a l dilatancy factor i s implied for each stress path or i e n t a t i o n . Paths of higher 5q/6p' may be observed to be associated with larger values of i n i t i a l dilatancy. The hypothesis that stress dilatancy i s v a l i d i n a l l paths of increasing stress r a t i o , from hydrostatic to f a i l u r e state does not appear to be supported by experimental evidence presented. Previous v e r i f i c a t i o n s of stress dilatancy were based on test results from conventional t r i a x i a l , simple shear and plane s t r a i n t e s t s . Of these, however, a hydrostatic stress state i s accessible only to the conventional t r i a x i a l t e s t . Both plane s t r a i n and simple shear tests i n i t i a t e from a state of one dimensional (K ) compression. Stress o ratios corresponding to K states are generally i n excess of 2. Hence o v a l i d i t y of stress dilatancy below a stress r a t i o of 2 could only be assesed on the basis of conventional t r i a x i a l test data. It may be noted (Figure 4.23) that the i n i t i a l dilatancy factor for conventional 97 F i g . 4.22a Strain Paths for Stress Paths of Incremental Stress Ratio of 2 86 Fig. 4.23 Relative Location of Different Stress Paths on a Stress Dilatancy Plot 101 t r i a x i a l paths i s located within close proximity of the upper and lower K £ v and bounds specified by stress dilatancy. This coincidence in conjunction with experimental l i m i t a t i o n s and uncorrected errors may have encouraged u n j u s t i f i a b l e extrapolation of experimental evidence i n previous studies i n support of stress dilatancy. 4.2.3 Further Remarks Experimental results presented above indicate that the stress dilatancy equation would not describe sand behaviour at small s t r a i n . An assembly of p a r t i c l e s subjected to external load would deform simultaneously i n r o l l i n g and s l i d i n g between grains as well as a consequence of e l a s t i c compression of the constituent grains. E l a s t i c compression and r o l l i n g would have d i r e c t i o n a l dependence on the causative s t r e s s . Whereas, s l i p deformation would require attainment of a threshold stress state. S l i d i n g within an assembly of p a r t i c l e s i s considered to occur between clusters rather than i n d i v i d u a l grains (Home, 1965). The size of s l i d i n g groups has been postulated to increase with density. In a medium and dense assembly, e l a s t i c deformation and r o l l i n g at unstable contacts would occur at small stress r a t i o . However, gross s l i d i n g would appear to be restrained u n t i l a large number of contacts reach l i m i t i n g equilibrium states simultaneously. At which time, the predominant mode of deformation becomes s l i d i n g . This view emanates from consideration of an assembly of r i g i d p a r t i c l e s i n regular packing wherein no deformation would occur as R i s Increased from i n i t i a l value of one u n t i l peak (Rowe, 1962). Simultaneous s l i d i n g would be i n i t i a t e d at a l l contacts upon attainment of peak stress r a t i o . Rowe's contention that stress 1 0 2 dilatancy would be applicable at a l l stages of R was questioned on the basis of this idealized framework (Roscoe and Schofield, 1964). In the case of sand, with decreasing density, a larger number of unstable contacts would exist and sl iding groups tend to be smaller. There would be more freedom for rotation and rearrangement and sl iding would be more localized. Since sl iding would not occur at a large number of contacts simultaneously, the overall stress ratio at which deformation due to sl iding becomes prominent would be lowered. Even at small stress rat io , therefore, both ro l l ing and sl iding become significant. Thus description of behaviour relative to a mechanism of sl iding alone would be inadequate. In this respect, previous observations relative to improved agreement with stress dilatancy upon reloading may be interpreted to be a consequence of reducing the relative significance of ro l l ing and formation of larger clusters. However, predominance of sl iding and development of larger clusters would in turn imply a larger magnitude of threshold stress ratio and thus less agreement with stress dilatancy. The linkage between cause and effect Is not straightforward and agreement or disagreement of experimental results with the stress dilatancy theory would appear to require cautious interpretation. At small stress ratio states, experimental results do not support previously reported straight line forms of the stress dilatancy relationship. In large measure, such disagreements with previous results stem from uncertainty in small deformation measurement, inadequate control of stress paths and unaccounted test errors. Improvements such as use of lubricated platten introduce bedding errors which, i f uncorrected, diminish the dilatancy term. This in turn encourages false alignment of results along a straight l ine . 1 0 3 4.3 Strain Separation for Elasto P l a s t i c Representation Idealizations of sand behaviour as either e n t i r e l y e l a s t i c or neglect of e l a s t i c s t r a i n s and consideration of s l i p components only have not been e n t i r e l y s a t i s f a c t o r y . There has therefore been int e r e s t i n modelling sand as an e l a s t o - p l a s t i c material. Fundamentally, the e l a s t o - p l a s t i c form of representation involves the separation, Independent analysis and re-assembly of e l a s t i c and p l a s t i c s t r a i n components. For an e l a s t i c - p l a s t i c material, the d i r e c t i o n of the nonrecovered vector i s a function of stress state and not the d i r e c t i o n of the stress increment vector. Pooroosharb et a l (1966) presented experimental results that show geometrically s i m i l a r p l a s t i c potential surfaces, c l o s i n g on the hydrostatic axis and f a i l u r e l i n e . Their work has been a basis for many subsequent model developments. However, i n spite of i t s claimed generality to small stress r a t i o response, i t has never been pointed out that actual v e r i f i c a t i o n for a p l a s t i c approximation of nonrecovered response was demonstrated only at a point close to i n c i p i e n t f a i l u r e and where e l a s t i c strains are n e g l i g i b l e . In view of i t s important implications, i t would appear to be of fundamental Interest to know as to whether the uniqueness of s t r a i n d irections to stress state hold or f a i l i n regions of small stress r a t i o . The following experiments are directed towards examination of this phenomenon. Results of loading to and unloading from increasing magnitudes of hydrostatic compression have been plotted i n s t r a i n space as shown i n Figure 4.24. An approximately linear s t r a i n path may be noted for accumulated residual s t r a i n . Such residual strains are generally considered to be associated with nonrecovered deformations. A linear. 105 s t r a i n path response f o r t o t a l , recovered and nonrecovered st r a i n s was noted by Rowe (1971) for constant stress r a t i o paths, including hydrostatic paths. A unique d i r e c t i o n of non recovered st r a i n s f o r a hydrostatic stress states implies r e l a t i v e independence of s t r a i n d i r e c t i o n s on mean normal s t r e s s . Hence p l a s t i c potentials when closing on the hydrostatic axis must maintain normality to this s t r a i n d i r e c t i o n as would be implied by the postulted geometric s i m i l a r i t y of s t r a i n p o t e n t i a l s . Figure 4.25 presents r e s u l t s i n s t r a i n space f o r loading and unloading to increasing l e v e l s of deviator stress along a conventional t r i a x i a l path and from a state of hydrostatic compression. For stress ratios maintained below 1.5, a d i s t i n c t nonrecovered s t r a i n increment d i r e c t i o n may be noted. If indeed nonrecovered deformations of sand were to be considered p l a s t i c , coincidence of nonrecovered s t r a i n paths at a stress point would be required. However, considering a hydrostatic stress state 0, Figure 4.26, from which hydrostatic and conventional t r i a x i a l stress increments are applied, two d i s t i n c t l y d i f f e r e n t non recovered s t r a i n directions r e s u l t . This would imply occurrence of in t e r s e c t i n g non recovered s t r a i n potentials at a stress point and that nonrecovered strains do not s a t i s f y requirements of c l a s s i c a l p l a s t i c i t y at low stress r a t i o states. Experimental observations advanced by Poorooshasb et a l (1966) i n support of a p l a s t i c c l a s s i f i c a t i o n of nonrecovered deformations i n sand are not comprehensive and only apply to near f a i l u r e stress states and at large s t r a i n . S l i p components of deformation are often considered to imply p l a s t i c s t r a i n . However, the occurrence of reverse s l i p on unloading 106 F i g . 4.25 Strain Paths for Conventional T r i a x i a l Loadings and Unloadings 107 o a. 6 0 0 5 0 0 4 0 0 3 0 0 200 100 a. UJ STRESS PATH NON RECOVERED STRAIN PATH ® TRIAXIAL COMPRESSION ® HYDROSTATIC COMPRESSION 100 200 300 4 0 0 500 600 C 3 '(KPa) F i g . 4.26 Comparison of Stress and Non-Recovered Strain Direction at the Same Stress Point 108 cannot be denied and r o l l i n g as well as s l i p would contribute to nonrecovered deformations i n sand. As discussed i n Section 4.2.3, e l a s t i c deformation of grains and r o l l i n g would maintain d i r e c t i o n a l dependence on the causative stress, whereas, s l i d i n g would depend on l i m i t equilibrium considerations and attainment of threshold stress r a t i o l e v e l s . As pointed out previously, Holubec's (1968) attempted v e r i f i c a t i o n of the e l a s t i c character of recovered strains was shown to be unsatisfactory (Merkele and Merkele, 1969; Coon and Evans, 1969) i n that incremental equations were not complete and t h e i r i n t e g r a l s were path dependent. The fin d i n g herein constitutes experimental evidence i n support of the c o r o l l a r y argument that nonrecovered strains are not p l a s t i c i n character. At extremely small s t r a i n s as encountered i n the lower l i m i t of strains i n resonant column testing, sand can r e a l i s t i c a l l y be i d e a l i z e d as e l a s t i c without need of e l a s t i c - p l a s t i c separation. However, at large s t r a i n s , experimental results from both continuum (Poorooshasb et a l , 1966; Lade and Duncan, 1976) as well as stress dilatancy considerations (Rowe, 1962; Barden and Khayatt, 1966) have shown that association of s t r a i n increment r a t i o s to stress states would be reasonably j u s t i f i e d and hence p l a s t i c c h a r a c t e r i zation. Indeed t h i s l a t t e r opinion holds true regardless of considering t o t a l or nonrecovered s t r a i n s . This would thus imply that at these stress states recovered s t r a i n s are s u f f i c i e n t l y small compared to s l i p strains and neither their i n c l u s i o n nor omission has much s i g n i f i c a n c e . In t h i s sense, the work of Rowe and Poorooshasb et a l are complementary even though they have d i f f e r e n t beginnings. Both recovered and nonrecovered s t r a i n s assume r e l a t i v e s i g n i f i c a n c e i n the s t r a i n range of 1 x 10 - 1 + to 1 x 10~ 2. Hence s t r a i n separation would be necessary to j u s t i f y an e l a s t o - p l a s t i c treatment. There are undoubtedly tremendous incentives to prefering an e l a s t o - p l a s t i c framework for s t r e s s - s t r a i n r e l a t i o n s h i p s . However, unless and u n t i l successful separation can be demonstrated, progress i n this d i r e c t i o n w i l l remain d i f f i c u l t . It would be f a i r to conclude that such evidence does not e x i s t , not from a lack of need or inte r e s t but perhaps because i t i s v i r t u a l l y impossible to achieve, as has been suggested by Zytynski, Randolph, Nova and Wroth (1978). 4.4 Concepts Based on Normalized Work On the basis of experimental observations and i n s p i r a t i o n from e a r l i e r work by Roscoe and Burland (1968); Moroto (1976) poroposed a new parameter for describing the shear deformation behaviour of sand. Expressing incremental work per unit volume done i n shear, for example i n a constant p' path, as <SW = p' 6e + q & (4.8) s v s and normalized by p' <SW —7-S= <5e + n 6e (4.9) 110 Moroto suggested that . dw W = Jb (4.10) s 1 a p' represents a new state parameter and thus independent of stress paths. This concept has been cen t r a l to some s o i l models proposed recently (Tobita and Yanagisawa, 1980; Varadarajan et a l , 1983; Ghaboussi and Momen 1984, 1982). Through further examination of experimental r e s u l t s (Momen and Ghaboussi, 1982) the slope, dW s has been indicated to be a material function dependent only on the physical c h a r a c t e r i s t i c s of the sand p a r t i c l e s . S has been found to be independent of r e l a t i v e density as well as mean pressure p'. It has further been postulated that normalized p l a s t i c work functions f o r di f f e r e n t stress paths would be related by functions of the Lode angle f o r stess path. If Indeed these suggestions hold true, discovery of t h i s new parameter for sand would be a major achievement. The following consideration i s intended to re-examine the possible usefulness of th i s concept. It would be convenient to begin by expressing Equation 4.11 i n the following equivalent form I l l de s - ^ + n (4.12) s As the sand approaches a c r i t i c a l state at <j> and thus large shear s t r a i n ; volume change would cease and Equation 4.12 would simplify to S = n (4.13) cv In stress dilatancy terminology the equivalent expression can be shown to be; 3 ( K -1) S = T K ^ T T ) <4'U> cv and this relationship would be v a l i d r e l a t i v e l y independent of density and mean normal stress. Both normalized work and shear- s t r a i n would continue to accumulate i n d e f i n i t e l y at this constant rate. Clearly, i n th i s context, S constitutes a material parameter. It may further be of interest to note that Equation 4.14 can be shown to describe reported r e s u l t s i n support of th i s parameter, S, without exception. What needs to be examined experimentally therefore i s the independence of S from density and mean normal stress l e v e l , p'; at small s t r a i n response regions. Small s t r a i n experimental r e s u l t s from constant mean normal stress paths at a r e l a t i v e density of 50 percent are presented i n the form of normalized work, Wg, against shear s t r a i n , as shown i n Figure 4.27. It may be noted that the relationship does not form a unique curve and S 112 2.5 to O Q LU N < 1.5 or o LEGEND 0.5 / 7 s% ' x ' \.*,0' XT V p' = 50 KPa X p' = 150 KPa • p' = 250 KPa O p' = 350 KPa • p' - 450 KPa D = 50 (%) 0.5 1.5 2 £ s x l 0 3 2.5 F i g . 4.27 The Influence of Mean Normal Stress on Normalized Work 113 i s shown to depend on mean normal s t r e s s . Furthermore, each constant p' curve i s non l i n e a r and S would not therefore be constant. Hence, S at small stress r a t i o does not represent a material parameter. However, a closer examination of Figure 4.27 suggests that constant stress r a t i o states but at d i f f e r e n t l e v e l s of p' form straight l i n e s through the o r i g i n . Since i n i t i a l dilatancy has been noted to remain r e l a t i v e l y constant and independent of p', Section 4.2.2, S may therefore be s p e c i f i e d as a function of n for corresponding stress paths, as implied by Equation 4.12. A means of determining s t r a i n increment r a t i o s for various stress paths w i l l be presented i n Chapter VI. Normalized work against shear s t r a i n for constant mean normal stress paths of 450 KPa but at r e l a t i v e densities of 30, 50 and 70 percent are presented i n Figure 4.28. It may be noted that states corresponding to the same density, data points not connected, maintain separate and non-linear rel a t i o n s h i p s . Hence, S would not remain unique and constant. Common stress states but at d i f f e r e n t r e l a t i v e densities f a l l along a straight l i n e . Overall, these experimental results demonstrate that the normalized work and shear s t r a i n r e l a t i o n s h i p , at small stress r a t i o , i s not unique and constant. It has been shown to be dependent on mean normal stress and r e l a t i v e density and would not be a material parameter as suggested. However, common stress r a t i o states along both d i f f e r e n t mean normal stress as well as density states trace l i n e a r r e l a t i o n s h i p s . This semblance of order has not been recognized previously. A l t e r n a t i v e p r e s c r i p t i o n of S i n small stress r a t i o regions w i l l be attempted with additional observations reported i n / o / / / / / / / f / / / / x \" X / // ' • LEGEND P x • D_=30 (%) / /• r y° O D r = 50 (%) * X D = 70 (%) J 1 1 ^ ns I 1.5 2 ^JJIQ3 2 5 4 4 5 The Influence of Relat ive Density on Normalized Work 115 Chapter VI. However, i n terms of i t s meaning at large s t r a i n regions, the normalized work concept does not appear to embody new fundamental advantages for e l a s t o - p l a s t i c representation. 4.5 Concluding Remarks In the preceeding experimental investigations, the s t r e s s - s t r a i n behaviour of sand at small s t r a i n has been shown not to be an extrapolation of e i t h e r large or very small s t r a i n response. Generally, e l a s t i c approximations of sand behaviour have been used i n the very small s t r a i n range. Considerating s l i p modes of deformation to be s i g n i f i c a n t , some common ground has also been indicated to e x i s t between p l a s t i c i t y and the large s t r a i n response of sand. Resemblences to e l a s t i c and p l a s t i c behaviour have been demonstrated i n regions where e i t h e r e l a s t i c deformation of grains or s l i p between sand p a r t i c l e s predominate o v e r a l l deformation, respectively. In the extreme cases wherein one component of deformation becomes predominant, neither the i n c l u s i o n nor omission of the other component makes a s i g n i f i c a n t d i f f e r e n c e . A l l s t r a i n s are considered to be e l a s t i c i n resonant column considerations. On the other hand, conclusions that were reached regarding the association of p l a s t i c s t r a i n increment directions and stress state by Pooroosharb et a l (1966) were previously established by Rowe (1962) assuming s l i p s t r a i n s to be approximated by t o t a l s t r a i n s . The mechanisms of sand deformation at small s t r a i n l e v e l s i s much more involved and complicated. Behaviour that has so far been demonstrated to apply for large s t r a i n conditions does not extrapolate to small s t r a i n response even when pr e v a i l i n g strains are presumed 116 separated. Concepts l i k e normalized work do not hold at small s t r a i n and assumed nonrecovered deformations maintain d i r e c t i o n a l dependence on the causative stress increment. Yet, large s t r a i n observations are usually assumed to be relevant for small s t r a i n considerations and i n situations wherein an assessment of nonrecovered deformations and associated pore pressure responses are desired. In small s t r a i n response, both e l a s t i c deformation of grains and s l i p between p a r t i c l e s assume r e l a t i v e s i g n i f i c a n c e . Neither can be regarded n e g l i g i b l e and separation has been easier to assume than v e r i f y experimentally. At the same time, the existence of r o l l i n g at. unstable contacts has also been recognized to be an a l t e r n a t i v e mode of deformation (Home, 1965), even though i t s contribution i s not e x p l i c i t l y recognized i n any form of representation. Over a l l , as suggested by Zytynski et a l (1978), there would appear to be no region or d i r e c t i o n of loading or unloading along which i s o l a t i o n of response would be possible. In what follows, the small s t r a i n response of sand w i l l be investigated without assuming separation of strains and thus considering t o t a l s t r a i n s . S t r e s s - s t r a i n relationships w i l l be examined i n s t r a i n and mixed s t r e s s - s t r a i n spaces. Important stress parameters of stress r a t i o , shear and mean normal stresses w i l l be varied systematically so as to allow isolated study of e f f e c t s . The behaviour of sand under proportional loading i s examined i n the following chapter. CHAPTER V - BEHAVIOUR OF SAND IN PROPORTIONAL LOADING Proportional loading paths have been of fundamental interest i n the study of s t r e s s - s t r a i n behaviour of sand. Stress components increase proportional to each other and thus the o v e r a l l o b l i q u i t y of applied stresses would remain fixed. F a i l u r e i n a f r i c t i o n a l material such as sand i s characterized by r e l a t i v e l y constant, thus l i m i t i n g , stress r a t i o states. Whereas i n other materials, f a i l u r e i s s p e c i f i e d by shear stress l e v e l and constant stress r a t i o loading would eventually lead to creep and frac t u r e . Below a threshold o b l i q u i t y of applied external loads, proportional loading i n sand induces s t i f f e n i n g and consolidation. With the o v e r a l l stress r a t i o remaining well below that corresponding to the material s l i d i n g f r i c t i o n angle, no tendency f o r major i n t e r p a r t i c l e s l i p would e x i s t . The associated i n t e r n a l change of geometry would l i k e l y r esult i n net contraction and corresponding s t r a i n s would be small. The response of sand to proportional loading w i l l be examined i n the f i r s t part of this chapter. A l l of the observations are based on tests performed at a r e l a t i v e density of 50 percent. As opposed to customary procedure, behaviour w i l l be investigated i n s t r a i n rather than stress space. In the second part of t h i s chapter, test r e s u l t s at 30 and 70 percent r e l a t i v e densities w i l l be reviewed. The influence of density on stress s t r a i n response w i l l be assessed. P o s s i b i l i t i e s for quantitative relationships within and across r e l a t i v e densities w i l l be investigated. 118 5.1 Behaviour at One Relative Density 5.1.1 Experimental Observations Results of several constant R path tests on specimens of Ottawa sand at a r e l a t i v e density of 50 percent are shown i n Figure 5.1. S t r e s s r a t i o , R, i s expressed as the r a t i o of a x i a l , a^, to r a d i a l s t r e s s a^,. T e s t s t h a t have R>1 are i n s t a t e s of compressional d e v i a t o r i c s t r e s s ( a ^ > o^,). The t e s t f o r which R=l corresponds to h y d r o s t a t i c l o a d i n g ( a ^ = a^) and s t r e s s paths with R<1 are i n e x t e n s i o n a l d e v i a t o r i c stress state (a' < a'). Overall stress ratios a r i n the tests range between 0.82 and 2. The re s u l t s have been presented i n the ' t r i a x i a l ' s t r a i n plane ( e & , /2"e"r). I t may be noted that constant R stress paths r e s u l t i n l i n e a r s t r a i n paths i n s t r a i n space. Such l i n e a r s t r a i n paths during constant R loading have also been observed by others (Barden et a l 1969; El-Sohby 1969; 1972; Rowe 1971). Since the s t r a i n paths are l i n e a r , i t follows that the incremental s t r a i n r a t i o s must be constant and equal to the t o t a l s t r a i n r a t i o s for a given stress path. A l l of the s t r a i n paths have positive slopes and are contained within the f i r s t quadrant of s t r a i n space. Thus both a x i a l and r a d i a l s t r a i n increments are contractant, regardless of compressional or extensional mode of loading. The r a t i o of incremental energy input i n each of the p r i n c i p a l axes of stress i s therefore also p o s i t i v e and constant. Along the pos i t i v e a x i a l s t r a i n axis, r a d i a l strains are zero. This s t r a i n path of zero l a t e r a l deformation i s associated with a stress path that i s commonly known as the K Q path. A x i a l strains are zero along the pos i t i v e r a d i a l s t r a i n a x is. A s t r a i n path oriented at 119 V2€rx I0 3 Fig. 5.1 Strain Paths for Proportional Loading and p' Equipotentials 1 2 0 an angle of 35 degrees to horizontal represents a path of equal a x i a l and r a d i a l s t r a i n increments. In F i g . 5.1 the observed s t r a i n path for hydrostatic compression does not coincide with this 35 degree l i n e . Deformation under hydrostatic loading i s such that the r a d i a l strains are greater than a x i a l s t r a i n s . The specimen thus possesses an inherent anisotropy, as has been observed previously (Arthur and Menzies, 1972; El-Sohby and Andraws, 1972; Oda 1972; Yamada and Ishihara 1979). The l i n e a r i t y of s t r a i n paths also Implies that anisotropy i s fixed by R and i s not altered by increasing hydrostatic stress (Rowe 1962). Strain increment r a t i o , defined as the r a t i o of a x i a l to /2x r a d i a l s t r a i n increment, e //2~ e , would be zero for horizontal s t r a i n a r paths. Whereas, the r a t i o would tend to i n f i n i t y for paths approaching v e r t i c a l . If these same rat i o s were expressed as tangent inverses, the ensuing angle, 9, w i i l range from 0 to 90 degrees. Such angles of s t r a i n increment ratios i n t r i a x i a l s t r a i n space of Figure 5.1 are plotted against corresponding stress r a t i o s i n Figure 5.2. A l i n e a r relationship may be noted which Is v a l i d i n regions of both compression and extension. Continuity of the relationship across compression and extension i s maintained by expressing extensional stress ratios as negative inverse of those i n compression. In t r i a x i a l t e s t s , the p r i n c i p a l stress axes are f i x e d . Because of a x i a l symmetry, = i n compression and o| = i n extension. Thus, the (o|» ^2 o^~) stress plane w i l l contain compression states when a\ = and = o^ ,. If R i s defined as ° a/ ap> then proportional loading paths i n compression would be described by —. On the other hand extension states w i l l 121 t • 5.2 R e l a t i o n s h i p Between S t r a i n Increment R a t i o and S t r e s s 122 remain i n the plane of (/2 o^, a^) for which = and = a^. - °r /2 Hence proportional loading paths would be represented by /2 — = ——.' a The two planes intersect along the space diagonal so that R =» 1 i s common to both. Similar representation of stress r a t i o was adopted i n the study of metal behaviour under proportional loading (Mehan, 1961). Because stress states l i e i n separate planes that Intersect along R = 1; separate expressions must be used i n the extension and compression sides. But, since the slope i s the same and R = 1 i s common to both compression and extension, either of the relationships can be derived from the other. Thus, from Figure 5.2 s t r a i n increment r a t i o s for compressional paths may be expressed by 6e — = tan [a + b (—)} (5.1) /26e /2 r and for extensional paths 6e -/2 a /26e r = tan [c + b ( — ) ] (5.2) For the result on Ottawa sand shown i n Figure 5.2; a = -28, b = 75 and c = 134 Since for R = 1 Equations 5.1 and 5.2 should y i e l d i d e n t i c a l values of 123 s t r a i n r a t i o , (a + b//2) = (c - /2b), which means that only two of the three constants a, b and c are independent. In Figure 5.1, i f points representing equal mean normal e f f e c t i v e stress p' along the l i n e a r s t r a i n paths are connected, the re s u l t i n g l o c i are str a i g h t l i n e s . Such straight l i n e l o c i may be seen to be p a r a l l e l for d i f f e r e n t values of p'. These straight l i n e s w i l l be refer$d to subsequently as equipotentials of p' . The gradient of equipotentials of p 1 , i . e . the s t r a i n path which i s normal to the d i r e c t i o n of equipotentials of p' , has a slope i|> = 43° for the sand tested. From Figure 5.2 i t may be seen that such a s t r a i n path w i l l r e s u l t from a stress path with R = 1.30. 5.1.2 Proportional Stress-Strain Relationships The deformation behaviour so f a r observed can be structured to form a simple proportional loading s t r e s s - s t r a i n r e l ationship that i s v a l i d for constant R paths i n the f i r s t quadrant of s t r a i n space i . e . consolidation paths. The foundations of th i s relationship l i e i n the observed l i n e a r i t y and p a r a l l e l o r i e n t a t i o n of equipotentials of e f f e c t i v e mean normal stress and the l i n e a r r e l a t i o n s h i p between stress r a t i o and 8. Thus, strains along a desired R path could be predicted using s t r e s s - s t r a i n relations f o r a known path. E s s e n t i a l features of Figure 5.1 are represented In a s i m p l i f i e d form i n Figure 5.3. 8^ and 8 are tangent inverses of s t r a i n increment r a t i o s along the given and a r b i t r a r y R paths, respectively, whereas i s associated with s t r a i n increment r a t i o s along a s t r a i n path coincident with the gradient of p'. Consider the given and a r b i t r a r y R paths at points of i d e n t i c a l mean normal stress p' and subjected to 124 F i g . 5.3 Geometric Features of Strain Paths and p' Equipotentials 125 equal Increments of mean normal stress, 6p*. Let the re s u l t i n g a x i a l s t r a i n increments for the given stress paths be Se . Then, from a i geometrical considerations, the corresponding s t r a i n increments, 6e , along the desired R paths may be shown to be 3. . a cos(S.-ib) / c P sin8 i r x (5.3) oe = — — — — * — — — — — * oe a s i n 6 ^ cos(8-^) a^ Since the volumetric s t r a i n increment 6e = Se + 26e (5.4) v a r <5e and noting that = tan 9 (5.5) /2Se r the volumetric s t r a i n increments i n the desired R paths become 6e - (1 + ^ ) Se (5.6) v Q a tan 8 Equations 5.3 and 5.6 together with the rel a t i o n s h i p i n Equations 5.1 and 5.2 have been used to develop the s t r e s s - s t r a i n predictions shown i n Figure 5.4. The known data base for these prediction constitutes the results of hydrostatic compression (R=l) t e s t . S t r e s s - s t r a i n predictions for both compressional and extensional constant R paths have been made and compared with observed s t r e s s - s t r a i n r e s u l t s . Even though the strains are very small, 126 Fi g . 5.4a Comparison of Measured and Predicted A x i a l Strain i n Proportional Loading Paths 127 F i g . 5.4b Comparison o f Measured and P r e d i c t e d V o l u m e t r i c S t r a i n P r o p o r t i o n a l L o a d i n g Paths 128 excellent agreement may be noted between the observed and predicted s t r e s s - s t r a i n response for both compressional and extensional modes. The proportional loading s t r e s s - s t r a i n behaviour of sand at one r e l a t i v e density i s therefore completely s p e c i f i e d i f results of two test s at d i f f e r e n t R values are known. One of such tests can be the simple hydrostatic compression test ( R = l ) . The second test could be ca r r i e d out at any other value of R , compressional or extensional. The re s u l t s of these two tests w i l l enable determination of a l l constants, namely a, b, c and ij> needed for a complete s p e c i f i c a t i o n of proportional loading behaviour. The s t r e s s - s t r a i n r e s u l t s of any one of the two tests can serve as the reference data base f or s t r a i n predictions under any R path loading. The r e l a t i o n s h i p expressed by Equation 5.3 has been derived from experimental r e s u l t s . This r e l a t i o n s h i p has an important implication regarding the r a t i o of energy density increment (incremental energy input per unit volume) between any two constant R paths. The increment In energy density 6Q r e s u l t i n g from an increment i n stress i s 60 = a'.&e + 2o'.6e (5.7) a a r r If the stress Increment i s along a constant R path for which ° a/ a^. = R» a ' = 3p * / ( R+2), and 2e /e = /2/tan6, Equation 5.7 can be written as 6 \ = [ R T T < R + ^ P * 6 e a ( 5 ' 8 ) i n which the subscript on Q refer s to stress increment along a constant R path. For a given R path, since tan6 i s constant, the expression i n 129 square brackets i n Equation 5.8 i s a constant along the entire stress path. The energy, density increment r a t i o of any two stress paths at i d e n t i c a l values of p' would thus be a constant multiple of the r a t i o of t h e i r a x i a l s t r a i n increments. This r a t i o of a x i a l s t r a i n increments may be seen to be a function of 8 values associated with the two stress paths and the angle ij> (Equation 5.3). Since both 8 values are fixed for the sp e c i f i e d stress paths and ij> i s constant, the r a t i o of the a x i a l s t r a i n increments i s constant along the entire stress paths. Hence the energy density increment r a t i o of the two R paths w i t h i d e n t i c a l mean normal stress h i s t o r i e s i s also constant. The r a t i o s of energy density increment at various R values to the energy density increment at reference hydrostatic stress r a t i o R = 1 are shown i n Figure 5.5. It may be seen that for each R, this r a t i o i s e s s e n t i a l l y constant, regardless of the l e v e l of p'. Furthermore, the magnitude of this r a t i o i s a minimum for R~1.30, which corresponds to the stress r a t i o y i e l d i n g the i|/ s t r a i n path coincident with the gradient of equipotentials of p' i n the s t r a i n space. The implication of the association of minimum energy density increment with s t r a i n vector i n the \> d i r e c t i o n i s not immediately clear at th i s time. 5.2 Extensions to Other Relative Densities The previous observations and s t r e s s - s t r a i n predictions under proportional loading were based on results obtained from tests on specimens at a r e l a t i v e density of 50 percent. Additional tests were made to investigate possible extensions to behaviour at other r e l a t i v e d e n s i t i e s . Two series of tests, one each on loose and dense specimens 130 z UJ o ~ tt >- o UJ —I o 2 or UJ cr <_> 1.5 1.0 0.5 Dr= 50% SYMBOL NANCE OF P ( kPa ) ° SO - 100 T 100 - 200 • 200 - 300 < 300 -400 o 400 - 500 0 0.6 0.8 1.0 1.2 1.4 1.6 STRESS RATIO , R 1.8 2.0 F i g . 5.5 V a r i a t i o n o f I n c r e m e n t a l Energy D e n s i t y R a t i o w i t h S t r e s s R a t i o 131 at r e l a t i v e densities of 30 and 70 percent, were ca r r i e d out. 5.2.1 Experimental Observations Test r e s u l t s on loose and dense specimens i n the form of Figure 5.1 are presented i n Figure 5.6. Once again constant R stress paths may be seen to r e s u l t i n l i n e a r s t r a i n paths for both r e l a t i v e d e n s i t i e s . Also equipotentials of mean normal stress seem l i n e a r and p a r a l l e l with e s s e n t i a l l y the same orientation for a l l r e l a t i v e d e n s i t i e s . This implies that i|> i s independent of r e l a t i v e density. As would be expected, for equal stress states the magnitude of contractant s t r a i n response decreased markedly with increase i n r e l a t i v e density. Results of tests on specimen of d i f f e r e n t r e l a t i v e densities but at equal stress r a t i o s are presented i n Figure 5.7. Under hydrostatic loading at R = 1 the slope 9 of the s t r a i n path for the dense sample i s ~35 degrees, which implies i s o t r o p i c behaviour. With decreasing r e l a t i v e density, inherent anisotropy increases progressively, as indicated by the slope of s t r a i n path which deviates more and more below 35°. It may also be noted that equipotentials of mean normal stress connecting s t r a i n paths of equal R but d i f f e r e n t r e l a t i v e densities are l i n e a r and p a r a l l e l . The gradient of these p' equipo-t e n t i a l s , however, i s not constant but depends upon the value of R. Relationships between stress r a t i o , R, and s t r a i n increment r a t i o s at d i f f e r e n t r e l a t i v e densities are shown i n Figure 5.8. As observed previously for a r e l a t i v e density of 50%, l i n e a r relationships between stress r a t i o and 9, the tangent inverse of the s t r a i n increment r a t i o , may also be noted for other r e l a t i v e d e n s i t i e s . The slope of 132 133 4 F i g . 5.7 The Influence of Density on S t r a i n Paths from Propor t iona l Loading 134 Fig. 5.8 Relationships Between Strain Increment Ratio and Stress Ratio at Different Relative Densities these r e l a t i o n s h i p s , however, decrease as the r e l a t i v e density i n c r e a s e s . Existence of increasing inherent anisotropy with decreasing r e l a t i v e density i s again indicated by the data points (deviation of 8 from 35°) corresponding to the hydrostatic, R = 1 loading. The relationships between 8 and r e l a t i v e density at several constant values of R are shown i n Figure 5.9. It may be noted that l i n e a r relationships exist between 9 and r e l a t i v e density at each value of R. The slope of these straight l i n e s decrease progressively with increasing R. The l i n e a r relationships observed i n Figures 5.8 and 5.9 when combined, r e s u l t i n a plane surface i n three dimensions with % r e l a t i v e density D r, stress r a t i o , R, and 9 as coordinates. The equation of th i s plane, which describes completely the contractant proportional loading behaviour of Ottawa sand i s , for compression D + A (—) + B6 + C - 0 (5.9) r /2 and f o r extension D + A (-g±-) + B8 + D = 0 (5.10) i n which A,B,C,D are constants. Since R = 1 i s common to both compression and extension, t h i s requires (A//2+C) = (-/2A+D), and thus there are only three independent constants. For Ottawa sand the values of the constants are; A = 189, B = -2.6, C = -115 and D = 287. The plane surface represented by Equations 5.9 and 5.10 for this sand i s shown i n Figure 5.10. This plane surface can be completely 136 Fig. 5.9 Relationships Between Str a i n Increment Ratio and Relative Density for Different Stress Ratios 138 determined by performing only three proportional loading tests on the sand. The magnitude of s t r a i n increment r a t i o under proportional loading can therefore be completely s p e c i f i e d under any R and D r, once the plane surface i n Figure 5.10 for the sand has been established. A relationship of s t r a i n increment r a t i o to stress r a t i o R i s also implied by the stress-dilatancy theory. The stress-dilatancy theory, however, neither enables precise numeric prediction of s t r a i n Increment r a t i o , nor takes e x p l i c i t account of the r e l a t i v e density as a s t a t e v a r i a b l e . F u r t h e r m o r e , as p o i n t e d out e a r l i e r , stress-dilatancy theory cannot, i n p r i n c i p l e , describe contractant deformation response of sand, which i s the subject of the present in v e s t i g a t i o n s . The a b i l i t y to predict s t r a i n increment r a t i o alone for any R value does not, however, enable determination of i n d i v i d u a l s t r a i n components. Determination of Individual s t r a i n components w i l l now be attempted by establishing energy density Increment relationships across r e l a t i v e d e n s i t i e s , s i m i l a r to those discussed e a r l i e r at one r e l a t i v e density. From examination of test r e s u l t s at a r e l a t i v e density of 50%, i t has been shown that the r a t i o of a x i a l s t r a i n increments of two R paths are constant, provided the mean normal stress h i s t o r i e s are i d e n t i c a l . The implication of that result was demonstrated to require the r a t i o of energy density increments to be constant a l l along the two stress paths (Figure 5.6). A s i m i l a r behaviour with regard to constancy of energy density increment r a t i o may be seen i n Figure 5.11 for r e l a t i v e densities of 30 and 70%. It would now be of i n t e r e s t to examine possible energy density increment r a t i o relationships between 139 CO z UJ _ o • 1.5 o 6 0 UJ 0 * 1.0 LU . _ J o ^ < 2 * 0 . 5 UJ or o (a ) D r= 3 0 % 1 SYMBOL RANGE OF p (kPa) 0 50-100 » 100 -200 • 200 - 300 1 300-400 o 400 - 300 0 . 6 0 . 8 1.0 1.2 1.4 1.6 S T R E S S RATIO , R 8 >-00 z UJ _ a •• ac UJ o Z 60 £ < z cr UJ UJ or o 0 . 5 (b) D r = 7 0 % 0 . 6 0 . 8 1.0 1.2 1.4 1.6 S T R E S S RATIO , R 1.8 5.11 Variation of Incremental Energy Density Ratio with Stress Ratio: a) ' Dr = 30% and b) Dr = 70% specimen having i d e n t i c a l stress h i s t o r i e s (p* and R) but d i f f e r e n t r e l a t i v e d e n s i t i e s . Such energy density increment ratios are shown plotted against stress r a t i o R i n Figure 5.12. At each r e l a t i v e density, the energy density increment r a t i o shown i s the r a t i o with respect to the energy density increment for D r = 50%. The results suggest that for any value of R the energy density increment r a t i o of two specimen having i d e n t i c a l stress h i s t o r i e s but d i f f e r e n t r e l a t i v e densities remain reasonably constant. Furthermore the magnitude of th i s r a t i o i s e s s e n t i a l l y the same regardless of the i d e n t i c a l R value for which the energy density increments are considered. Average li n e s have been drawn i n Figure 5.12 through data points for each energy increment r a t i o considered, i n order to determine the constant values of these r a t i o s . Values of the constant energy density r a t i o s i n Figure 5.12 are now plotted against the inverse r a t i o of the corresponding r e l a t i v e densities i n Figure 5.13. A l i n e a r relationship with a slope equal to unity i s obtained. Thus 6 Q 1 D r 2 - r z (5.11) 6 (V °rl i n which subscripts on Q and D associate corresponding energy and r e l a t i v e density states D ^ and • Equation 5.11 may be stated as follows: Given two specimen at r e l a t i v e densities and proportion-a l l y loaded under i d e n t i c a l R to mean e f f e c t i v e stress p'; i f i d e n t i c a l stress increments 6p' were to be applied to each specimen, energy 1 4 1 SYMBOL RANGE Of p (kPO> < CC >-»-tn z >-a ct UJ 2.0 1.5 1.0 a ? SO -100 •00 - 200 » 0 - 300 S C O - 4 0 0 400 - 500 •8Q3O/8Q50 •8Q.n/8Q SO'^50 2 0.5 CC -g 8Q70/8Q50 0 0.6 0.8 1.0 1.2 1.4 1.6 STRESS RATIO . R 1.8 F i g . 5.12 Relationships Between Incremental Energy D e n s i t y R a t i o w i t h Stress Ratio 142 A 5.13 Relationships Between Ratios of Energy Density Increment and Relative Density 143 density increments of 6Q^ and 6Q^ would take place. The r a t i o of these energy density increments i s equal to the inverse r a t i o of th e i r r e l a t i v e d e n s i t i e s , regardless of the magnitude of the i d e n t i c a l R value under which proportional loading occurred. Using Equation 5.8, the energy density increment r a t i o may be written i n terms of s t r a i n increments as 6e (R + al /2 tan9. ST 6 £a2 < R + tan6 2> (5.12) D , 6 £ a l < R + S e / and from Eq. 5.11 =» — (5.13) r l /2 6 £a2 <R + tS5e > i n w h ich ( ° " e a l » and ( < S e a 2> 92^ a r e a s s o c i a t e d with r e l a t i v e d e n s i t i e s , D^and respectively and the stress r a t i o R. 5.2.2. Proportional Stress-Strain Relationships E q u a t i o n 5.13 enables p r e d i c t i o n of a x i a l s t r a i n increments at any desired r e l a t i v e density from a known s t r e s s - s t r a i n r e l a t i o n s h i p at another r e l a t i v e densi-ty, provided the stress h i s t o r i e s (p', R) of two samples are i d e n t i c a l . E a r l i e r , Equation 5.3 was shown to enable predictions of a x i a l s t r a i n increments under any stress r a t i o from a known s t r e s s - s t r a i n relationship at another stress r a t i o , provided the two samples have Ide n t i c a l r e l a t i v e d e n s i t i e s . It i s thus possible to p r e d i c t a x i a l s t r a i n increments under any r e l a t i v e d e n s i t y D f and stress r a t i o R by a simple superposition of the i d e n t i c a l r e l a t i v e density process (Equation 5.3) and i d e n t i c a l stress h i s t o r y process (Equation 5.13). Considering the i d e n t i c a l r e l a t i v e density process, the a x i a l s t r a i n increment S e ' under R at D i s given by (Equation 5.3) ct 171 s i n 9' c o s ( 9 i - ^ . 6 e i = i i n - 9 7 c o 8 ( 9 ' - » ) * 6 £ a i ( 5 ' U ) i n which Se . i s the known a x i a l s t r a i n increment at D , and R. and 9. a i r i i i and 9' are associated with s t r a i n increment ra t i o s at r e l a t i v e density D ^ under R^ and R respectively. Both 9^  and 8' can be evaluated for t h e i r respective stress r a t i o and r e l a t i v e densities from Equations 5.9 or 5.10. Now following next a constant stress h i s t o r y process, the d e s i r e d a x i a l s t r a i n increments at D and R are related to S e ' by r a J (Equation 5.13). •2" Se r i R + tan9' •I Se ' a (5.15) R + tan9 i n which 9 i s associated with s t r a i n increment r a t i o at D r under R. Substituting f o r S e ' from Equation 5.14 into Equation 5.15 we get 145 Se sine , cos(e 1-^) sine^^ cos(e'-i|0 Se , (5.16) a i Since s t r a i n increment r a t i o Se /Se (=^2tan8) i s uniquely prescribed i n the plane surface by Equation 5.9 and 5.10 once R and are s p e c i f i e d , the r a d i a l s t r a i n increment S e y (or volumetric s t r a i n increment Se^) can be r e a d i l y c a l c u l a t e d . Hence a complete stress-s t r a i n response under any R at can be predicted from a known response under a known R^  and "r£« In Equation 5.16 the three terms i n the square brackets may be considered to r e f l e c t respectively the influences of (1) r e l a t i v e density, (2) variation i n anisotropy due to changes in relative density and (3) the effect of stress path. Equation 5.16 has been used to develop stress-strain predictions for specimen at 30, 50 and 70 percent r e l a t i v e densities (Figure 5.14, 5.15 , 5.16). Predictions are made for proportional loading under hydrostatic as well as i n compressional and extensional modes and compared with the measured response. The known data base for these predictions consists of hydrostatic loading (R = 1) at a r e l a t i v e density of 50 percent. I t may be seen that excellent agreement exists between predicted and observed response i n each case. The agreement between predicted and measured a x i a l s t r a i n response for loose sand i n extensional loading (Figure 5.15a) may not be viewed as satisfactory. However, considering the extremely small magnitude of strains, the agreement can be regarded rather good. I t may be pointed out, that the stress-strain relationship proposed does not require any appeal to material isotropy, which Is generally needed i n most constitutive models. The effect of inherent 146 Fi g . 5.14 Comparison of Measured and Predicted Volumetric Strains for Hydrostatic Loading at Var ious Relative Densities 147 0 I 2 € Q X | 0 3 0 1 2 3 4 5 6 7 8 9 € y x to 3 F i g . 5.15 a) & b) Comparison of Measured and P r e d i c t e d S t r a i n s f o r P r o p o r t i o n a l L o a d i n g i n E x t e n s i o n Mode and a t V a r i o u s R e l a t i v e D e n s i t i e s : a) A x i a l S t r a i n and b) V o l u m e t r i c S t r a i n 148 € y X |0 3 F i g . 5.16 a) & b) Comparison of Measured and Predicted Strains for Proportional Loading i n Compression Mode at Various Relative Densities: a) A x i a l Strain and b) Volumetric Strain 149 anisotropy in sand, which is considered to be associated with the one dimensional sedimentation process, is inherently contained within the proposed relationships. Furthermore, unlike most other material models, the proposed model takes account of relative density in the form of an Independent state variable, enabling stress-strain predictions from one relative density state to another. 5.2.3 Required Parameters In a l l , four parameters A, B, C (or D) and angle i|> are required for the overall relationship. Constants A, B and C (or D) determine the unique plane surface from which strain increment ratio (or 9) for any desired R and can be determined. In addition, angle I|J is needed for evaluating axia l strain increments for the desired R and D f (Equation 5.16). A l l four constants can be determined from three t r i a x i a l tests. Two of these constant R tests must be at the same relative density in order to determine parameter i|>. The third test must be at another value of relative density. For simplicity two tests at different relative densities can be the conventional hydrostatic compression tests (R = 1). The three tests together thus provide three points required in R, D f and 6 space for defining the equation of the plane surface (Equation 5.9 or 5.10). The results of any one of the three tests can be used as the reference data base for prediction of s t r e s s strain behaviour under any R path at any D . 150 CHAPTER VI - NON-PROPORTIONAL STRESS PATHS Experimental study of small s t r a i n behaviour of sand i n proportional loading paths i n Chapter 5 suggested a consistent pattern of response. The test results led to a framework capable of predicting deformations due to contractant proportional loading across r e l a t i v e densities and stress r a t i o . In this chapter small s t r a i n response i s investigated i n non-proportional stress paths i n which the o v e r a l l stress r a t i o does not remain f i x e d . Stress r a t i o can vary along a path with changes i n either or both shear and mean normal s t r e s s . In the f i r s t part of this chapter, non-proportional t o t a l loading response, wherein none of stress r a t i o , mean normal stress or shear stress decrease, i s examined. The influence of increasing or decreasing stress r a t i o , R, under alternate conditions of constant shear and constant mean normal stress, with stress reversal, on small deformation response i s investigated i n the second part. 6.1 T o t a l Loading Paths States of mean normal stress, shear stress and stress r a t i o are important i n the s p e c i f i c a t i o n of stress state for sand. These three stress parameters are of course not independent as s p e c i f i c a t i o n of any two would determine the t h i r d . However, a change i n state, i n the sense of loading and unloading, cannot be prescribed without ambiguity i n situations wherein anyone of these parameters i s decreasing. In order to f a c i l i t a t e systematic study, only t o t a l loading paths i n which none of the three s,tress parameters decrease w i l l be considered i n the following. On the compression side these stress paths cover a range 151 from compression along the hydrostatic axis to a constant mean normal stress paths, a l l originating from a state of hydrostatic compression. The general objective i s to examine relationships between p a r a l l e l stress paths and possible association of stress and s t r a i n d i r e c t i o n s . 6.1.1 P a r a l l e l Loading Paths Experimental investigations i n Chapter 4 have shown that d i f f e r e n t stress paths from a common consolidation state result i n corresponding d i f f e r e n t s t r a i n paths. The e f f e c t of consolidation stress l e v e l and density on s t r a i n paths i s now examined from observation of test results derived from p a r a l l e l stress paths orig i n a t i n g from d i f f e r e n t consolidation states. Widely used incremental l i n e a r e l a s t i c approaches only require representation of p a r a l l e l conventional t r i a x i a l r esults i n analytic form and at only one density. The sub-class of p a r a l l e l stress paths considered herein cover a broad range of stress paths including the conventional t r i a x i a l path. Alternatives for improvement of current incremental e l a s t i c procedures are considered on the basis of experimental observations at small s t r a i n . 6.1.1.1 E f f e c t of Consolidation Stress on Str a i n Paths S t r a i n response i n a series of conventional t r i a x i a l stress paths i n i t i a t i n g from various hydrostatic consolidation stress levels i s shown i n Figure 6.1.a. It may be noted that a l l s t r a i n paths maintain a common i n i t i a l orientation regardless of the l e v e l of i n i t i a l consolidation s t r e s s . The i n i t i a l slope of this s t r a i n path i s -4.4. A slope of -2 represents zero volumetric s t r a i n . Hence incremental 152 € r xl0 3 F i g . 6.1a Strain Paths for Conventional T r i a x i a l Stress Paths from Different Consolidation States 153 volumetric s t r a i n s f o r these stress paths are contractant. For each l e v e l of confining stress, non l i n e a r i t y of s t r a i n paths and d i l a t a n t tendencies did not commence u n t i l s t r a i n l e v e l s corresponding to stress r a t i o s i n excess of about 2 were reached. S t r a i n response i n constant mean normal stress paths also resulted i n a common and i n i t i a l l y l i n e a r s t r a i n path which i s independent of consolidation stress l e v e l , Figure 6.1.b. This common s t r a i n path i s oriented at a slope of -2.4. Thus the contraction rate was smaller for constant p' paths than for the corresponding conventional t r i a x i a l paths. Cl e a r l y , the increase i n mean normal stress that i s associated with conventional t r i a x i a l paths contributes to contraction i n addition to that r e s u l t i n g from shearing. It may be noted that even though both conventional t r i a x i a l and constant mean normal stress paths lead to i n i t i a l s t r a i n paths that result i n net volumetric contraction, r a d i a l s t r a i n increments remain expansive. A l i n e a r s t r a i n path and expansive r a d i a l s t r a i n may also be observed for a stress path which followed an incremental stress r a t i o , (6a{/So^), of 4 and from an I n i t i a l confining stress 03* = 50 kPa, as shown i n Figure 6.I.e. However, the contraction rate i n t h i s stress path, slope = - 5.2, i s greater than for the conventional t r i a x i a l path because of i t s larger rate of mean normal stress increase with shear. Stress paths In which the incremental stress r a t i o was held to 2 were also followed from d i f f e r e n t l e v e l s of hydrostatic compression. Once again, the s t r a i n path i n each t e s t , Figure 6.1.d, follows the same d i r e c t i o n which i s independent of the i n i t i a l l e v e l of consolidation pressure. In these s t r a i n paths, both a x i a l and r a d i a l s t r a i n components are, however, contractant and the s t r a i n paths have a pos i t i v e slope of about 4.5. Between these stress paths and that LEGEND p1 ~ 150 KPa O p' = 250 KPa 6 5 4 3 2 £ r x l0 3 F i g . 6.1b Strain Paths for Constant Mean Normal Stress Paths from Different Consolidation States 155 156 corresponding to a stress increment r a t i o of 4, there would exist an equivalent K path for which l a t e r a l s t r a i n increments would be zero, o In general, contractant volumetric s t r a i n i n sand can result from independent increases i n shear as well as mean normal stress. Even though these two effects cannot be i s o l a t e d when they occur simultaneously; the trend i n s t r a i n paths observed i n Figures 6.1a, b, c and d suggests an increasing rate of volume change with stress paths of decreasing incremental stress r a t i o . This implies an increasing tendency for volume contraction as the stress paths progressively deviate from a constant p' d i r e c t i o n towards a hydrostatic stress d i r e c t i o n . At moderate stress l e v e l s , hydrostatic compression alone i s not e f f e c t i v e i n bringing about a major change i n the density and structure of sand (Youd, 1972). Thus, the sense of inherent anisotropy would not be altered by hydrostatic compression. This view i s consistent with the experimental results presented i n Figures 6.1a, b, c and d which show that p a r a l l e l stress paths that originate from d i f f e r e n t hydrostatic compression states share a common s t r a i n path. However, as shown i n Figure 4.1; the s t r e s s - s t r a i n response s t i f f e n s with increasing compression stress l e v e l s . A unique l i n e a r s t r a i n path thus implies that s t i f f e n i n g as a consequence of compression occurs to the same degree i n both p r i n c i p a l s t r a i n d i r e c t i o n s . The s t r a i n path corresponding to a proportional loading of R = 2, i s also shown i n Figure 6.1.d. It may be noted that this s t r a i n path i s very close to the unique path associated with the constant incremental stress r a t i o path of 2. A sand specimen cannot exist i n a 157 Fig. 6.1d Strain Paths for Stress Paths of Constant Incremental Stress Ratio of 2 158 stress free state. It i s therefore not possible to i n i t i a t e proportional loading from zero confining stress. This physical l i m i t a t i o n and the necessary experimental approximation to proportional loading may explain the small difference between these s t r a i n paths. The r e s u l t s i n Figure 6.1.d thus show an equivalence between proportional loading paths and incremental stress r a t i o paths i n that both r e s u l t i n the same s t r a i n path. Consequently, since proportional loading does not conform to stress dilatancy Rowe (1971), constant incremental stress r a t i o paths, at small l e v e l s of o v e r a l l stress r a t i o would also not conform to stress dilatancy, even though such stress paths are associated with increasing stress r a t i o s , as required by stress dilatancy. 6.1.1.2 E f f e c t of Density on S t r a i n Paths The association of l i n e a r stress and s t r a i n paths was also explored for other r e l a t i v e d e n s i t i e s . Test results for various p a r a l l e l stress paths and d i f f e r e n t densities are shown i n Figure 6.2.a,b,c,d. It may be observed that for a given type of p a r a l l e l stress paths, separate s t r a i n paths are followed for each density. The separation between s t r a i n paths corresponding to i d e n t i c a l stress paths but at d i f f e r e n t densities may be noted to increase as stress path orientations tend towards hydrostatic compression. Thus the influence of density on s t r a i n paths appears to vary with stress path d i r e c t i o n . However, such a corresppndence between stress and s t r a i n paths can be shown to be a r e f l e c t i o n of material anisotropy by comparing the response of an i s o t r o p i c and cross i s o t r o p i c materials subjected to hydrostatic and constant p' stress increments. 159 F i g . 6.2a The Influence of Relative Density on I n i t i a l S train Paths for Conventional T r i a x i a l Compression 160 F i g . 6.2b S t r a i n Paths for p 1 = 450 kPa Tests at Different Relative Densities 161 Fi g . 6.2c St r a i n Paths for R Relative Densities = 1.67 Stress Paths at Different 163 As noted i n section 5.2.1 and also shown by the r e s u l t s for hydrostatic compression i n Figure 6.2.d, sand specimens formed by plu v i a t i o n turn progressively from being anisotropic to i s o t r o p i c with increasing i n i t i a l density. Hence the observed v a r i a t i o n of s t r a i n paths with density appears to be mainly a consequence of d i f f e r e n t anisotropies. Clearly, i f an i s o t r o p i c loose sand were to be formed, i t s s t r a i n path during hydrostatic compression would coincide with that for dense sand. However, attempts to form loose i s o t r o p i c samples by a l t e r n a t i v e procedures were not found successful. It therefore may be necessary to consider changing anisotropy and i n i t i a l density to be concurrent phenomena; i f so, the observed difference i n s t r a i n paths may be j u s t i f i a b l y linked to changes i n density. Figure 4.17 and 4.21 show that s t r a i n paths that have an i n i t i a l negative slope t r a n s i t i o n from a constant to changing dilatancy. This change over occurs at a higher stress r a t i o with increasing density. Thus, the threshold stress r a t i o at which the l i n e a r i t y of s t r a i n paths terminate increases with density. As discussed previously, deformation in loose sand takes place i n smaller clusters than would be i n denser states (Home, 1965). Hence, a l i m i t i n g equlibrium state would need to be achieved at a r e l a t i v e l y few contacts to induce s i g n i f i c a n t s l i d i n g i n loose sand and the o v e r a l l stress r a t i o at which s t r a i n paths become non l i n e a r would therefore be lowered accordingly. 6.1.1.3 Stress Ratio Equipotentials At low l e v e l s of stress r a t i o , s t r a i n paths have been shown to be associated with stress paths and independent of stress r a t i o . Hence 164 p a r a l l e l stress paths at d i f f e r e n t confining stresses w i l l y i e l d p a r a l l e l s t r a i n paths. A family of such s t r a i n paths o r i g i n a t i n g from a hydrostatic stress state and for conventional t r i a x i a l tests at a r e l a t i v e density of 50 percent are shown i n Figure 6.3.a. Points representing equal stress r a t i o states along the s t r a i n paths are found to l i e on straight l i n e s when connected. These l i n e s w i l l be referred to as stress r a t i o equipotentials. A progresssive increase i n the slope of stress r a t i o equipotentials with stress r a t i o l e v e l and convergence toward an approximate common o r i g i n may be i n f e r r e d . This 6e 33 l i n e a r i t y and convergence to a common point implies - j r ; — = const along ab two p a r a l l e l stress paths a and b for a change 6R applied at stress r a t i o state R. Similar stress r a t i o equipotentials may be observed for deformation response under constant mean normal stress and constant 6aJ incremental stress r a t i o , -g—r = 2, paths as shown i n Figure 6.3.b and 6.3.c, respectively. Corresponding stress r a t i o equipotential l n d i f f e r e n t paths do not, however, maintain the same ori e n t a t i o n . These experimental observations of behaviour regarding deformations under p a r a l l e l stress paths can be linked within a coherent framework. Consider a sand specimen at a stress state (<*a, o^) subjected to an increment of stress (6a', So') along a nonproportional loading path 3 TC such that the incremental stress r a t i o 6a'/6a' = r i s constant. This a r stress Increment would induce a corresponding s t r a i n increment (6e , 3. 6e^). The r e s u l t i n g energy Increment per unit volume would be expressed by Equation 5.7 as F i g . 6.3a Conventional T r i a x i a l Strain Paths and Stress Ratio Potentials i n Strain Space 166 Fi g . 6.3b Stress Ratio Potentials i n Strain Space for Constant Mean Normal Stress Paths F i g . 6.3c Stress Ratio Potentials i n Strain Space for Constant Incremental Stress Ratio of 2 Stress Paths 168 SO = o'Se + 2 a'Se (5.7) x a a r r and since R = —p ; Equation (5.7) can be written as r 6e <5Q = a' 6e (R + 2 -j-£) (6.1) x r a oe a In the experimental results presented previously, i t has been shown 6e that at small stress r a t i o , — , i s a c h a r a c t e r i s t i c for a given stress a path. Therefore the term within the brackets i n Equation (6.1) may be considered to represent a path variable, which i s r e l a t i v e l y independent of confining stress l e v e l . Thus, the r a t i o of energy increments along two p a r a l l e l stress paths, say a and b i n Figure 6.3.a, with i d e n t i c a l R history w i l l be (5Q r a b r b a b i n which a and b refer to path l a b e l . As discussed above, the r a t i o of a x i a l stress increments i n p a r a l l e l stress paths that have the same stress r a t i o h i s t o r y remains constant. Hence the energy increment r a t i o along such p a r a l l e l stress paths would be proportional to the r a t i o of corresponding confining stresses. In conventional t r i a x i a l paths, i s constant along the stress path. Therefore, under the assumed conditions of i d e n t i c a l stress 169 r a t i o h i s t o r y and p a r a l l e l conventional t r i a x i a l paths; the energy density increment r a t i o between two stress paths would remain constant, Along other p a r a l l e l stress paths, current r a t i o s of confining stress at equal stress r a t i o states would remain unchanged and equal to the r a t i o of the i n i t i a l consolidation stresses. This i s because as the current R translates to R + 6R along a stress path prescribed by r = -=—r ; a new confining stress of (a' + 6a') i s established. 6 a ' ' s r r r a* + So' However, R = cf'/o"' and R + 6R =» — s — • . , ; from which 6a' can be ' a r a' + 6a' ' r r r expressed i n terms of o^, R, 6R and r as <Sa* = a'5R r (6.3) r (r-(R+6R)) and a' + 6a' - a' . , ( ^ ~ R ^ . . (6.4) r r r (r-(R+6R)) v Therefore, so long as r, R and 6R between p a r a l l e l paths are fixed, the r a t i o of the updated confining stesses w i l l remain equal to the r a t i o of the consolidation confining stresses. Hence the incremental energy r a t i o f o r two p a r a l l e l stress paths that have the same stress r a t i o h i s t o r y would also remain unchanged for stress paths other than the conventional t r i a x i a l t e s t . Stress r a t i o against input energy density at various l e v e l s of confining stresses are shown i n Figures 6.4.a, b and c for conventional t r i a x i a l , constant p' and constant 6aV6o^ = 2 p a r a l l e l stress paths. 170 0 0.5 1.0 1.5 ENERGY DENSITY, Q (KPa) F i g . 6.4a Energy Density with Stress Ratio i n Conventional T r i a x i a l Paths 171 Fig. 6.4b Energy Density with Stress Ratio i n Constant Mean Normal Stress Paths LEGEND Fig.6.4c Energy Density with Stress Ratio i n Constant Incremental Stress Ratio of 2 Stress Paths 173 Input energy density normalized by corresponding input energy density along paths i n i t i a t i n g from a hydrostatic consolidation state of 150 kPa are plotted against stress r a t i o i n Figures 6.5.a, b and c. It may be noted that for each stress path and confining stress, the associated energy r a t i o s are r e l a t i v e l y constant. This lends support to the above development which i s based on the unique association of s t r a i n paths with stress d i r e c t i o n . When these constant energy ra t i o s i n Figures 6.5.a, b, c associated with each confining pressure are plotted against a normalized value of confining pressure i n log - log plot, Figure 6.6, the data points f a l l close to a straight l i n e which suggests a power rela t i o n s h i p that i s independent of stress path. This result thus provides a more generalized basis for comparing the energy increment performance of p a r a l l e l stress paths. A possible use for this observation i s discussed i n the following section. 6.1.1.4 Quantitative Relationships Between P a r a l l e l Paths The results shown i n Figure 6.6 represent a unique relationship between energy density and confining stress r a t i o s along p a r a l l e l stress paths. This relationship would be described by an expression of the form (6.5) where a and b i d e n t i f y p a r a l l e l stress paths that originate from consolidation states al and oL ; and d i s the slope of the l i n e shown 174 a S 6 or g < 5 CC (T :^ 550 KPo C^ '^  450 KPa L U 2 4 LU * or o >. _. A A _ (T^ 350 KPa 55 z L U a >- 2 g _ A A A A A "^3 - 2 5 0 ><Pa L U Z L U I | _ ___________________________________________ (T3'=I50 KPa 0 r « 50 % -y 17 x x X 0,'= 50 KPa I T I I ' » 1.2 1.4 1.6 1.8 2.0 STRESS RATIO, R F i g . 6.5a Normalized Energy Density with Stress Ratio for Conventional T r i a x i a l Stress Paths 175 o 1 10 cT o p'= 450 KPa < or LU LU CC O p' = 350 KPa CO z LU O >-or LU z LU p'= 250 KPa p'= 150 KPa 1.0 1.4 1.6 1.8 STRESS RATIO, R 2.0 D_ = 50 % F i g . 6.5b Normalized Energy Density with Stress Ratio for .Constant Mean Normal Stress Paths 176 2.5 i a o *0 2.0 Pj' = 250 KPa or ui S ui or o 1.5 >. 1.0 CO Ul a >-O 0.5 or ui z Ul I.I I x I Dr = 50 % 1.2 1.3 1.4 1.5 STRESS RATIO, R 1.6 Pj= 150 KPa -X p.= 50 KPa F i g . 6.5c Normalized Energy Density with Stress Ratio for Constant Incremental Stress Ratio of 2 Stress Paths 177 F i g . 6.6 Average Normalized Energy Density with Confining Stress Ratio for A l l Stress Paths 178 i n Figure 6.6. Using Equations 6.2 and 6.5, corresponding a x i a l s t r a i n increment r a t i o s along p a r a l l e l paths can be expressed as (6.6) which implies that the constant d can be estimated from a x i a l s t r a i n and confining stress r a t i o s without having to evaluate actual energy density increments. Simple s t r e s s - s t r a i n r e l a t i o n s h i p s that are usually used i n incremental e l a s t i c approximations were examined i n Chapter 4. These methods also constitute procedures for r e l a t i n g conventional t r i a x i a l r e sults at various confining stress l e v e l s and hence involve relationships between p a r a l l e l stress paths. However, the use of large s t r a i n results to describe small s t r a i n response was shown to be i n c o r r e c t . These ad d i t i o n a l small s t r a i n observations In Figure 6.6 for p a r a l l e l paths are now u t i l i z e d to seek further improvements to current procedures of incremental e l a s t i c representations of s o i l behaviour. Considering conventional t r i a x i a l test r e s u l t s , i n which = o\j and constant, a x i a l s t r a i n increments would be expressed as (6.7) where E i s tangent modulus at ( a l , R or a.) and 179 &a'a = <5R . (6.8) Therefore 6 e a = 1~ * 6 R ' a3 ( 6 ' 9 ) Substituting Equation 6.9 into Equation 6.6; the following r e l a t i o n s h i p i s obtained ai d-1 ol E T , 6R ffl = E ^ - sir <6-10> °3b E t a a3b 6 R b Considering the same stress r a t i o history i n both paths a and b; ( i . e . 6R a = "SRjj) would be expressed as 3a If experimental data was available along any stress path, then i t would be possible to represent E f c using parameters derived from a small s t r a i n transformed hyperbolic plot as i n Figure 4.2.b. From the available data, E^ and the slope m for the small s t r a i n , 1 x 10 _ l + to 1 x 10~ 2, range can be determined and E f c at any would be expressed by 180 E = E, (1 - m.o.) 2 (6.12) t I d or E t = E i (1 - mx'y ( R - l ) ) 2 Thus, for a given test "a", at confining stress o' ; tangent moduli J SL E i n small s t r a i n range would be spec i f i e d by E - E. (1 - m a I . ( R - l ) ) 2 (6.13) ta i a a Ja On substituting the above relationship i n Equation (6.11); the following r e l a t i o n s h i p r e l a t i n g tangent modulus between two paths a and b would be obtained E t b = ^r]2"* • E i a • ( 1-V3a ( R" 1 ) ) 2 ( 6' 1 4 ) 3a In this expression, a desired tangent modulus, E ^ , at a stress state (a' , R) i s determined from a known tangent modulus at a stress state 3b ( a ' a , R). The expression for tangent modulus given by Equation (6.14), bears some resemblence to the following very popular relationship reported by Duncan and Chang (1970) and Byrne and Eldridge (1982). a R.(l-sin«|>)(a1,-a') , . P . (1 - f - , , , 3 ) 2 E a *• 2o\j sin<() ; (6.15) where; P^ = atmospheric pressure (o|, o^) = current stress state $ = Is the f a i l u r e f r i c t i o n angle corresponding to Kg = the modulus number which represents an for a confining stress equal to atmospheric pressure and divided by P a n = the modulus exponent K and n represent the intercept and slope of a log-log l i n e a r f i t E between E./P and o'/ p > implying 1 a J a i = Cr"] 1 1 ' E i P a ( 6 - 1 6 ) E i P a Since Kg = - , Equation (6.16) can also be written as a E i = LT - P • K E • P a ( 6 ' 1 7 ) a which i s i d e n t i c a l to the relationship o r i g i n a l l y suggested by Janbu (1963) from review of experimental r e s u l t s . 182 According to Equation 6.16, the i n i t i a l modulus E^ for an ar b i t r a r y confining pressure i s determined from a reference i n i t i a l modulus, E^p^ at a confining stress equal to atmospheric pressure. This statement i s equivalent to Equation 6.11 provided the reference confining stress o^ a i s made equal to atmospheric pressure and n = 2-d. A value of d = 1.43 may be determined from Figure 6.6 and thus n = 0.57. Such a value for n i s very close to the mid range of reported n values for sand (Duncan et a l , 1980). Hence, the expressions f o r i n i t i a l tangent moduli given by Equaiton 6.14 and Equation 6.15, when R = 1, ( i . e . loading from an i n i t i a l hydrostatic stress-state) are of the same form. However, E^ from Equation 6.14 i s based on small s t r a i n r e s u l t s and with reference to a confining stress at which actual tests were performed. Whereas, E^ that would be given by Equation 6.15 i s based on large s t r a i n r e s u l t s and i s referenced to a confining s t r e s s , P , 3. which usually corresponds neither to the data base nor the pressures fo r which predictions are to be made. In both cases, however, a functional relationship between i n i t i a l modulus, E^, and confining stress i s implied to e x i s t . A useful further development now would be to r e l a t e the second hyperbolic parameter, m, to confining stress but without resorting to f a i l u r e conditions and large s t r a i n response. The squared term i n Equation 6.14 contains the stress r a t i o , R, at which E , Is desired. It also contains the slope m i n transformed tb a hyperbolic plot and confining stress at which the material behaviour has been established experimentally. In this a l t e r n a t i v e form, m represents the best possible f i t of data within the small s t r a i n range of i n t e r e s t . To f a c i l i t a t e comparison with 6.14, the squared term i n Equation 6.15 can be rewritten i n the following form. (1 -R f(l-sin* ) ( o-*-0£) 2 (6.18) 2 a* sintj) and (1 " m a' 3 ( R - l ) ) (6.19) i n which (0^-03) u l t = ultimate asymtotic value of Oj-o 3 and m = slope of transformed hyperbolic plot based on large s t r a i n data. Thus as was the case for Equation 6.14, the squared term i n Equation 6.15 contains the stress r a t i o , R, at which a tangent modulus i s desired. However, a current confining stress, 0 ^ , i s used i n Equation 6.15 instead of a reference confining stress The slope m i n Equation 6.14, corresponds to results at the reference confining stress a^ a and would remain unchanged. Whereas, the slope m for Equation 6.15 would correspond to the current confining stress 0^ and would thus vary with the l e v e l of 0^. In the presently adopted procedures m i s made to vary by adjusting <|> with reference to current o^, as i n the case of Equation 6.15. The advantage of using a fixed reference slope and confining stress within the square brackets of Equation 6.21 i s that the product of confining stress and slope m remain constant. This i s i n the same sense of decreasing m with increasing confining pressure as would be implied by current procedure. 184 The relationships that have so far been determined from experimental observations can be used to express small strain response. Given d, which would be determined from two tests, 'a and b', and also E. and m from a transformed hyperbolic plot for test data 'a' at a ia a confining stress a' ; tangent moduli E , can be determined. The stress Ja tb state in test b at which E ^  i s evaluated would be specified by o^ a and R. Equation 6.14 was used to predict the deviator stress axial srain response shown in Figure 6.7. E^ a for = 150 KPa and d = 1.43, (Figure 6.6), were used to make predictions of axial strain response under of 50 KPa and 350 KPa. Actual stress-strain response for these stress paths is shown in Figure 6.7 by data points. The observed agreement with prediction is very satisfactory and incorporation of this further refinement in current hyperbolic stress-strain models may prove very useful. This is particularly important because the development proposed does not make use of failure or near failure parameters for describing the deformation response of sand at small strain. Conceptually, there is no basis for linkage of small deformation response and failure conditions in any material. As the stress ratio regime increases, and on approaching failure states use of alternative parameters within this or previous procedure would lead to an improved predictive capability. Thus, as indicated previously, a dual hyperbolic representation of stress strain response has been made possible. The use of failure parameters for describing dynamic behaviour has also not been found satisfactory. Consequently, modified hyperbolic and reference strain concepts were introduced to enable more re a l i s t i c 185 F i g . 6.7 Comparison of Experiment and Prediction i n Conventional T r i a x i a l Paths 186 representation (Hardin and Drenevich, 1972). Separate consideration of small s t r a i n response from dynamic tests along l i n e s of development si m i l a r to that considered herein would appear to lead to simpler procedures for characterization of dynamic parameters using either stress or s t r a i n c r i t e r i a . 6.1.2 Quantitative Relationships Between Stress and S t r a i n Directions At small s t r a i n , p a r a l l e l non proportional loading paths have been shown to maintain the same s t r a i n d i r e c t i o n , Figure 6.1.a,b,c,d. For such paths, actual s t r a i n increments have also been i n t e r r e l a t e d i n terms of confining stress and input energy density. Prescription of s t r a i n d irections i n terms of stress directions would be a further development. For proportional loading, Chapter 5, stress r a t i o and s t r a i n increment d i r e c t i o n were associated by the li n e a r r e l a tionship shown i n Figure 5.2. Furthermore, proportional and corresponding constant incremental stress r a t i o paths have been found to maintain a common s t r a i n d i r e c t i o n , Figure 6.1.d. Hence by expressing stress directions i n a more general incremental form, i t i s possible to relate common s t r a i n directions to both proportional and nonproportional loading stress paths. Such relationships at a fixed r e l a t i v e density of 50 percent are shown i n Figure 6.8.a. Expressions for stress and s t r a i n axes i n Figure 6.8.a are similar to those used i n studies of proportional loading of metals by Rees (1981), except that incremental rather than t o t a l stress ratios are u t i l i z e d and parameters have been adopted to s u i t t r i a x i a l test data. In Figure 6.8.a, the applied 187 F i g . 6.8a Incremental Stress and Strain Ratio Relationships at 50% Relative Density 138 conditions of hydrostatic loading, So'/So' = 1, correspond to a £i r horizontal axis stress function value of zero whereas the v e r t i c a l s t r a i n function response i s negative. This negative ordinate i s a correct r e f l e c t i o n of the recognized sense of inherent anisotropy i n a pluviated medium dense sand. It may be noted that the results shown i n Figure 6.8.a f a l l along two intersecting l i n e s with a t r a n s i t i o n a l gently curved segment within the i n t e r s e c t i o n . Incrementally proportional loading test results on loose sand, Figure 6.8.b, also show behaviour s i m i l a r to that observed for medium dense sand. The influence of density i s r e f l e c t e d by the d i f f e r e n t orientations of the inte r s e c t i n g l i n e s , as well as d i f f e r e n t magnitude of zero intercept. Key concepts and trends based on results i n Figure 6.8.a and b are shown In Figure 6.9. For a given r e l a t i v e density, a l l stress increments corresponding to contraction i n both a x i a l and r a d i a l s t r a i n f a l l on the steeper state l i n e A and below a s t r a i n function value of about one, which corresponds to a K q or equivalent K q path. Strain paths belonging to lower s t r a i n function values would be contained within the f i r s t quadrant of s t r a i n space, (both Se , Se ; p o s i t i v e ) , and the correspondence between stress and s t r a i n directions would remain f i x e d . Stress paths that are oriented steeper than a K q stress d i r e c t i o n have also been shown to be associated with a unique s t r a i n d i r e c t i o n within a sizeable i n i t i a l stress r a t i o range, Section 6.1.1.1. For such paths, u n t i l a threshold stress r a t i o , whose magnitude varies mainly with r e l a t i v e density, i s mobilized; stress and s t r a i n path relationships would be described by state l i n e B. The r e l a t i v e l y smooth t r a n s i t i o n between l i n e s A and B appears larger for loose as opposed to dense sand. As stated previously, because loose 189 F i g . 6.8b Incremental Stress and Strain Ratio Relationship at 30% Relative Density 190 F i g . 6.9 Trends and Features i n Incremental St r a i n and Stress Ratio Relationships 191 sands deform i n smaller c l u s t e r s ; l o c a l s l i p , lack of f i t and collapse would be more l i k e l y and would occur over a wider range of stress r a t i o s than i n dense sand. As the threshold stress r a t i o i s exceeded along the f a i l u r e paths, s t r a i n directions begin to change even though the stress d i r e c t i o n i s s t i l l f i x e d . Thus the s t r a i n function would begin to increase with the stress function remaining constant. At this time, the association of stress state and s t r a i n increment r a t i o i n the form of stress dilatancy becomes more relevant as the incremental s t r a i n rate progresses to zero, maximum and back to zero rate of volume change. For a given r e l a t i v e density, l i n e s A and B therefore describe an association of stress and s t r a i n d i r e c t i o n s . These state l i n e s would s h i f t lower and to the right with decreasing r e l a t i v e density, (see Figures 6.8.a and 6.8.b). A dense specimen would tend to be i s o t r o p i c under hydrostatic loading and i t s state l i n e A would pass through the o r i g i n . As discussed previously and comparing Figures 6.8.a and b, inherent anisotropy as r e f l e c t e d by r e l a t i v e density predominates consolidation s t r a i n paths. The slope of A and i t s intercept are functions of D f; whereas i n i t i a l s t r a i n paths of f a i l u r e stress paths, state l i n e B, i s r e l a t i v e l y unaffected by r e l a t i v e density. The orientation of state l i n e s i s a function of r e l a t i v e density and perhaps other material and f a b r i c parameters. U n t i l further establishment of basic relationships, at least three tests would be required to determine two points on each of l i n e s A and B at a given r e l a t i v e density. Association of s t r a i n and stress directions i s by no means an end. Actual s t r a i n increments corresponding to stress increments along the path have to be extracted. To some extent this would be possible using the relationship shown i n Figure 6.6, but only so long as s t r a i n 192 response i n a p a r a l l e l path i s known. Future studies of behaviour across stress paths might enable a more general representation. However, since inherent anisotropy of sand has an important influence on small s t r a i n response, rotation of p r i n c i p a l axes of stress and s t r a i n may have a major impact. Hence, further study of factors not examined herein and assembly of more experimental data i n follow up studies of small s t r a i n phenomena using t o r s i o n a l hollow cylinder equipment would be more b e n e f i c i a l to the development of comprehensive rel a t i o n s h i p s between stress and s t r a i n . 6.2 Other Non Proportional Paths Considerations were so far r e s t r i c t e d to conditions of t o t a l loading. A decrease i n any of p', q, and n (=q/p') did not occur. The paths so f a r considered have been l a b e l l e d as t o t a l loading paths. In other stress paths, where one or more of these stress components decrease, the p r e s c r i p t i o n of loading or unloading becomes ambiguous. Out of keeping with this general c l a s s i f i c a t i o n , the small s t r a i n behaviour of sand i n two other nonproportional paths Is examined i n this section. In the f i r s t part, experimental results under conditions of constant shear, q, are examined. The second part focuses on experimental observations for conditions of stress reversal under constant mean normal stress, p'. 6.2.1 Constant Shear Stress Paths Stress paths i n which a constant shear stress was maintained are associated with an increasing mean normal stress and decreasing stress r a t i o or a decreasing mean normal stress and increasing stress r a t i o . 193 Hence both loading and unloading occur simultaneously and the stress increments constitute an Increase or decrease only i n mean normal s t r e s s . Hydrostatic loading paths are unique i n that both q and n remain constant at a l l times. When the mean normal stress was increased at constant shear i n both compression and extension, and thus decreasing stress r a t i o , very l i t t l e change i n shear s t r a i n was detected. In volumetric and shear s t r a i n space, the s t r a i n paths for such tests are seen to be l i n e a r and nearly v e r t i c a l (Figures 6.10.a and b). Also, when mean normal stresses were reduced at constant shear and thus increasing stress r a t i o , no change i n shear s t r a i n was observed i n i t i a l l y as evidenced by an e s s e n t i a l l y v e r t i c a l s t r a i n path (Figures 6.11.a and b) and u n t i l a l i m i t i n g stress r a t i o was approached. The magnitude of volume swelling that occured without shear s t r a i n was more for low shear stress l e v e l s , Figure 6.11.a. However, the l i m i t i n g stress r a t i o at which changeover from a near v e r t i c a l s t r a i n path was i n i t i a t e d was about 1.9 for compression and 1.6 for extension. With further increases i n stress r a t i o , but s t i l l at constant shear stress, on the compression side; the i n i t i a l swelling continues In volume expansion with increasing shear s t r a i n , Figure 6.11.b. Whereas i n extension mode, the i n i t i a l swelling Is arrested and i s followed by volumetric contraction with shear s t r a i n . This phenomena i s shown i n an expanded scale i n Figure 6.12 wherein volume contraction and shear s t r a i n i n extension mode may be noted to precede volume expansion with shear s t r a i n or d i l a t i o n . The experimental re s u l t s presented i n Figures 6.10, 11 and 12 show that the sense of increasing or decreasing stress r a t i o along constant 1 9 4 4 0 0 CONTRACTION SWELLING 300 to O x >• B 2 X y\ = 0.56 B, o>|=052 O 2 A, t>|=0.8l A , X 0.87 B , • "^=0.53 A 3 • >| = 0.82 200 A, B, q = 178 KPa , CONSTANT A 2 B 2 q = 191 KPa , CONSTANT A 3 B 3 q = 180 KPa , CONSTANT D r = 5 0 % 100 50 150 £ 8 xl0= 250 F i g . 6.10a Shear and Volumetric Strain States i n a Constant Shear Stress Path for Decreasing Stress Ratio and Increasing Mean Normal Stress Conditions i n Extension Mode 195 CONTRACTION SWELLING B • 0.44 1 l T|=0. .67 q s 147 KPa, CONSTANT D r = 5 0 % 700 6 0 0 500 g > 4 0 0 3 0 0 2 0 0 - 7 0 0 - 6 0 0 - 5 0 0 - 4 0 0 d 8 xl0 5 F i g . 6.10b Shear and Volumetric S t r a i n States i n Constant Shear Stress Path with Decreasing Stress Ratio and Increasing Mean Normal Stress i n Compression Mode EXTENSION C A A, B,C, = 176 KPa, CONSTANT A 2 B 2 C 2 :o- 2 * 170 KPa, CONSTANT A 3 B 3 C 3 : c j 3 = 115 KPa, CONSTANT D r = 50 % CONTRACTION 700 SWELLING A| £>^ 0.52 - J . B-liy0.55 n=o.eo 1 MV0.50 >|=0.59 600 o K if 500 400 1 300 -800 -700 -600 -500 -400 £,xl0 5 -300 -200 -100 g. 6.11a Shear and Volumetric Strain States i n Constant Shear Stress Paths with Increasing Stress Ratio and Decreasing Mean Normal Stress i n Extension Mode 6.11b Shear and Volumetric Strain States i n Constant Shear Stress Paths with Increasing Stress Ratio and Decreasing Mean Normal Stress i n Compression Mode F i g . 6.12 Volume Contraction i n a Constant Shear Stress Path Under Increasing Stress Ratio and Decreasing Mean Normal Stress i n Extension Mode 199 shear stress paths does not encourage shear s t r a i n development so long as n or R i s maintained below a l i m i t i n g value, appropriate for the density and sense of loading. Inspection of proportional loading test r e s u l t s , Chapter 5, however, shows development of shear s t r a i n with increasing shear and at constant stress r a t i o states. This again implies that shear stress and not stress r a t i o i s instrumental for the development of shear s t r a i n when stress states are below a l i m i t i n g stress r a t i o . At the same time, fixed s t r a i n path orientations are associated with proportional loading paths and with nonproportional loading paths at low levels of stress r a t i o . Whereas, at constant shear and nonproportional paths that ultimately lead to f a i l u r e , non l i n e a r i t y and changes i n s t r a i n path d i r e c t i o n develop when l i m i t i n g stress r a t i o are exceeded. Hence inherent anisotropy i s preserved Independent of shear stress state, when stress r a t i o i s held constant or contained to small values. The development of stress induced anisotropy i n sand i s therefore primarily dependent on stress r a t i o and not shear stress l e v e l . The tendency for contraction that was observed i n the extension mode, Figure 6.12, would contribute to positive pore pressure and reduced strength i n an undrained condition. Even though conventional t r i a x i a l compression i s associated with an increase i n mean normal stress whereas conventional t r i a x i a l extension involves a decrease; the net volume change at peak, strength was Invariably found to r e f l e c t more volume expansion on the compression side (Green 1969). This somewhat paradoxical phenomena would appear to be due to the intermediate contraction that was observed, Figure 6.12, while i n extension. Under undrained conditions, this tendency for contraction 200 and related positive pore pressure lead to a reduced strength in extension as opposed to compression mode, as shown by the experimental results of Chern (1984) and Cheung (1984). Even more, i t would imply a reduced potential for contractant volume change with only compression side stress histories as opposed to those involving stress reversal. Sand formed by pluviation possesses inherent anisotropy because the major principal axes of grain orientation tend to be inclined disproportionately toward the horizontal. In considering conventional t r i a x i a l paths, predominant s l i p would occur at contact planes oriented at 45 + ^ to horizontal in compression and 45 - y in extension. Therefore, because of the sense of inherent anisotropy and orientation of slip planes, extensional shearing would encourage more closer packing and thus contraction. Such a difference between extension and compression modes is a reflection of inherent limitations of the t r i a x i a l test rather than fundamental stress-strain behaviour of sand in that predominant slip planes in the two modes of shearing are entirely separate. However, when rotation of principal axes is permitted and predominant slip planes remain the same for both senses of shearing, such as would be in simple shear tests, separate types of volumetric strain response would not develop. The symmetry of undrained simple shear response as opposed to the nonsymmetry of undrained triaxal results is a clear indicaton of this phenomenon. The conventional t r i a x i a l test therefore seems to be less than desirable for the study of cyclic behaviour of sand and such limitations should ! perhaps be considered in evaluation of test results. 201 6.2.2 Constant p' Paths From a review of the test data i n constant p' loading paths i n compression, corresponding s t r a i n paths were observed to be r e l a t i v e l y independent of p' and remained l i n e a r up to a stress r a t i o of about 2, Figure 6.3.b. Results of compression side shear unloading at constant p' as well as stress reversal and continued shearing i n extension are shown i n Figure 6.13. It may be noted that the s t r a i n paths follow the same d i r e c t i o n both i n compression shear unloading and extension shear loading. Si m i l a r l y unloading from an extensional state and extending into the compression mode at constant p' gives r i s e to s t r a i n paths with e s s e n t i a l l y the same d i r e c t i o n (Figure 6.13). Shear stress and not stress r a t i o was observed to be mainly responsible for development of shear s t r a i n at low values of stress r a t i o . However, for comparable changes i n shear stress, a larger magnitude of shear s t r a i n accumulated In extensional as opposed to compressional mode, Figure 6.13.a and b. In both compression and extension modes, the magnitudes of recovered strains were much less than the respective loading counterparts. The rate of volume change with shear s t r a i n was greater i n compressional unloading and extension loading, (steeper s t r a i n paths), as compared to extension unloading and compression loading. The sense of shear s t r a i n increment changes i n concert with the sense of shear stress increment. However, the sense of volumetric s t r a i n increment due to shearing remains contractant whether or not the shear stress i s being applied or retracted. Shear volume change only accumulates and does not decrease for both loading and unloading senses of shear stress. Although s t r a i n increment r a t i o s and thus s t r a i n directions of compression loading and extension unloading and visa versa were found COMPRESSION C, B A A EXTENSION D =50% r • >j=0.62 C. n=o A— CONTRACTION SWELLING A,B, C, P.- 3 4 0 KPa, CONSTANT B 2C 8 p2= 440 KPa, CONSTANT B 3C 3 p3= 220 KPa, CONSTANT 700 600 500 g > <0 400 300 200 T\=0.89 -600 -500 -400 -300 -200 -100 100 200 Fig. 6.13a Strain Paths for Extension Side Shear Unloading and Compression Side Shear Loading at Constant Mean Normal Stress o tsi Fig. 6.13b Strain Paths for Compression Side Shear Unloading and Extension Side Shear Loading at Constant Mean Normal Stress 1 0 o CO 204 to be i d e n t i c a l , magnitudes of actual s t r a i n increments vary. Within a region about the hydrostatic stress axis and small stress r a t i o states, s t r a i n d irections depend on stress directions regardless of stress reversal and sudden rotation of p r i n c i p a l axes of stress, as happens when the stress path crosses the hydrostatic axis. However, softening or s t i f f e n i n g appears to occur simultaneously and i n the same sense r e l a t i v e to both a x i a l and r a d i a l s t r a i n componments preserving s t r a i n p r o p o r t i o n a l i t y . The stress s t r a i n r e s u l t s of compressional unloading and extensional loading at constant p' are shown i n Figure 6.14.a. Results for compression loading and extension unloading are presented i n Figure 6.14.b. St r e s s - s t r a i n responses to v i r g i n extension and compression loadings are also shown i n Figures 6.14.a and b, respectively. It may be noted that compared to response i n v i r g i n loading, a stress h i s t o r y i n the opposite side has the e f f e c t of softening response on the other. In contrast, reloading without stress reversal results i n a s t i f f e r response as compared to v i r g i n loading, Figure 4.9. However, i t i s common knowledge that i n a v i r g i n h y s t e r e i s i s loop, reloading i n the same sense as v i r g i n loading invariably results i n a softer stress s t r a i n response when compared to v i r g i n loading. There are currently two contrasting positions r e l a t i v e to the Influence of stress reversal on s t r e s s - s t r a i n response. Arthur (1971) and Thurairajah (1973) observed that shearing i n one d i r e c t i o n softens deformation response i n the other. In their studies, stress reversal was I n i t i a t e d a f t e r mobilization of large s t r a i n and from a near f a i l u r e stress state. Tatsuoka and Ishihara (1974) argued that the 205 F i g . 6.14a Relationships Between Shear Stress and Shear Strain for Compression Side Unloading and Extension Side Loading of Shear Stress at Constant Mean Normal Stress 206 Fi g . 6.14b Relationships Between Shear Stress and Shear St r a i n for Extension Side Unloading and Compression Side Loading of Shear Stress at Constant Mean Normal Stress 207 observations of Arthur and Thurairajah are due to stress reversal from near f a i l u r e states and attendant changes of i n t e r n a l structure and grain o r i e n t a t i o n as a consequence of stress induced anisotropy. They presented experimental results i n support of stress reversal and stress h i s t o r y having no influence on response on the opposite side so long as low amplitude shear stress levels are being considered. The experimental r e s u l t s presented herein do not support the findings of Tatsuoka and Ishihara. At the same time, stress reversal i n these r e s u l t s was accomplished at small stress r a t i o and small s t r a i n l e v e l s . Hence, i t would be d i f f i c u l t to j u s t i f y agreement with the findings of Arthur and Thurairajah on the same basis of a change i n i n t e r n a l structure due to stress induced anisotropy. Stress reversal from both extension and compression modes introduces a d i s t i n c t change i n s t r e s s - s t r a i n response. Loading following stress reversal and reloading without stress reversal are both i n i t i a t e d from a state of residual shear s t r a i n . When the sense of the residual s t r a i n i s the same as the d i r e c t i o n of loading, such as in the case of reloading without stress reversal, a s t i f f e n i n g response i s observed. Whereas, the opposite i s true when the sense of residual s t r a i n i s opposite to the d i r e c t i o n of loading, as i n the case of loading following stress reversal. These observations would appear to be described by the deformation mechanism of linked springs and f r i c t i o n blocks proposed by Zytynski et a l (1978). At small s t r a i n , both s l i d i n g and e l a s t i c deformation of the grains occur with r e l a t i v e importance. On ap p l i c a t i o n and removal of a disturbing force i n one d i r e c t i o n , a l l of the e l a s t i c energy stored would not be released on account of counter s l i d i n g resistance. 208 Consequently, the magnitude of disturbing force necessary to i n i t i a t e s l i d i n g i n a d i r e c t i o n opposite to the sense of the residual s t r a i n would be less than would otherwise be required i f motion were to be i n i t i a t e d from v i r g i n loading. On the other hand, a large disturbance would be required to i n i t i a t e s l i d i n g on reloading and i n the same sense as the residual s t r a i n . • 209 CHAPTER VII - SUMMARY AND CONCLUSIONS This research i s a fundamental i n v e s t i g a t i o n of sand behaviour at small s t r a i n . Test equipments and procedures were c r i t i c a l l y evaluated and improved to enable precise load a p p l i c a t i o n i n d i f f e r e n t stress paths, accurately measure small deformations and ensure reproduction and consistency of test r e s u l t s . The experimental programme was designed to systematically examine some common frameworks as well as experimental evidence i n support of fundamental assumptions necessary fo r incremental e l a s t i c , e l a s t o - p l a s t i c and p a r t i c u l a t e concepts for modelling sand behaviour. New experimental observations are presented. A l t e r n a t i v e fundamental i n t e r p r e t a t i o n and s t r e s s - s t r a i n relationships are proposed on the basis of the experimental findings. The present form of the s t r e s s - s t r a i n model proposed i n t h i s research handles contractant proportional loading but includes r e l a t i v e density as a separate parameter. Some common ground between proportional and nonproportional loading behaviour i s indicated and future extension of the reported development may be possible through further experimental study. A value of i n i t i a l Young's modulus i s often used as a key parameter for representing the nonlinear s t r e s s - s t r a i n response of sand a n a l y t i c a l l y . I n i t i a l moduli, E , determined from resonant column J max tests are compared with i n i t i a l moduli from conventional t r i a x i a l tests and conditions of v i r g i n loading, E^; unloading E ^ and reloading, E. The test results show that E i s unattainable from v i r g i n i r. max loading and that suitable i n i t i a l moduli for characterizing large 210 s t r a i n response are generally d i f f e r e n t from E . However, i n i t i a l max Young's moduli from unloading, E^ , are close to E m a x , r e l a t i v e l y independent of the deviator stress l e v e l from which unloading i s i n i t i a t e d and number of cycles of loading. On subsequent reloadings, moduli, E. , remain intermediate between E. and E . Moduli evaluated ' i r * i max at common stress points vary depending on the stress path followed i n reaching the stress state. The stress dilatancy equation does not characterize the behaviour of sand at small s t r a i n . Nonrecovered s t r a i n directions are found to depend on stress d i r e c t i o n which leads to in t e r s e c t i o n of p l a s t i c potentials. This i s i n contrast to large s t r a i n behaviour where s t r a i n increment directions depend only on stress state and are independent of stress Increment d i r e c t i o n . As well, the concept of normalized work and suggested fundamental parameter for sand i s not v a l i d for describing small s t r a i n response. Because s t r a i n Increment r a t i o s are associated with stress increment ratios at small s t r a i n , proportional loading paths are uniquely related to l i n e a r s t r a i n increment d i r e c t i o n s . In s t r a i n space, mean normal stress equipotentials are p a r a l l e l with e s s e n t i a l l y the same orientation for d i f f e r e n t r e l a t i v e d e n s i t i e s . The r a t i o of energy density increments between two proportional loading paths that have i d e n t i c a l mean normal stress h i s t o r i e s remains r e l a t i v e l y constant. These observations lead to a framework capable of characterizing proportional loading response within and across r e l a t i v e d e n s i t i e s . 211 P a r a l l e l stress paths i n i t i a t e d from different levels of hydrostatic compression result i n p a r a l l e l s t r a i n paths. For stress path orientations below an overall stress ratio of 1/KQ» st r a i n paths remain linear and p a r a l l e l , essentially without l i m i t . However, i n stress paths leading to f a i l u r e , s t r a i n paths become nonlinear at higher stress ratio states. The magnitude of stress ratio above which s t r a i n path l i n e a r i t y terminates appears to increase with r e l a t i v e density. In nonproportional loading paths, stress ratio equipotentials along i n i t i a l linear s t r a i n path segments are linear and appear to radiate from a common point. I n i t i a l segments of p a r a l l e l stress paths can be normalized, with respect to the corresponding hydrostatic compression stress, u t i l i z i n g this r e s u l t . The normalized relationship can be considered common for a l l paths considered. Paths of constant shear stress, increasing mean normal stress and decreasing stress ratio are found to generate volumetric and no shear st r a i n . However, when constant shear paths i n decreasing mean normal stress and increasing stress r a t i o are followed, i n i t i a l swelling i s observed. The magnitude of rebound decreases with shear stress l e v e l . After i n i t i a l swelling, response i s different i n compression as opposed to extension modes. On the compression side, i n i t i a l swelling with zero shear stra i n i s followed by d i l a t i o n . In extension mode, i n i t i a l swelling with no shear s t r a i n i s followed by contraction and then d i l a t i o n . Constant mean normal stress paths result i n unique s t r a i n increment directions. Compression and extension side shear loadings 212 are associated with d i f f e r e n t s t r a i n d i r e c t i o n s . However, s t r a i n paths for compression side shear loading are i d e n t i c a l to paths of extension side shear unloading and those for extension side shear loading are the same as for compression side shear unloading. For comparable levels of shear stress Increments, more shear s t r a i n i s generated and the rate of volumetric s t r a i n with shear s t r a i n during compression side shear unloading and extension side shear loading i s much higher than for extension side shear unloading and compression side shear loading. Within the confines of small s t r a i n considerations, accumulated levels of i n d i v i d u a l s t r a i n components r e f l e c t a dependence on s t r e s s - s t r a i n h i s t o r y . 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