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Characterisation and modelling of a uniformly graded, well-rounded coarse sand Rouse, Pascale C. 2005

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CHARACTERISATION AND MODELLING OF A UNIFORMLY GRADED, WELL-ROUNDED COARSE SAND By P A S C A L E C. R O U S E Civil Engineering, University of Chile, 2003 B . S c , University of Chile, 2001 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (CIVIL ENGINEERING) THE UNIVERSITY OF BRITISH COLUMBIA September 2005 © Pascale C Rouse, 2005 A B S T R A C T Strength parameters and intrinsic properties of sand are very important for engineering design. It is well known that such properties depend on factors including grain size distribution, fines content, mineralogy and grain shape. Recently, several authors have sought to establish a relation between the constant volume friction angle, as well as the maximum and minimum void ratios, and the roundness of the sand grains. Little information exists on well-rounded sands. A series of triaxial tests were performed on a uniformly graded, well-rounded coarse sand (R=0.81). Interpretation of the results confirms the available relations, and provides a link between the data on commonly found sands (0.2<R<0.7) and glass beads (R=1.0). Additionally a comparison of drained and undrained triaxial test data indicates § M C = § P T * (j>cv, (J)cv = § M O i § M 0 = (j)PT + 3° and (j)cv is very similar to § r e p. The behaviour of the well-rounded sand is modelled using the critical state soil mechanics constitutive model, NorSand. The results show that the parameters needed for the model fall within the expected range of values, and that NorSand captures both the drained and undrained response of the sand in triaxial testing. However, the volumetric coupling coefficient N, shows an unusual value indicative of unusual dilation of this sand. To account for this, a modification of the flow rule is proposed. In addition, the NorSand constitutive model is implemented in the commercially available software « FLAC. For this, the standard method used within FLAC, F=0, will not work. But by solving F = 0 it is possible to introduce any critical state soil mechanics (CSSM) constitutive model. This approach is described in detail for two selected CSSM models, Original Cam Clay and NorSand. ii TABLE OF CONTENTS Page Abstract ii List of tables vj List of figures vii List of symbols x Acknowledgments xiii CHAPTER I: INTRODUCTION 1 1.1. Purpose of the study 2 1.2. Organization of the thesis 3 CHAPTER II: LITERATURE REVIEW 4 2.1. Behaviour of sands 4 2.1.1. General stress-strain behaviour of soils 4 2.1.2. Triaxial testing 5 2.1.3. Triaxial behaviour of sands 10 2.2. Definition of friction angles for sands 12 2.2.1. Angle of repose (§rep) 12 2.2.2. Critical state or steady state friction angle ((|)cv) 13 2.2.3. Peak friction angle O p ) 13 2.2.4. Maximum contraction friction angle (<))Mc) 15 2.2.5. Phase transformation friction angle (<j>PT) 16 2.2.6. Maximum obliquity friction angle ((j)Mo) 16 2.3. Definition of roundness and sphericity 17 2.3.1. Sphericity 17 2.3.2. Roundness 19 2.4. Critical State Soil Mechanics 22 2.4.1. General plasticity concepts 22 -Elasticity: 22 - Yield surface, F: 22 -Normality: 22 iii - Plastic potential surface, Q: - Hardening rule: - Dilatancy: 2.4.2. The critical state 2.4.3. Original Cam Clay formulation 2.4.4. Modified Cam Clay formulation 2.4.6. Stress - dilatancy 2.5. Critical state model for sands 2.6. Critical state models for sands and NorSand 2.6.1. The state parameter and overconsolidation 2.6.2. NorSand formulation 2.6.3. Determination of NorSand parameters CHAPTER III: MATERIAL PROPERTIES 58 3.1. Description of the soil 58 3.3.1. Extreme void ratios 60 3.3.2. Angle of repose 63 3.3.3. Roundness and sphericity 63 3.2. Triaxial testing program 67 3.2.1. Description of the triaxial equipment 67 3.2.2. Data acquisition system 67 3.2.3. Calibration of the system and measurement resolutions 67 3.2.4. Preparation and testing of the specimen 69 3.3. Triaxial test results 70 3.3.1. Test description 71 3.3.2. Void ratio determination 71 3.3.3. Triaxial test results 73 3.3.4. Shear modulus 77 CHAPTER IV: ANALYSIS OF RESULTS 78 4.1. Friction angles results from triaxial test program 78 4.1.1. Constant volume friction angle (t)>cv) 78 4.1.2: Peak friction angle (t|)p) 81 23 24 24 27 30 34 39 43 47 47 48 55 iv 4.1.3. Maximum contraction friction angle (tbMc) 82 4.1.4. Phase transformation friction angle (<j)pT) 83 4.2. Comparison of Badger sand properties with literature 86 4.3. Analysis of NorSand results 89 4.3.1. Comparison between triaxial data and NorSand model for Badger sand 90 4.3.2. Determination of NorSand parameters 98 - Critical state parameters 98 - Plastic parameters 98 CHAPTER V: SUMMARY AND CONCLUSIONS 104 R E F E R E N C E S 109 APPENDIX A 117 APPENDIX B 147 v LIST OF TABLES page Table 2.1: Effect of angularity and grading on peak friction angle 13 Table 2.2: Categor ies of roundness 20 Table 2.3: Stress-di latancy relations 41 Table 2.4: Relat ions of M; after different authors 42 Table 2.5a: Constitutive models for sands 45 Table 2.5b: Constitutive models for sands 46 Table 2.6: Summary of equations for NorSand 54 Table 2.7: Propert ies needed to descr ibe the NorSand constitutive model and their typical range for sand 55 Table 2.8: Summary of C S S M constitutive models 57 Table 3.1: Properties of the grain s ize distribution curve 58 Table 3.2: e m a x calculations for Badger sand 60 Table 3.3: e m i n calculations for Badger sand 61 Table 3.4: Relation between t for a 9 5 % confidence and s ize "n" of a sample 65 Table 3.5: Sphericity of Badger sand 65 Table 3.6: Roundness of Badger sand 66 Table 3.7: C o d e s and characterist ics of tests 71 Table 4.1: Relative dilatancy index for Badger sand 81 Table 4.2: Rate of dilatancy for Badger sand 82 Table 4.3: Maximum contraction friction angle for Badger sand 82 Table 4.4: P h a s e transformation friction angle for Badger sand 84 Table 4.5: Different friction angles for Badger sand 86 Table 4.6: Va lues of roundness and void ratio for various sands 87 Table 4.7: Friction angles for Ottawa sand and Fraser River sand 88 Table 4.8: Parameters summary of NorSand for Badger sand 103 Table 5.1: Summary of Badger sand parameters 106 Table A . 1 : System of units used by F L A C . 121 Table A .2 : Parameter summary for NorSand model 143 vi L I S T O F F I G U R E S page Figure 2.1: Effective stresses on a soil element 6 Figure 2.2: Drained and undrained stress path for compression triaxial test 11 Figure 2.3: Drained and undrained stress-strain response for (a) loose sand specimens (b) dense sand specimens 11 Figure 2.4: Contributions to shear strength of granular soils 14 Figure 2.5: Maximum obliquity friction angle and phase transformation friction angle 17 Figure 2.6: Measurement of pebble diameters 20 Figure 2.7: Relations between roundness and minimum and maximum void ratios 21 Figure 2.8: Mohr-Coulomb yield surface with an associated flow rule 25 Figure 2.9: Zones on the q-p' diagram 25 Figure 2.10: Definition of normality 26 Figure 2.11: Difference between rate and absolute definitions of dilatancy 26 Figure 2.12: Shape of the CSL 33 Figure 2.13: CSL in the e-ln(p') space 33 Figure 2.14: Work done in a quadrilateral element 36 Figure 2.15: Original Cam Clay yield surface 36 Figure 2.16: Modified Cam Clay yield surface 38 Figure 2,17: Illustration of consistency condition 38 Figure 2.18: Four-step loop to solve plasticity models 39 Figure 2.19: Nova's stress-dilatancy rule 42 Figure 2.20: Comparison of stress-dilatancy relations 43 Figure 2.21: Difference between state parameter and overconsolidation, infinity of NCL 50 Figure 2.22: NorSand image condition on yield surface (ys) 53 Figure 2.25: Definition of %: D p versus \\i 53 Figure 3.1: Grain size distribution curve of Badger sand 59 Figure 3.2: Microscope photograph: well-rounded particles 59 Figure 3.3: e m a x and e m i n determinations 62 Figure 3.4: Correlations between minimum and maximum void ratio 62 Figure 3.5: Angle of repose of Badger sand 64 Figure 3.6: Sphericity and roundness after Santamarina and Cho (2004) 66 Figure 3.7: Triaxial apparatus 68 Figure 3.8: Schematic diagram of UBC triaxial apparatus 68 Figure 3.9: Sand sample before testing (dummy sample) ' 72 Figure 3.10: Shape of the specimen after testing (CDP100) 72 vii Figure 3.11: Triaxial (CD) test results: a) deviator stress and b) volumetric strain 74 Figure 3.12: Triaxial (CU) test results: a) deviator stress and b) excess pore pressure 75 Figure 3.13: Triaxial test results: void ratio 76 Figure 3.14: Triaxial test results: stress path 76 Figure 3.15: G m a x in function of mean stress for Badger sand 77 Figure 4.1: D m i n versus r|m a x for CDP50, CDP100 and CDP150 80 Figure 4.2: Stress dilatancy plot for CDP100 80 Figure 4.3: Relation between angle of repose and maximum contraction friction angle 83 Figure 4.4: Relation between roundness and extreme void ratios 88 Figure 4.5: Relation between roundness and constant volume friction angle 89 Figure 4.6: Comparison of deviator stress and volumetric strain between triaxial data and NorSand model for CDP50 91 Figure 4.7: Comparison of deviator stress and volumetric strain between triaxial data and NorSand model for CDP100 92 Figure 4.8: Comparison of deviator stress and volumetric strain between triaxial data and NorSand model for CDP150 93 Figure 4.9: Comparison between CDP100 and NorSand model for a greater hardening 94 Figure 4.10: Comparison of deviator stress pore water pressure between triaxial data and NorSand model for CUP100 95 Figure 4.11: Comparison of deviator stress pore water pressure between triaxial data and NorSand model for CUP150 96 Figure 4.12: Comparison of deviator stress pore water pressure between triaxial data and NorSand model for CUP100, for R =2 97 Figure 4.13: Critical State Line and void ratios 100 Figure 4.14: Dilation at peak strength, D m i n , as a function of the state parameter 100 Figure 4.15: Stress-Dilatancy comparison between NorSand and CDP100 102 Figure 4.16: Zoom of stress-dilatancy for CDP100 102 Figure A. 1: General flow chart 119 Figure A.2: Example of FLAC model 120 Figure A.3: FLAC zone divided in 4 triangles; triangular element with velocity vectors; nodal force vector 120 Figure A.4: Positive shear stress (or strain) convention and associated distortion 122 Figure A.5: Force acting on a mass 128 Figure A.6: Flow chart Model Modified Cam Clay fish function 135 Figure A.7: Original Cam Clay fish function flow chart 139 Figure A.8: Comparison of deviator stress and volumetric strain for Original Cam Clay, between VBA code and FLAC results 140 vii i Figure A.9: NorSand fish function flow chart 144 Figure A.10: Comparison of deviator stress and volumetric strain for NorSand, between VBA Code and FLAC results for a loose state 145 Figure A.11: Comparison of deviator stress and volumetric strain for NorSand, between VBA Code and FLAC results for a dense state 146 ix LIST O F S Y M B O L S C c : coefficient of curvature Cu: coefficient of uniformity X- dilatancy rate parameter D: dilatancy QABS. absolute dilatancy ( D A B S = sv/sq • D : • ' 1 ' rate of dilatancy (D = sv / sq ) peak dilatancy DR: relative density D E 5 : the 85% size D60: the 60% size Dso: the 50% size D30: the 30% size D15: the 15% size D10: the 10% size dV: change of volume e: void ratio ec: critical void ratio maximum void ratio &min- minimum void ratio ex: strain in principal direction 1 s2: strain in principal direction 2 s3: strain in principal direction 3 volumetric strain (sv-el+s2+s3) shear strain (sq -2/3(£]-s3)) 1- effective stress ratio (rj- q/p') E: Young's modulus F: yield surface G: elastic shear modulus H: hardening parameter le- void ratio difference In- NorSand elastic property X IR: relative dilatancy index K: slope of the elastic line in the e-ln(p') space K: Rowe's constant K: bulk modulus A: slope of the critical state line in the e-ln(p') space Xs: plastic multiplicator n void ratio at p' = 1kPa M: critical effective stress ratio Mf. M at image condition N: Nova's volumetric coupling parameter v: Poisson's ratio p': mean effective stress Pc': - mean effective stress at critical state Pen': mean consolidation pressure of the current yield surface Pi': p' at image condition W- state parameter Wf- state parameter at image condition v: specific volume ^repose-angle of repose <t>cv'-constant volume friction angle <t>9: peak friction angle </>MC- maximum contraction friction angle <t>pj: phase transformation friction angle <j>MO- maximum obliquity friction angle q- deviator siress invariant Q: plastic potential R: roundness, overconsolidation ratio S: sphericity c r / ; effective stress in principal direction 1 cr 2 ' ; effective stress in principal direction 2 cr 3 ' : effective stress in principal direction 3 0: Lode angle u: pore water pressure V: volume xi V0: initial volume Vs: volume of solids W: work Notes: The superscripts "e" and "p" refer to elastic and plastic, respectively The subscript "tc" refers to triaxial compression The symbol refers to "rate of or "increment" xii ACKNOWLEDGMENTS I would like to thank my supervisors, Dr. Dawn Shuttle and Dr. Jonathan Fannin, for their time and guidance during this study. Without their support and advice this study would not have been possible. I would also like to thank Dr. Mike Jefferies for his personal communication and very useful comments, and Dr. John Howie for reviewing this manuscript and providing helpful input. Thanks to my colleagues in the Graduate Soil Mechanics Laboratory at UBC for their useful discussions and help. Finally, I would like to thank Pablo for his support and love during this process and his understanding when working until late and on weekends. xiv C H A P T E R I: INTRODUCTION The engineering study of soils requires a knowledge of the stress-strain behaviour of the material and its intrinsic properties, such as maximum and minimum void ratios and strength parameters. The constant volume friction angle is of particular interest, given the importance of this parameter to engineering analyses. In the case of coarse-grained soils, factors including grain size distribution, fines content, mineralogy and grain shape all influence the intrinsic properties such as void ratio and friction angle. Recent efforts have sought to correlate the influence of the grain shape, quantified by an index value of roundness (R), on these intrinsic properties of sands. Several authors have found that the minimum and maximum void ratios, as well as the constant volume friction angle, decrease as the roundness of the particles increases. Most of the studies make reference to sub-angular to rounded sands (0.1<R<0.7), and to perfectly spherical glass beads (R«1). It transpires that few data are published for the intervening grain shape given by well-rounded sands. In addition, numerical modelling is more commonly used in engineering design, as well as for behaviour prediction of soils and structures. Different constitutive models of varying complexity have been developed in order to capture soil behaviour. These models have been implemented in simple codes such as Visual Basic for Applications (VBA) or in more complex commercially available software. This study addresses the influence of grain shape on the behaviour of sands: more specifically, it examines the influence of roundness on properties such as minimum and maximum void ratio, and the angle of shearing resistance. Additionally, the NorSand constitutive model is evaluated for its ability to capture the behaviour of a well-rounded sand. Given the focus of the research, a laboratory testing program was undertaken on a uniformly graded, well-rounded coarse sand, referred to as "Badger sand", in order to determine its properties. The roundness of Badger sand was quantified, and its strength properties established from analysis of the response to drained and undrained loading in triaxial compression. The findings were related to, and then compared with, available data for sands in the literature. The comparison makes reference to the state of the art on the quantification of various friction angles used to characterize the response to shearing, including the constant volume friction angle, phase transformation friction angle and maximum contraction friction angle, among others. 1 The NorSand constitutive model was used to predict the behaviour of Badger sand. NorSand has been able to capture the behaviour of sands such as, Erksak sand (Jefferies and Shuttle, 2005), Brasted sand (Jefferies and Shuttle, 2002) and Ticino sand (Shuttle and Jefferies, 1998) which are more angular sands than Badger sand. An objective of this study is to verify that NorSand is also able to capture the behaviour of well-rounded sands, such as Badger sand. 1.1. Purpose of the study The main objectives of this study are: • Measure physical properties of Badger sand grains, such as roundness, and the grain assembly, such as minimum and maximum void ratio. • Determine strength parameters of Badger sand, including the angle of repose, failure envelope friction angle, phase transformation friction angle, maximum contraction friction angle and constant volume friction angle. Correlate the roundness of Badger sand with its extreme void ratios, and constant volume friction angle, and contrast those findings with relations for sands in the literature. • Compare the relative magnitude of the different friction angles obtained for Badger sand, and examine the implications for characterization of well-rounded sands. Determine the elastic, plastic and critical state properties of Badger sand for modelling purposes. • Capture the behaviour of Badger sand with the NorSand constitutive model within FLAC. To meet these objectives, a triaxial testing program was undertaken and the results obtained used to establish parameters for the sand that were verified through comparison with findings reported in the literature. Finally, parameters needed for the NorSand constitutive model are obtained. 2 1.2. Organization of the thesis In Chapter II a literature review is presented explaining the drained and undrained behaviour of sands, defining the different friction angles mobilized during laboratory testing and introducing two concepts for characterization of the grain shape, namely, roundness and sphericity. In addition, the concept of critical state soil mechanics (CSSM) is defined and a description provided for three constitutive models (Original Cam Clay, Modified Cam Clay and NorSand) that are based on the C S S M . Chapter III describes the sand used for this research and the program of laboratory testing undertaken in the graduate geotechnical laboratory at UBC. Results obtained from the laboratory testing program are also presented. A discussion and analysis of results is presented in Chapter IV. First is a discussion of the parameters obtained from tests on the well-rounded badger sand described in Chapter III, followed by a comparison of those results with values for other sands found in the literature. Finally the NorSand model is used to fit the tests, and the parameters needed for modelling purposes are obtained and discussed. A summary of these findings and the companion conclusions is given in Chapter V. In addition the implementation of NorSand within FLAC (Fast Lagrangian Analysis of Continua) is explained in Appendix A. First a brief description of the FLAC program is presented, followed by an explanation of the steps taken to code the NorSand model. 3 CHAPTER II: LITERATURE REVIEW The shear strength of a sand depends on several factors, including the type of loading (triaxial compression, triaxial extension, plane strain), type of drainage (drained or undrained), initial state (loose or dense), and initial fabric. Recently, it has been shown that the so-called intrinsic characteristics of sands are also a function of the shape of the grain particles. In order to capture the behaviour of sands, several constitutive models have been proposed. In this chapter, first, a general description of sand behaviour is given, followed by a definition of the various friction-related angles mobilized during the shearing of a sand specimen, after which the concepts of roundness and sphericity are reviewed. Then a description is provided of different constitutive models, starting with Original Cam Clay, followed by Modified Cam Clay, which are the first Critical State Soil Mechanics constitutive models, and finally a description of NorSand, the first CSSM constitutive model able to capture the real behaviour of sands. 2.1. Behaviour of sands 2.1.1. General stress-strain behaviour of soils The general stresses and strains on a soil element are shown in Figure 2.1. Stresses acting perpendicular to the planes in the 1,2,3-direction are an' , a 2 2' and a3 3', respectively. The stresses oy are determined using the suffix i to indicate the direction of the normal stress to that plane and the suffix j to indicate the actual direction of the stress. The stress matrix can be written as: <7 1 2 ' °"13' c r 2 1 ' <J22 °"23 ' 2.1 ° 3 l ' °Z2 °"33'_ where aVj' is symmetrical and c r 1 2 ' = c r 2 1 ' , o - 1 3 ' = <7 3 1' and <T 3 2 '= <7 2 3 ' , hence six independent components are needed to describe the stresses on a soil element. Similarly, the stress increment is written as: 4 • • • cr 1 12 13 • • • 22 23 • • • i 32 a 33 where cnf is symmetrical and is the difference in effective stress due to the load increment applied to the element. When the load increment is applied, the displacement of the element is decomposed in three parts comprising, displacement, rotation and distortion, and the corresponding strain increment can be written as: £ 11 S 12 S 1 £ 21 ^22 E 2 ^ 31 ^ •*-> ^ i '32 ° 33 2.3 where EJJ' is symmetrical. It is possible to find the principal stresses and strains in three orthogonal planes, where no shear stress or strain occurs. The stress matrix can be written as: 0-1 0 0 0 0 2.4 where 1, 2 and 3 are the principal stress directions. The matrix for the stress increment and strain increment can be written under the same form, but the principal directions are not necessarily the same as the principal stress directions. 2.1.2. Triaxial testing Compression triaxial tests are widely used to characterize the strength of soils. A cylindrical specimen is typically subjected to an hydrostatic confining pressure (cr0'= a3') and then sheared by means of an axial load (o-0'+A<r'= a-.,'). Axial displacement is measured as the specimen deforms; volumetric strain is measured if the test is performed under drained conditions, or pore water pressure if the test is undrained. 5 Figure 2.1: Effective stresses on a soil element As the sample is loaded, it is possible to obtain a relation between stress and strain. The slope of the stress-strain curve gives the Young's modulus (or elastic modulus), given by: 2.5 When the element in Figure 2.1 is subjected to an isotropic stress, the volumetric change, s„ is: dV 2.6 where V 0 is the initial volume of the element and dV is the change in volume. 6 The general strains in an elastic element having constant isotropic elastic stiffness are given by: s2 = — (- vAcr1'+A<T2'-v /Ao"3') 2.7 £ 3 = — (- vA<r^-vAcr2'+Acr3') where, v is the Poisson's ratio. The volumetric strain is the sum of the strains in the three principal directions, and is given by: sv = + s2 + e3 2.8 From the principal stress matrix (Eq. 2.4), the mean effective stress, p', and deviator stress, q, are defined by: g-= ^[{a:-crzf +{a2-a3f + (a3'-a,f\ Stress and strain invariants should be work-conjugate, meaning that the stresses and strains components and the invariants can be used interchangeably in the equation of work. Hence: Work done per unit volume -qs g + p'sv = (T\ + <y'2 s2 + a'3 s3 2.10 Where sq and sv are the shear strain increment and volumetric strain increment, respectively. Special care is needed when choosing the shear strain component in 3-D. As the volumetric and shear strains are separated, it is necessary to find a work conjugate expression for the shear strain increment, as the volumetric strain increment is already a strain invariant (Eq. 2.8). Rearranging Equation 2.10, the shear strain increment is given by Equation 2.11: 7 where = {2a^-a2'-03'/3) ar>d s 2 and s 3 are found by cycling rotation. By replacing the principal stresses, the work-conjugate shear strain can be written as: )£*- 2 sin <?£-> + 2.12 Where 6 is the Lode angle. For triaxial compression, 6 = 3 0 ° . This thesis invokes stresses and strains found in triaxial compression, where <J2'=(T3'. Hence, the mean effective stress and deviator stress are given by: Similarly, the volumetric strain for triaxial compression is given by: 2.14 The total mean stress and deviator stress are given by: p = p'+u 2.15 q = q' where u is the pore water pressure. The bulk modulus, K, is defined as the ratio between the change in mean stress and change in volumetric strain. It represents the change of size at constant shape and is given by: where p' is the mean effective stress rate (from Eq. 2.13) and sv is the volumetric strain rate (from Eq. 2.8). The shear strain is given by: 2.17 The shear modulus represents the distortional part, ie, the change in shape at constant volume and it is independent of drainage conditions. G is given by: 3G = 2.18 Finally, the stress ratio, r| is defined by: i = q/p' 2.19 Figure 2.2 shows a typical stress-path in the q-p' space for the drained and undrained triaxial compression test. Replacing the deviator stress in the mean effective stress, the equation for the drained stress-path is: P = 2.20 This equation corresponds to a slope of 1:3 for drained triaxial compression tests (A -> C ). For an undrained condition, as the sample is loaded, the pore water pressure starts increasing, and the stress path follows A -> B. The difference between the drained and the undrained mean stress (D -> B ) is equal to the pore water pressure, u. At failure, a relation is found between the mean effective stress and the deviator stress given by: q = Mp' 2.21 9 The friction angle in triaxial compression, <j>tc, assoc iated with M is given by: sin<ztto = 3M 6 + M 2.22 2.1.3. Triaxial behaviour of sands Figure 2.3 shows a typical response of (a) loose and (b) dense sand specimens, under drained and undrained loading. For the loose spec imen, the void ratio-confining pressure combination is above the critical state line, which will be defined in details later in this chapter, (point A) . If the specimen is tested in the drained condit ion, the stress-path follows a 1:3 slope and ends in the critical state line (from A to C) . For an undrained condition, the stress-path goes from A to B. The difference between the drained and the undrained stress-path corresponds to the pore pressure generated during undrained shear. A s seen in Figure 2.3 the drained strength is higher than the undrained strength. For dense sand spec imen, the drained stress path follows the same 1:3 s lope and exhibits a peak friction angle, and then goes back (following the same path) and stops at the critical state (point F). The undrained stress path ends at point E showing, in this case , a higher strength than the drained case . Finally in the u-lnp' space, a loose spec imen starts above the critical state line (point A) and f inishes at C for a drained condition and at B for an undrained condition. A dense sample starts at the left of the critical state line (point D), shows a decrease in void ratio as the spec imen contracts and then an increase in void ratio as the specimen dilates and ends up in the critical state line (point F). 10 Projection of CSL q=Mp' A < • p' Po'= 03' Figure 2.2: Drained and undrained stress path for compression triaxial test («): ' ' W) Figure 2.3: Drained and undrained stress-strain response for (a) loose sand specimens (b) dense sand specimens (from Mitchell and Soga, 2005) 11 2.2. Definition of friction angles for sands As sand specimens are sheared, different friction angles are mobilized at each state; peak or residual friction angle for dense sands, critical state friction angle, and phase transformation, among others. Some authors have given conflicting definitions to these angles. Hence some confusion exists when trying to determine them from laboratory testing. The intent of this section is to define each of these angles and clearly identify them in order to be able to obtain the values from laboratory testing programs. Consider first the most commonly reported friction angles, namely, angle of repose, constant volume friction angle and peak friction angle. Thereafter a description follows of the maximum contraction friction angle, phase transformation friction angle and maximum obliquity friction angle. 2.2.1. Angle of repose (§ r e p ) Little information is found in the literature about the angle of repose. Some authors have given definitions and have tried to establish which factors influence its value. After the Dictionary of Earth Sciences (2003) the angle of repose is defined as the "maximum angle at which unconsolidated material can stand. Various factors determine this angle, including particle size and angularity, the degree of interlocking between particles, and pore water pressure." In order to measure the angle of repose, dry material is deposited as loose as possible through a funnel and the angle of the resulting heap is then calculated. Miura et al (1997) performed several tests to determine the angle of repose (0rep) of four granular materials (two sub angular sands and two rounded materials, glass beads) and the factors influencing its value. They found it decreases as the volume of material used increases, increases with the time elapsed to form the heap and increases with the roughness of the base plate upon which the heap is formed. The latter finding was corroborated by Chik and Vallejo (2005). The relation between the angle of repose and the angle of internal friction of granular materials is somewhat intriguing. Miura et al (1997) demonstrated that "the angle of repose of sand is equivalent to the angle of internal friction mobilized under low confining pressure within a thin surface sliding layer which is at the loosest state." This statement is in accordance with the observation of Lambe and Whitman (1979) that the angle of repose is approximately equal to the internal friction angle for very loose sands at very low confining pressures. 12 2.2.2. Critical state or steady state friction angle ((j)Cv) Roscoe et al (1958) defined the critical state as "the state at which a soil continues to deform at constant stress and constant void ratio". Similarly, Poulos (1981) defined the steady state as "the state in which a mass of particles continuously deforms at constant volume, constant normal effective stress, constant shear stress and constant velocity". Independent of the given definition, the critical state or steady state friction angle can be seen as a material property; it is a combination of the particle-to-particle friction angle and the degree of interlocking that can occur with zero overall volume change during continued shear deformation. This angle is related to the critical void ratio, e c (Lambe, 1979). Negussey et al (1988) performed ring shear tests on Ottawa sand, a tailings sand, granular copper, lead shot and glass beads in order to determine the effects of initial density, particle size and shape, gradation and confining stress, on the value of <j>cv. They found that value of the critical state friction angle is independent of the previously mentioned factors. This finding is supported by Verdugo (1992) who found that the critical state friction angle is independent of the initial state of the soil. Also Chu (1995) concluded from his tests on Sydney sand that the critical state friction angle is independent of the initial void ratio, but claimed it is a stress level dependent parameter, specially at low confining pressures; the higher the confining pressure, the lower the critical state friction angle. 2.2.3. Peak friction angle ((j)p) A peak friction angle is obtained from tests on dense samples (as shown in Figure 2.3). Grain properties, including angularity and grading, exert an influence on the values of (|>p (see Table 2.1). A more angular and well graded soil has a higher peak friction angle. Table 2.1: Effect of angularity and grading on peak friction angle (from Lambe and Whitman, 1979) Shape and Grading Loose Dense Rounded, uniform 30° 37° Rounded, well graded 34° 40° Angular, uniform 35° 43° Angular, well graded 39° 45° 13 The peak friction angle is the sum of the interparticle friction, rearrangement, crushing and dilation as shown in Figure 2.4 (Mitchell and Soga , 2005). This angle is not a material property s ince it depends on the density of the sample and stress-path (Bolton, 1986). Peak friction angle increases when the void ratio and the mean effective stress decrease. A lso , the peak friction angle is related to the rate of dilation of the soil. By correlating data of peak friction angle and constant volume friction angle, for 17 soils, Bolton (1986) found a relation between relative density (Dr) and the mean confining pressure, expressed as the relative dilatancy index, l R , given by: lR = Dr(Q - In p')- R 2.23 where values of Q=10 and R=1 give the best fit. Further in triaxial compression he found that the peak friction angle and the constant volume friction angle are related through 2.24. 3 /c 2.24 46-42 : 38. 4.34 ' 39/ 26 "Dilation. Interlocking Densest | True Friction Packing ? ^Rearrangement, Fabric":Development, ^•:Grushing;!estimated); Critical. .Void Ratio. T To Zero -t-'26' 30 34 38 Porosity : / i { % ) :42-Figure 2.4: Contributions to shear strength of granular soils (from Mitchell and S o g a , 2005) ' In the figure, <j>'crit corresponds to <j>Cv, <l>'m corresponds to <t>'p, and f f corresponds to the interparticle friction angle 14 2.2.4. Maximum contraction friction angle (<j>MC) The term "characteristic state" was proposed by Luong (1980) to define the point where the volumetric deformation passes from contraction to dilation, ie, sv = 0. This state was also defined as "maximum contraction state" by Negussey et al (1988). Kirkpatrick (1961) found that the maximum contraction friction angle is unique for each sand. However, Negussey et al (1988) found that this angle is not unique and varies with density and confining stress level; this last finding is further confirmed by Chu (1995). The stress ratio at the characteristic state is given by: where <j>MC is the friction angle mobilized at the characteristic state or maximum contraction state and it is an intrinsic property of the material tested. The subscript "MC" (for maximum contraction) was chosen instead "CS" (characteristic state) for a better distinction with the critical state or constant volume friction angle defined later in this chapter. It is attributed to the point at which interlocking between particles ceases and the disruption of interlocking starts. The characteristic state is independent of the initial void ratio, anisotropy and grain size (Luong, 1980). More studies performed by Luong in monotonic triaxial tests, showed that the characteristic state is also independent of the stress path imposed on the specimen. For drained triaxial tests under constant confining pressure, the characteristic state separates two regions in the behaviour of sands. A "subcharacteristic" region (or contraction), where the stress ratio rpq/p' is smaller than T | M C , which is the stress ratio at maximum contraction, and a "surcharacteristic" region (or dilation), where TJ > r/MC . In the subcharacteristic region, as the soil is loaded, the change in volume is due to interlocking of the sand particles. In the surcharacteristic region, Luong states that "the characteristic state merges with the critical state", a claim that is discussed, together with the concept of a phase transformation angle, in Chapter V with reference to the results obtained in this research. 2.25 15 2.2.5. Phase transformation friction angle ( § P J ) The state at which the response of the sand, in undrained tests, transforms from positive excess pore water pressure to negative excess pore water pressure is termed phase transformation by Ishihara et al (1975), see Figure 2.5. It corresponds to the maximum excess pore water pressure. The phase transformation state is very important for assessing the risk of liquefaction and limited liquefaction in loose (contractive) sand samples. Several authors have shown the stress ratio at the phase transformation point is independent of the initial void ratio (Vaid and Chern, 1985), confining stress and loading, and hence it is unique for a sand. This finding is confirmed by Negussey et al (1988) who showed that the phase transformation friction angle is independent of the density of the specimen. However, in contrast Chu (1995) showed that the phase transformation state changes with the initial confining stress and the initial void ratio of soil. Finally Luong (1980) suggested that the maximum contraction friction angle established in drained tests is the same as the phase transformation friction angle from undrained tests. These findings are confirmed by Chu (1995). However, Negussey et al (1988) suggested that this condition is valid only for loose specimens. 2.2.6. Maximum obliquity friction angle ((j>Mo) The failure envelope or maximum obliquity friction angle corresponds to the state where the effective stress ratio is maximum, ie, (a^ '/<r3 ' ) m a x . In undrained tests, at strains larger than that at the phase transformation friction angle, the stress path climbs up and approaches the maximum obliquity line tangentially until the end of the test. It has been shown that the maximum obliquity friction angle is dependent on the angularity of the sand grains and the depositional structure (Al Hattamleh, 2005). Vaid and Chern (1985) observe the failure envelope "represents the upper limit of all undrained effective stress paths, excluding cases when liquefaction occurred". They suggested that the maximum obliquity friction angle is unique for each sand and that it is independent of the initial state before undrained shearing. Vaid and Eliadorani (1998) suggested that the maximum obliquity state in drained triaxial tests is the instant of instability. Figure 2.5 shows a schematic representation on the maximum obliquity friction angle and the phase transformation friction angle. 16 p' Figure 2.5: Maximum obliquity friction angle and phase transformation friction angle 2.3. Definition of roundness and sphericity In this section, two concepts are introduced; sphericity (a 3-D phenomenon) and roundness (a 2-D phenomenon). These terms have been in use for many years, and different definitions can be found in the literature. 2.3.1. Sphericity From the Dictionary of Earth Sciences (2003), "Sphericity can be determined by examining the relation between the long (L), intermediate (I), and short (S) axes of the particle, the maximum projection Sphericity, yj, being given by the expression: 2.26 17 Krumbein (1941) observed sphericity (S) as: "Fundamentally the shape is a measure of the ratio of the surface area of a particle to its volume. For practical purposes this ratio is difficult to measure, and the actual measurement is expressed in terms of the ratio of the volume of the particle to the volume of its circumscribing sphere. The cube root of this ratio is called the sphericity of the particle." He considered the long (a), intermediate (b) and short (c) axes as mutually perpendicular intercepts, as shown in Figure 2.6, which yields: Hx/ehbc [be S = 3 V / " = 3 — 2.27 E q . 2.26 and 2.27, for a volumetric calculation of sphericity, are different. However, taking the particular cases of a perfect sphere and an elongated grain, the values of S are equal and both between 0 and 1. For a perfect sphere: L = l = S = a = b = c , s - ^ f l - l « d r - £ l - i For an elongated grain: L = 21 = 4 S and a = 2b = 4c, S-sjl-0.5 and „ = 3^=0.5 Santamar ina and C h o (2004), defined sphericity as a two-dimensional problem: "Sphericity is quantified as the diameter ratio between the largest inscribed and the smallest circumscribing sphere". This latest expression was defined by Mitchell and S o g a (2005) as an "Inscribed circle sphericity" given by: 2.28 18 2.3.2. Roundness From the Dictionary of Earth Sciences (2003), the Roundness Index (RI) is defined as "the average radius of curvature of the corners of a particle, divided by the radius of the maximum inscribed circle for a two-dimensional image of the particle." Krumbein (1941) defined the roundness (R) of a particle as "a measure of the curvature of the corners and edges expressed as a ratio to the average curvature of the particle as a whole, independent of its form. For practical purposes the "average curvature" is expressed in terms of the inscribed circle drawn on a projection of the particle in a plane." Santamarina and Cho (2004), defined roundness as: Roundness is quantified as the average radius of curvature of surface features relative to the radius of the maximum sphere that can be inscribed in the particle." The three previous definitions are the same and are given by: 2.29 max Other definitions for roundness are also available but are less used; they are presented for completeness in Equations 2.30 and 2.31 (Mitchell and Soga, 2005): Radius of curvature of the most convex part 2.30 0.5(longest diameter through the most convex part) Radius of curvature of the most convex part : 2.31 Mean radius Powers (1953) proposed a classification table with six intervals to define the roundness of particles, as shown in Table 2.2. 19 Figure 2.6: Measurement of pebble diameters. Left, the b-axis in position; center, the c-axis in position; right, the pebble in perspective (from Krumbein, 1941) Table 2.2: Categories of roundness (from Powers, 1953) Grade Terms Class Interval Geometric Means Very Angular 0.12-0.17 0.14 Angular 0.17-0.25 0.21 Subangular 0.25-0.35 0.30 Subrounded 0.35-0.49 0.41 Rounded 0.49-0.70 0.59 Well Rounded 0.70-1.00 0.84 Several authors have studied the influence of particle shape on material properties such as minimum and maximum void ratios. It has been observed that e m i n and e m a x decrease with increasing roundness of the particles. Superimposed in Figure 2.7 are the relations found by Youd (1973), Shimobe and Moroto (1995) and Santamarina and Cho (2004). All three of them obtained the values of roundness using the method described by Powers (1953), corresponding to Equation 2.29. Inspection shows that the values found by Santamarina and Cho define an upper limit for the void ratios and the values of Youd define a lower limit. 20 Youd (1973) reported the values of e m i n , e m a x and R for specimens of Ottawa Sand, Del Monte White Sand, Monterey Sand, Lapis Lacustre Sand and Crushed Basalt, all of them having a coefficient of uniformity C u = 1.4. Shimobe and Moroto (1995) found a relation between e m a x and R for uniform samples with a coefficient of uniformity Cu < 2 based on results obtained with materials as sand, gravel, glass beads and sand mixed with glass beads, given by: e m a x = 0.642 /r 0 3 5 4 2.32 Based on observations of 54 specimens of angular-crushed and natural-rounded sands, Santamarina and Cho (2004), proposed a relation between the roundness of the particles and the values of e m i n and e m a x , defined in Equation 2.33. * 0.154 „ 0.082 e m a y =0.554 + and e m i n = 0.359 + 2.33 1.6 0.2 0.0 1 1 1 1 0 0.2 0.4 0.6 0.8 1 R o u n d n e s s , R Figure 2.7: Relations between roundness and minimum and maximum void ratios 21 2.4. Critical State Soil Mechanics Soil exists in a large variety of densities, going from very loose to very dense, and the stress-strain and volumetric behaviour depend on those variations. Many constitutive models for soils exist where for each density, the model has to recalibrate, (ie, each soil density has a different parameter set). It is more convenient to introduce a constitutive model able to capture and predict the behaviour of soils at different void ratios. This section introduces the concept of Critical State Soil Mechanics (CSSM) and two of the most important and well known constitutive models, Original Cam Clay (Roscoe, Schofield and Thurairajah, 1963) and Modified Cam Clay (Roscoe and Burland, 1968). For convenience, before discussing C S S M consider first general concepts of plasticity. 2.4.1. General plasticity concepts All elasto-plastic models comprise four main components: i) elastic properties, ii) a yield surface which is the locus of stresses leading to plastic strain, iii) a hardening/softening rule, which determines the size of the yield surface, and iv) a flow rule which specifies the direction of the plastic strain increment vector. Each of these terms is considered separately together with other plasticity concepts. The definitions are given below: - Elasticity: All plasticity models assume elasticity at very small strains. Elasticity is associated with recoverable strains and a 1:1 relation between stresses and strains, ie, if we know the strain level we also know the stress level. - Yield surface, F: In simple words, the yield surface is the interface between an elastic and an inelastic region. Yielding is associated with the occurrence of an irrecoverable, ie plastic, volumetric strain increment e% and an irrecoverable, plastic, shear strain increment e% . 22 It was proposed that the yield surface must intersect the NCL in order to have an end, or cap. A cap implies that the model yields in isotropic compression. Hence, the Mohr-Coulomb criterion could not be seen as a yield surface. Finally a curved yield surface which closes was proposed as for example, Figure 2.15. That is the start of critical state soil mechanics. - Normality: Taking a point A on the yield surface, as shown in Figure 2.9, there exist three options. When a stress increment is applied, the first option is that a stress increment combination moves toward E in the elastic zone. In this case all strains are elastic and no plastic work is done. The second option is that the stress increment combination moves from A toward P into the plastic zone. In that case plastic work is done and energy is dissipated. Finally, the third option is that the stress increment makes A move toward Y, along the tangent to the yield surface. In this case no work is dissipated. Equation 2.34 is the definition of normality. W = p,ef+qef! =0 q_ 2.34 Figure 2.10 shows the normality condition. With normality, the principal stress and principal strain increment are aligned and the net strain increment vector is normal to the yield surface at the stress state corresponding to the present yielding. - Plastic potential surface, Q: For each combination of stresses a vector of plastic strains can be drawn. The curves formed by orthogonal plastic strain increment vectors : are called plastic potentials. After Poorooshasb et al (1966), "the plastic potential curves for a given value of e should form a family of geometrically similar curves." The plastic strain increments are defined as: 23 where A.s is a scalar which depends on the mode of yielding and determines the incremental magnitude and Q is the plastic potential surface. If the yield surface (F) and the plastic potential surface (Q) are the same, then the material is obeying the postulate of normality or associated flow. - Hardening rule: The hardening rule defines how the size of the yield surface changes with increasing plastic shear strain If the yield surface becomes larger, the material is hardening and if the yield v j surface becomes smaller, the material is softening. - Dilatancy: Stress-di latancy (or flow rule) relates the st resses and strains increments of a soil . Dilatancy was first defined by Reynolds in 1885 as "dilatancy consists in a definite change of bulk, consequent on a definite change of shape or distortional strain, any disturbance whatever causing a change of volume and generally dilatation. ... A medium composed of grains of any possible shape, possessed this property of dilatancy, so as long as one important condition was satisf ied... . Th is condition is, that the medium should be continuous, infinite in extent, or the grains at the boundary should be so held as to prevent a rearrangement commencing". Two definitions of dilatancy are widely used in North Amer ica , the absolute and the rate of dilatancy. Jefferies and Been (2005) defined the absolute dilatancy as "dilation is the change in volumetric strain incurred s ince the initial condition" given by: 2.36 24 The rate of dilatancy is defined as "dilation is the ratio of rate (or increment) of volume change with rate (or increment) of shear strain and given by D . This is shown in Figure 2.11. When normality was applied to the Mohr-Coulomb yield surface, dilatancy larger than the ones measured in real soils were found, given that the dilation angle was equal to the friction angle, as shown in Figure 2.8. 25 Figure 2.11: Difference between rate and absolute definitions of dilatancy (adapted from Jefferies and Been, 2005) 26 2.4.2. The critical state Before the soil mechanics was established as a subject, Reynolds (1885), found that by placing sand and water into "thin india-rubber balloons", the sand tended to change in volume as the bag was squeezed. This observation was reproduced by Casagrande (1936) at MIT in Cambridge MA, who showed experimentally doing drained shear box tests in sands, that "dense sand expands and very loose sand reduces its volume". In addition, Casagrande found that irrespective of the initial density of the sand, loose and dense sands ended up at the same ultimate shear resistance (r u / () and void ratio. He defined this ultimate void ratio as the "critical void ratio". Later on, Roscoe et al (1958) took Casagrande's idea of "critical void ratio" and defined the critical state of soils as "the state at which a soil continues to deform at constant stress and constant void ratio". Two ideas appeared from this definition, the first is that at the critical state the soil reaches a constant void ratio (or the "critical void ratio") and the second is that this constant void ratio does not change as shearing continues. 10 years later, supervised by Casagrande, Castro (1969) preformed stress controlled triaxial tests on loose sand samples achieving systematically a steady state of deformation at the end of the tests. He defined this state as "steady state line". Finally, Poulos (1981) formalized Castro's idea and defined the steady state as "the state in which a mass of particles continuously deforms at constant volume, constant normal effective stress, constant shear stress and constant velocity". The locus of mean effective stress and void ratio combination at the steady state is referred as the steady state line for undrained tests. There has been much discussion on whether the critical and steady state are the same (Alarcon-Guzman et al (1988), Oyenuga and Tisot (1989), Been et al (1991), Chu and Lo (1992), Verdugo (1992), and Negussey and Islam (1993), among others). Review of the literature seems to suggest that the only difference between them is in how they are measured; Poulos (1981) suggested that the steady state is measured in undrained tests on fully saturated contractive sand samples and the critical state is obtained from drained tests on dilative sand samples. Been et al (1991) showed that for Erksak 330/0.7 sand, the critical and steady state are the same. This finding is confirmed by Chu (1995). For critical state soil mechanics, it is assumed that the critical state and the steady state are the same and unique (as discussed later in this Chapter). The concept that when sheared, soils reach the already defined critical state locus is the framework for CSSM. C S S M models are based on two ideas, formally termed axioms: 27 - Axiom 1: "A unique locus exists in q, p', e space such that soil can be deformed without limit at constant stress and constant void ratio; this locus is called the critical state locus (CSL)". This axiom can be mathematically represented by Equation 2.37. In simple words, Eq. 2.37 means that there exists (3) a function C, at a constant mean effective stress, such that (B ) the volumetric strain rate and ( A ) the change in volumetric strain rate must be zero in order to be in the CSL for every (V ) shear strain. 3C(e,q,p'). 3sv=0^ev=0 2.37 p'=0 where C( ) is the function that defines the CSL which at the critical state has a unique value function of the void ratio and the mean effective stress. This condition is equivalent to the one proposed by Roscoe et al (1958) in terms of dilatancy, where the dilatancy and the rate of dilatancy must be zero(D = 0) and (D = 0) at the critical state. The Second Axiom is stated as: - Axiom 2: "The CSL forms the ultimate condition of all distortional processes in soil, so that all monotonic distortional stress state paths tend to this locus". e -» ec as sq - > oo V tj > M 2.38 This condition is equivalent to Casagrande's idea that as shear strains develop, the void ratio tends to the critical void ratio. It is usual to idealise the critical state locus (CSL) as a straight line in the e-lnp' space, however this is not a condition. It is just a mathematical simplification; the critical state locus can be curved. The CSL in the e-lnp' space is usually written as: ec=r-A\npc' 2.39 where r corresponds to the void ratio at p' = 1kPa, X is the slope of the CSL and pc' is the mean effective stress at the critical state. This is shown in Figure 2.13. 28 Data on Erksak sand (Been et al 1991) and Toyoura sand (Ishihara, 1993) suggest that the CSL is not a straight line in the e-ln(p') plane. Figure 2.12 shows data on Erksak sand where the critical state line curves at around 1MPa. Been et al (1991) suggested that this break point is given by some breakage of the grains. Verdugo (1992) discusses this idea by showing that if the critical state line is plotted in a bi-linear manner, the curvature of the CSL happens at a lower stress. In addition Verdugo shows that the critical state line is linear at higher stresses when plotted in a linear scale. The shape of the critical state line is not of real importance for the void ratio-mean stress combinations used in practice. For C S S M the important condition is that the CSL has to be unique. Authors such as Been et al (1991), Verdugo (1992) and Ishihara (1993) found an apparent unique CSL for Erksak and Toyoura sand. However, some authors have suggested that the CSL is not unique and depends on triaxial compression or extension, or initial fabric (for example, Vaid et al, 1990, Negussey and Islam, 1994). Been (1999) discussed the arguments given by the different authors in order to clarify the non-uniqueness of the CSL. He suggests that special care is needed when defining the critical state. The confusion arises when considering the critical state at the phase transformation point, defined by Ishihara et al (1975) also called pseudo or quasi steady state (Alarcon et al, 1988 and Zhang and Garga, 1997). As well, Jefferies, 1999, states that the critical state is defined by dilatancy equal zero but also the rate of dilatancy must be zero, and that choosing the phase transformation point as the critical state is a mistake given that this second condition is not satisfied. This topic is not the intent of this thesis, but it is important to recognize that the uniqueness of the CSL is not universally accepted. For the development of C S S M constitutive models, a unique CSL has been assumed. As shown in Figure 2.2, in the stress space, the CSL is also unique and defined as: q = Mp' 2.40 where, for triaxial condition, M is given in terms of the friction angle by Eq. 2.22. Note that uniqueness requires a unique locus, not constant M in the rt-plane or with p'. It is important to state that constant M does not mean a constant friction angle. For constant friction angle M varies with the Lode angle, 6, which represents the intermediate principal stress, a2 • For triaxial compression, 0 has a value of 30° and for triaxial extension 0 = -30°. Plotting the Lode angle against the dilatancy for plane strain condition, it is found that 9 is not constant. Hence, there is no unique M p s (Jefferies and Shuttle, 2002). The question is how M varies with the Lode angle. Data on Brasted sand shows that for plane strain, M varies between the Mohr-29 Coulomb criterion and the Matsuoka-Nakai criterion. For constant friction angle, the Mohr-Coulomb criterion is given by Equation 2.41. . M = (3S)/[cos 0(1 + 6/M ( c) - Jz sin&\ 2.41 For triaxial tests, there are different ways to find the value of M. For loose sand specimens, M is found as the value of the stress ratio at the end of the test. However, for dense sand specimens, as they dilate, the critical state can be reached only at very large strains. A different approach to find the critical stress ratio, M, for dense samples was proposed by Bishop (1966). M can be obtained by plotting the maximum stress ratio, r / m a x against the dilation, D m i n 2 , of several drained tests. At the critical state Dmin=0, hence, taking the best fit between the points and projecting to Dmin=0, M is obtained as the value of rjmax at Dmin=0. 2.4.3. Original Cam Clay formulation Original Cam Clay was first introduced by Roscoe, Schofield and Thurairajah (1963) and is based on plastic work dissipation. Consider an element where the only stresses acting on it are the principal effective stresses, oV, a 2 ' and a 3 ' . If the sides of this element shorten by the strains E I , E 2 and E 3 , respectively, due to the applications of these stresses (Figure 2.14), then the work per unit volume is given by: W = a/£1 + a2 E2 + cr3'£3 2.42 Here, two assumptions have been made. The first is that the effective stresses are constant during the strain increment and that the change of area and volume of the element is not affected by the small strains. Using work conjugate stress and strain invariants: • • • • • Work done per unit volume=g£p + p'ev = a\ e^ + a'2 £2 + <*'z £z 2-10) and translating Equation 2.42 for a triaxial condition, the total rate of working per unit volume on the soil skeleton can be divided in an elastic and a plastic component as: 2 Peak dilatancy is D m i n because of the dilation-negative convention 30 W =We+Wp =qeq+p,£v 2.43 where the subscript "e" means elastic and "p" means plastic. Here, the elastic part of the work is stored as strain energy (as long as the soil remains in the the K-line (Figure 2.13)) and the plastic part is dissipated. Another assumption is made here for Original Cam Clay, all deviatoric energy is dissipated, ie, the elastic shear modulus is infinite. In Figure 2.13, assuming a movement from A to Q, there is a plastic part ( A to P ) and an elastic part ( P to Q). The change in volumetric strain can be written as: • _ °A - e g ^ e 2.44 1 + e 1 + e Considering the equation of the K-line as: e = eK - * - lnp ' 2.45 where eK is the void ratio for a particular K-line taken at 1 kPa. Differentiating Equation 2.45 and replacing in Equation 2.44, «• 1 *, sv = p' 2.46 v p' 1 + e The elastic bulk modulus is given by: *=1^ 2.47 P' K As shown in Equations 2.8 and 2.17, the volumetric and shear strains are linear and can also be decomposed in elastic (P to Q ) and plastic (A to P) components. Hence, the plastic rate of working can be written as: Wp=q£P+p'£P 2 4 8 31 Dividing by the mean effective stress and the plastic shear strain increment in order to make the plastic work rate dimensionless, we have that the dimensionless plastic work rate is just the sum of the plastic dilation rate (Dp) and the stress ratio (q/p'), as shown in Equation 2.49: 2.49 Original Cam Clay assumes that the dimensionless plastic energy dissipation rate is constant and equal to M. ' / * Equation 2.49 is consistent with the idea that at the critical state, dilatancy ,£?/ eg , is equal to zero and the stress ratio r| is equal to M. Hence, rearranging Equation 2.49 gives a stress-dilatancy relation for Original Cam Clay: In order to know when the plastic yielding occurs, a yield surface is needed. To derive the yield surface, two assumptions are needed; normality and the previous stress-dilatancy relation From the definition of stress ratio q = rjp' (Eq. 2.19) and taking the differential, the change in shear stress can be written as: Dp =M-rj 2.50 (Eq. 2.50). q = p'r]+rjp' 2.51 Where rj is the change in stress ratio. From normality: 2.52 32 <~i Load controlled,' compress ion a Strain coni roted, compression <5 Strafn-cofitrollsd, extension 10 100 p' (kPa) 1.000 10000 Figure 2.12: Shape of the CSL (from Jefferies and Been, 2005) 33 Substituting Equation 2.51 in Equation 2.52 and rearranging: ••£- + _ J L _ = 0 2.53 Substituting Equation 2.50 in Equation 2.53 and integrating, ln(p.) +JL = C 2.54 The value of the constant C can be determined by substituting the value of In(p') when r p M (ie, the critical state), the yield surface for Original Cam Clay can be written as: -2- = 1-ln M 2.55 where pc' is the pressure where the current elastic line meets with the CSL. pc' indicates the size of the yield surface. The shape of the yield surface for Original Cam Clay is shown in Figure 2.15. Equation 2.55 can be written as: F = In p'M 2.56 The condition F=0 means that the stresses are on the yield surface. 2.4.4. Modified Cam Clay formulation Original Cam Clay was the first mathematical model able to predict the behaviour of clays. Only three material properties are needed which makes it very simple to use. However, in the late 1960's some problems were found with the predictions given by OCC. First of all, for a condition of isotropic consolidation, OCC predicts shear strains. Secondly, OCC overestimates the values of the strain increments for changes of the stress ratio at small strains and finally OCC seems to overpredict the values of the coefficient of earth pressure at rest (Roscoe and Burland, 1968). 34 Because of that Modified Cam Clay was developed. Nowadays, Modified Cam Clay is the most widely used advanced constitutive model. Commercial codes as ABAQUS, PLAXIS, CRISP and FLAC have MCC as built-in constitutive models. To obtain the expressions for MCC, Roscoe and Burland (1968), proposed a new expression for the work dissipation. Assuming that under isotropic stress there is no shear strain and that at the critical state the plastic volumetric strain is equal to zero, they defined: ( ' } 2 ( . \ 2" ep + Ms? 2.57 V J K ) Equation 2.57 ensures that at the critical state the work dissipation rate with respect to total plastic strain is equal to the isotropic state for a given mean effective stress (Burland, 1965). Combining Equation 2.57 and the work equation for OCC (Eq. 2.48), qsp+p'ep =p' ( ' ") 2 2" £P + Msp 2.58 V J V ) and rearranging, the stress-dilatancy relation for Modified Cam Clay is given by: „P sp M2-n2 The same procedure used to find OCC yield surface can be used to find the yield surface of Modified Cam Clay. Applying normality and integrating Equation 2.59, we have: p' M —,=—2 5- 2.60 P c M2+TJ2 MCC yield surface is an ellipse passing through the origin and the size is controlled by p c 3 5 Figure 2.14: Work done in a quadrilateral element Figure 2.15: Original Cam Clay yield surface 36 Figure 2.16 shows the shape of Modified Cam Clay yield surface. This new yield surface takes into account the shear distortion without plastic volumetric change. Work hardening or softening plastic models change the size of their yield surface with plastic strain. The stresses must stay on the yield surface during plastic strain or else the stress point would be in illegal space (ie, beyond the yield surface). The consistency condition is the name given to the condition that a loading stress starts and finishes on the yield surface, and is given by: This is shown in Figure 2.17. Differentiating the equation of the yield surface, F, (Eq. 2.55), the consistency condition is given by: rj = M\ Pc_ Pc' Bl p' 2.62 The first term of the previous equation is the hardening rule as it controls the size of the yield surface via pc'. It is obtained by differentiating the CSL: e^r-AtoPc1 (Eq. 2.39) Pc' (1 + e)gf A- K 2.63 Figure 2.18 shows a four steps-loop in order to solve plasticity models. 37 Figure 2.16: Modified Cam Clay yield surface 3 8 Impose plastic strain increment Add elastic strain increments Calculate plastic yield surface hardening Derive new stresses using consistency condition Figure 2.18: Four-step loop to solve plasticity models 2.4.6. Stress - dilatancy The stress-dilatancy definition was presented earlier and the relations for Original Cam Clay (Eq. 2.50) and Modified Cam Clay (Eq. 2.59) have already been discussed; OCC has a linear relation between stress and strains and MCC has a nonlinear form. However this topic must be discussed in more detail. Taylor (1948) examined shear box tests on sands, and first attempted to separate the strength of soils into the true friction angle between particles, 0M, and the dilation produced by the shearing. <f>M was found to be a function of the mineralogy of the material, the properties and roughness of the surface and the size of the load per particle. Rowe (1962) by means of idealized packing of rods and spheres, proposed a stress-dilatancy (for triaxial compression) relation given by: 2.64 39 where K is given by: 1 + s i n ^ K = - —f- • 2.65 1 - s in^ f where is an angle of soil friction, given by: flfl < fa < <j>cv. Rowe showed that fa was equal to <zS/( at the minimum porosity and increased towards <j>cv at the maximum porosity. When Rowe's theory was applied to real soils, it was found that when K was computed using fa = $ , the predicted strains were too small, and when K was computed using fa = 0CV, they were too large. Hence, K has to be related to fa and not to $ f l because K has to evolve with strain, ie, has to relate the stress ratio to strain increment ratio. Rewriting Equation 2.64 in terms of the stress ratio (q/p'), e\ _ 37(2 + K) -9 (K- l ) • 2 /7 (K- l ) -3 (2K + l) 2 6 6 £q Using the expression of the friction angle, (j), as a function of M (Eq. 2.22) and rewriting Equation 2.65 as, 3 - M . the stress-dilatancy relation proposed by Rowe can be written as: 9(M - n) ~ 9 + 3M - 2Mrj 2.68 As well, Wroth (1965) performed simple shear tests on 1mm steel balls and assumed a work dissipation postulate. He suggested that work has to be done by some external force in order to deform a sample. This energy has to be enough to supply three components; a frictional component, a volume change component and a pressure change component. In his work, the relation between the input and the output work is related to a constant M which is given by Equation 2.22 but the angle of friction is <j>cv . 40 Based on experimental observations, the simple O C C dilatancy D p = M-r| did not match sand well. Nova (1982) proposed a new stress-dilatancy relation given by: Dp = ^LZ± 2.69 1-A/ He introduces a new material property, a volumetric coupling coefficient, N, found by plotting the maximum stress ratio against the minimum dilatancy. The slope of the line is equal to (1-N) as shown in Figure 2.19. Values of N have been found to be equal to 0.37 for Erksak sand and 0.30 for Brasted sand. It is important to note that for N=0 the previous flow rule reduces to the Cam Clay one. Recently, Li and Dafalias (2000) proposed a stress-dilatancy relation given by: Dp =D0(Mj:-T])/M 2.70 Replacing D 0 by M/(1 - N), the previous equation is the same as the flow rule proposed by Nova, but using a changing Mi rather than a constant M. In this case, Mj is used to follow the same principle than Rowe with his evolving friction angle (j>f. Table 2.3 shows a summary of the described flow rules. Table 2.3: Stress-dilatancy relations (from Been and Jefferies, 2004) Theory Relation Cam Clay Dp Mj - ij Modified Cam Clay Dp = {M2 -7]2}/2T] Nova 1982 Dp = (M - ?/)/(l - N) Li and Dafalias 2000 Dp =D0(Mj -V)IM Rowe 1962 Dp = 9 (Mj - 77)/(9 + 3Mj - 2M,ij) The idea of a M, in the flow rule that evolves with strain has been adopted by many recent authors. Table 2.4 shows the expressions found for the changing Mj by different authors. 41 Table 2.4: Relations of Mj after different authors (from Been and Jefferies, 2004) Originator Relation Manzari and Dafalias 1997 Mj = M + m [(/ Li and Dafalias 2000 Mj = M exp(rm//) Jefferies and Shuttle 2002 Mj = M - . The NorSand constitutive model will be used in this thesis, then the relation obtained by Jefferies and Shuttle, 2002, will be explained in details later in this chapter. Figure 2.20 shows the comparison of these four flow rules for a value of M,=1.25. Figure 2.19: Nova's stress-dilatancy rule (from Been and Jefferies, 2004) 42 Figure 2.20: Comparison of stress-dilatancy relations 2.5. Critical state model for sands Many constitutive models must be calibrated at the in situ void ratio. Hence, every time density changes, another set of parameters is needed and the model has to be re-calibrated. In CSSM models, the parameters are not associated to any particular value of initial void ratio. Hence the same set of parameters is able to capture the behaviour of soils regardless of the initial void ratio. However, C S S M as Original Cam Clay and Modified Cam Clay primarily work for "well-behaved, insensitive" clays (Lade, 2005) and they are not able to predict dilation and strain softening as it occurs in dense sands. Hence to model sands, new constitutive models were developed. The intent of this section is to briefly describe the main model groups and the new constitutive models for sands, between the 1970's and 1993 where Jefferies developed NorSand, the first CSSM for sands. Lade (2005) divided the available constitutive models in eight groups. The characteristics of the best known groups will be briefly described next. 4 3 Elastic models are probably the simplest models available, and the simplest elastic models are based on isotropic elasticity. Isotropic elasticity is defined by a single value of Young's Modulus and a single value of Poisson's Ratio. More complex elastic models are the Transversely isotropic, Orthotropic and Hyperbolic (in these type of models, principal strains are in the same direction as the principal stress increments). Elastic-plastic models comprise elasticity, a yield surface, a hardening rule and a flow rule. These models are associated with irrecoverable strains. The most popular plasticity model is Mohr-Coulomb, followed by Tresca and Von Mises. Only Mohr-Coulomb is suitable to model sands. Critical state models have already been presented and discussed. Typical critical state models are Original Cam Clay and Modified Cam Clay for clays, and NorSand for sands. Double hardening models comprise two yield surfaces; one yield surface yields in shear and the other yields in isotropic compression. A summary of the constitutive models for sands is presented in Table 2.5a and Table 2.5b. In Table 2.5a the different attributes of the models, such as shape of failure surface and yield surface, associated or non-associated flow rule and type of hardening parameter are presented. In Table 2.5b the capabilities of each model, such as, prediction of drained softening, large stress reversals, cyclic loading, pore pressure and 3D behaviour are presented. In addition the type of tests needed to obtain the input parameters and the number of parameters needed for the model are presented. 44 Table 2.5a: Constitutive models for sands (extracted from Lade, 2005) Model .Failure surface Yield surface Plastic potential' Hardening , parameter Hyperbolic Mohr-Coulomb NA NA NA Mohr-Coulomb Mohr-Coulomb = failure surface Non-associated None DiMaggio-Sandler Curved extended von Mises Elliptical cap Associated Plastic volumetric strain Lade and Duncan Smooth triangular, conical Smooth triangular, conical Non-associated Plastic work Darve Mohr-Coulomb NA NA NA Hypoplastic Smooth triangular, conical NA NA NA Fuzzy set plasticity Smooth triangular, conical Smooth triangular, conical & oct. plane Non-ass. and associated Plastic work Lade Smooth triangular, conical Smooth triangular, conical & sphere, cap Non-ass. and associated Plastic work PLAXIS hardening Mohr-Coulomb Curved Mohr-Coulomb & ellipt. cap Non-ass. and associated Plastic vol. & shear strain MONOT Smooth triangular, conical Smooth triangular, conical & sphere, cap Non-ass. and associated Plastic vol. & shear strain MIT-S1 Smooth triangular, conical Distorted lemniscate Non-associated Plastic vol. & shear strain Single hardening Smooth triangular, conical Tear-drop shaped Non-associated Plastic work Sinfonietta classica Smooth triangular, conical Tear-drop shaped Non-associated Plastic vol. & shear strain Disturbed state concept/hierarchical Smooth triangular, conical Tear-drop shaped Non-ass. or associated Plastic vol. & shear strain Generalized plasticity Smooth triangular, conical Elliptical cap Non-associated Plastic vol. & shear strain NorSand Smooth triangular, conical Bullet-shaped (OCC) Associated Plastic work 45 Table 2.5b: Constitutive models for sands (extracted from Lade, 2005) Model Drained softening Laige stress 5 ^reversal -Cyclic loading Realistic pore pressures Realistic 3D behaviour Experiments for parameter determination Number of parameteis Hyperbolic No No No No Y e s , away from failure 3 triaxial compression 7 Mohr-Coulomb No No No No No 3 triaxial compression 5 DiMaggio-Sandler No No No No No 3 triax. comp. , 1 Iso. comp 10 Lade and Duncan No No No No Y e s 3 triaxial compression 9 Darve No Y e s Y e s No No Triax. comp & ext., cyclic tests 16 Hypoplastic Y e s Y e s Y e s Y e s Y e s Triax. comp & Ko-comp. 8 Fuzzy set plasticity Y e s Y e s Y e s Y e s Y e s Triax. comp. & Iso. C o m p . 13-20 Lade Y e s No No Y e s Y e s 3 triax. comp., 1 Iso. comp. 14 PLAXIS hardening No No No Y e s Y e s 3 triax. comp., 1 Iso. comp. 11 M O N O T No Y e s Y e s Y e s Y e s 3 triax. comp. , 1 Iso. comp. 21 MIT-S1 Y e s Y e s Y e s Y e s Y e s C U Triax. comp. C D Triax. comp. Iso./Ko-comp. Ko-comp. with meas. lateral stresses 13 Single hardening Y e s Y e s Y e s Y e s Y e s 3 triax. comp., 1 Iso. comp. 12 Sinfonietta classica No No No Y e s Y e s 3 triax. comp., 1 Iso. comp. 17 Disturbed state concept/hierarc hical Y e s No No Y e s Y e s , requires 3D test 3 triax. comp. , 1 Iso. comp. 15 General ized plasticity Y e s Y e s Y e s Y e s Y e s 2 C D or C U triax. comp. & 1 cyclic test 15 NorSand Y e s , for O C soil No Y e s , undrained Y e s Y e s 2 C D triax. comp 2 C U triax. comp. with Iso. comp. 7 46 2.6 Critical state models for sands and NorSand Critical state constitutive models such as Original Cam Clay and Modified Cam Clay are able to capture the behaviour of soils, mostly clays, at different densities. However, those models are not able to predict the dilation and softening as it occurs in dense sands. NorSand is a C S S M constitutive model, developed by Jefferies in 1993, able to capture the softening and dilatancy behaviour of sands by introducing a new parameter, the state parameter and by postulating an infinity of normal consolidation loci (NCL). NorSand is used later in the thesis to simulate triaxial compression tests. This model was chosen because it is convenient given that it needs only few parameters (7) compared with other more advanced soil models, as shown in Table 2.5a and Table 2.5b. Also, as with all C S S M models, NorSand is developed from ideas rather than curve fits, which enables new concepts and ideas to be tested. However, sometimes some fits may not be as good as from models based on curve fits, as the predictions are constrained by its rules. The idea of Infinity of NCL was first introduced by Ishihara et al (1975). They suggested that given that in a sand an infinity of void ratios can be obtained, an infinity of NCL exist. Jefferies and Been (2000) supported the idea with experimental data. Four samples of Erksak sand were tested at different void ratios. They performed load/unload tests for each sample and found that each isotropic loading line can be seen as a true NCL. Figure 2.21 shows the concept of infinity of NCL. Infinity of NCL can be viewed as infinity of yield surfaces and each NCL can be viewed as a hardening law for an associated yield surface. This means that instead of one NCL, as used in O C C and MCC, with an infinity of NCL, the void ratio e and the CSL are uncoupled in NorSand. Hence, as any soil can exist over a wide spectrum of NCL, the void ratio is no longer sufficient to define the state of a soil, two parameters are introduced in NorSand, the state parameter, vy, and the overconsolidation ratio, R. 2.6.1. The state parameter and overconsolidation The use of void ratio or relative density does not tell us how the soil is going to behave, given that dilation or contraction depends not only on the void ratio, or relative density, but also on the stress level. That is why the state parameter is so useful in determining whether the soil sample will dilate or contract when sheared. Been and Jefferies (2004) showed that samples with similar relative densities and different stress levels behaved differently, while samples with same state 47 parameter but different void ratios and stress level, had similar behavior, confirming that the behaviour of sands under shear loading depends on the state parameter. The state parameter, relates the void ratio of the soil in e-ln(p') space with the void ratio at critical state. \y is defined as the distance between the actual void ratio and the void ratio at the critical state for the same p', and is given by; y/ = e-ec 2.71 where e is the actual void ratio and e c is the critical void ratio at the same p'. If the state parameter is positive, then the soil sample will contract with shear and if the state parameter is negative the soil will dilate with shear. From stress-dilatancy theory and experimental observation it is evident that peak dilatancy is related to the state parameter. This will be shown later when the parameter x will be defined and determined for the NorSand constitutive model. While NorSand was the first C S S M for sand, many models are now based on the state parameter, implicitly or not, such as Manzari and Dafalias (1997), Wan and Guo (1998), Gajo and Wood (1999) and Collins and Muhunthan (2003). For completeness the over-consolidation ratio, R, is also defined. R represents how close the soil is to its yield surface. It is important to note that here, R is defined using the mean stress, typical for constitutive models, and not the vertical stress, more usual in engineering practice. Figure 2.21 shows the difference between overconsolidation and state. 2.6.2. NorSand formulation This section introduces the original 1993 version of NorSand for triaxial compression. The later modifications incorporated into the simulations shown in Chapter V are described later. The NorSand and Original Cam Clay formulations are very similar; they both lie in stress dilatancy. NorSand uses the plastic shear strains to control the hardening rule. As explained before, this does not work with MCC because for a q=0 condition there is no shear strain. 48 NorSand uses the two axioms of CSSM described above. However with the introduction of the state parameter, Axiom 2 can be re-stated as: y/ -> 0 as sg - > o o 2.72 The deduction of the yield surface equation is the same as for Cam Clay. From Nova's flow rule and substituting in Equation 2.53, P' | (1-M)>7 = 0 p' M - rjN 2.73 Integrating Equation 2.73, it is possible to obtain again Equation 2.54, by taking N=0. In this case, the integration constant C is taken as the value of p' when rpM. Here, p' is taken as the image condition (denoted by pi', see Figure 2.22 ) and it is used to scale the size of the yield surface, similar to the use of pc', to size the yield surface in OCC. Hence, M = 1 - In IP/ 2.74 The introduction of p,' is required as with the introduction of the state parameter, the yield surfaces no longer intersect the critical state (see Figure 2.22) as they did with O C C . For convenience, the yield surface is written as: F = ln Pi' p'M 2.75 for N=0 and as: M_ N 1 + (/V-lj-?_ V 11 Pi' ( N / N - 1 ) 2.76 for N*0. 49 Sample in over consolidated state In p Figure 2.21: Difference between state parameter and overconsolidation, and infinity of NCL (from Jefferies, 1993) The original version of NorSand (Jefferies, 1993) uses Nova's flow rule (Eq. 2.69) D ' = f ^ (Eq.2.69) 1 - N To relate the image stress with the soil's void ratio, NorSand uses the Second Axiom of critical state theory. Rearranging the equation of the yield surface and differentiating, for N=0, the consistency condition is given by: 50 From Axiom 2, the hardening law expressed in dimensionless form is: Pi' = H BL P' max V EL P' 2.78 where, for N=0 and: for N*0. exp ( -Dmin /M) 2.79 (pr V P / m ax = (l + ^ , . / V / M ) ( N - 1 / w ' 2.80 It has been found that there exists a strong relation between the minimum dilatancy, Dpin (ie, value close to the peak stress ratio) and \\i„ the value of \\i at the image state, shown in Figure 2.23 and given by: where % is a model property and it usually lies between 2.5 and 4.5 for triaxial compression (with a common value of 3.5), the image state is related to \\i by: i//, = i// + A \n(p, 'Ip') 2.82 Rearranging the yield surface equation (Eqs. 2.75 and 2.76), for N=0: 51 = exp(rj/M -1) 2.83 and for N*0: (pr) V 1-A/77/M) ( « - l ) / N 2.84 I 1-/V j From Equation 2.81 and substituting the value of x by 3.5, Equation 2.79 can be written as: ^ - =exp(-3.5^,/M) ^ J m a x 2.85 for N=0 and for N*0 as: p', = (l + 3.5y/,/V/M)(w~ 1/W) 2.86 By reproducing experimental data, a better fit has been found if the hardening rule depends on the shear stress level using an exponential function (Jefferies and Shuttle, 2002), ^ = H e x p ( l - / 7 / M | V P' Vmax V EL P'J 2.87 where H is a model parameter, which is a function of the state parameter. In NorSand, isotropic elasticity is assumed. Elasticity parameters are defined by a shear rigidity lr and a constant Poisson's ratio as: K=l 2(1+K) 2.88 p' r 3(1 - 2K) Table 2.6 shows a summary with the equations of the original version of NorSand. 52 53 Table 2.6: Summary of equations for NorSand (from Jefferies and Been, 2005) Aspect of NorSand Equations Internal model parameters y/j = y/ + A ln(p, '/p') where y/ = e - ec Critical state e c = r - A In p' Yield surface F = In Pi' + - 2 - - 1 for N=0 p'M F = n- M_ N V * Pi' (W/W-1) for N*0 Hardening rule jL = H exp(l - ri/M rPi' V P /max = e x p ( - j ^ / M ) forN=0 ^ P /max (1 + xviN I / w ) ( w ~ 1 / w ) for N*0 V P /max Stress dilatancy M -TJ 1 - /V Elasticity fr p' A modification of the original version of. NorSand was suggested (Jefferies and Shuttle, 2002) where the N parameter was equal to zero. The incorrect sand dilatancy from. N=0 is accounted for by allowing M to evolve with shear strain, similar to the idea of Rowe where <j)f evolves from ^ to <j)cv with shear strain. Rearranging Equation 2.69 the expression for Mj is given by: M, = M + A/Dmin 2.89 Substituting the relation for maximum dilatancy (Eq. 2.81) in Equation 2.89, 54 M, = M/{n-XtcN\v,l\/Mlc) 2.90 Typical values of N and %tcare found to be around 0.3 and 3.5 respectively. Hence Mj is finally given by: M, = M ( 1 - V , | / M t e ) 2.91 This later modification of the flow rule is used in this thesis. 2.6.3. Determination of NorSand parameters In order to describe the NorSand constitutive model, 7 properties are needed and 2 initial soil state parameters. The properties needed to describe the model are shown in Table 2.7 and the procedure to obtain them is summarized next. More detail is given in Jefferies and Shuttle, 2005. Table 2.7: Properties needed to describe the NorSand constitutive model and their typical range for sand (from Jefferies and Been, 2005) Property Typical Range Remark C S L r X 0.9-1.4 0.01 - 0.07 Altitude of CSL, defined at 1 kPa Slope of CSL, defined on base e Plasticity M,c H X 1.2-1.5 50 - 500 2 .5-3 .5 Critical friction ratio, triaxial compression as reference condition Plastic hardening modulus for loading Relates minimum dilatancy to v|/ at D m i n Elasticity Ir V 200 - 800 0.1-0.3 Dimensionless shear rigidity Poisson's ratio Initial Soil State R -0.3-0.15 Overconsolidation ratio (R=1 for NC soils) State parameter 55 The critical state parameters in the e-ln(p') space, r and A. are obtained from drained and undrained tests on very loose reconstituted sand samples at different confining pressures. In that case, the samples will contract and reach the CSL in the e-ln(p') space at strains within the limits of the triaxial device. As the critical state of sands is an intrinsic state and does not depend on the initial soil structure, using reconstituted samples is fine. Loose samples are preferred instead of dense samples, given that these later dilate and do not reach the critical state, sometimes at 25% strain and are prone to localize. M t c is obtained from the end of test condition of the drained tests and can be confirmed by the stress-dilatancy plots. A couple of very dense drained test are necessary to estimate the maximum dilatancy of the soil (Dmin). As the incremental elastic strains are zero at peak strength, from a stress-dilatancy plot (r| vs Dm i n), it is possible to obtain Dm i n. As the critical state line is known, as well as the initial void ratio of the samples, the state parameter is easy to find using y/ = e - ec. The plastic hardening parameter, H, is obtained by fitting the results obtained by the drained triaxial tests and iterating to have the best match. H is fabric dependent and hence, not an intrinsic parameter. Based on previous calibrations H is found to be either a constant value or linearly related to the state parameter, vy, under the form H = a - by/ . X t c is a constant for each sand and is defined as the slope of the plot plastic dilation at peak strength and the initial state parameter from drained tests on dense samples, ie Dmin = XtcVi • The elastic property Ir is calculated by obtaining the shear modulus from bender elements during the triaxial tests. If bender elements are not included unload-reload cycles can be performed. The Poisson's ratio is assumed to be between 0.15 to 0.2. Finally, Table 2.8 shows a summary of the CSSM constitutive models. 56 Table 2.8: Summary of C S S M constitutive models C S S M model Yield surface Hardening rule Stress-di latancy rule Original C a m Clay = 1 - l n M p\' _ (l + e K P Pc' *•-< DP =M-rj Modif ied C a m Clay p' M2 Pc' M2+rj2 p' c ' _ (l + e K P Pc *--K DP M2-n2 NorSand (1993) * - = 1 - l n M (JL) KPI J — = H exp(l - TJ/M) pr \ fp/'V ^ P /max I P JJ Dp = M-n 1 - /V NorSand (2002) -2- = 1 - ln M, — = H exp(l - TJ/MJ pr \ H) -\ P /max I P ' J J £ P D = MT -rj 57 C H A P T E R III: M A T E R I A L P R O P E R T I E S In order to characterize the sand and establish parameters needed for the NorSand constitutive model, a series of tests were performed in the Graduate Geotechnical Laboratory at UBC. First, a description of the sand is given with reference to attributes of the grains and grain size distribution curve. Second, the strength of the sand is reported with reference to triaxial tests on reconstituted specimens. 3.1. Description of the soil The soil used for this study is a silica sand that was used by Raju (1995) for geosynthetic pullout tests, and produced by the Badger Mining Corporation. For convenience, this sand is subsequently referred to as "Badger Sand". Figure 3.1 shows the grain size distribution of Badger sand. The sand is a uniformly graded coarse sand with no fines, a coefficient of uniformity of 1.3 and a coefficient of curvature of 1.1. After the Unified Soil Classification System (USCS), Badger sand classifies as S P - a poorly graded sand. Particle sizes are between 0.6 mm and 2 mm, with a D 5 0 of 0.87 mm. The particles shape is well-rounded (Figure 3.2). The specific gravity is G s = 2.65 and the color varies from light brown to dark brown. Table 3.1 summarizes the properties of the grain size distribution of Badger sand. Table 3.1: Properties of the grain size distribution curve Soil Description D., (mm) D (mm) D -(mm) D30 I D 1 S I D 1 0 (mm) ; (mm) : (mm) (Dro,Dv.) c, !Plo • / D .o ' D, 0 ) Uniformly graded with little or no fines 0.95 0.91 0.87 0.83 0.76 0.72 1.3 1.1 58 59 3.3.1. Extreme void ratios The minimum and maximum void ratios were determined according to ASTM D 4253-93 and ASTM D 4254-91, respectively, as shown in Figure 3.3. In each case, knowing the volume, the weight and the specific gravity of the sand, it is possible to calculate the void ratio. The maximum void ratio was obtained using Test Method B of ASTM D4254. This method is applicable to soils for which 100%, by dry mass, of the particles pass a 19 mm (3/4-inch) sieve. A thin-walled tube of inside diameter of about 0.7 times the inside diameter of the mold is placed inside the mold and the dry sand is poured inside the tube. The tube is then quickly removed allowing the particles to slide inside the mold and not to drop. Upon leveling the surface, the mass of the mold plus soil is recorded. This procedure was repeated 6 times, yielding an average e m a x = 0.69 (see Table 3.2). Table 3.2: e m a x calculations for Badger sand Test Mass of mold + soil (g) Mass of soil (g) (g/cm") 1 8046.0 4431.0 1.56 0.70 2 8060.9 4445.9 1.57 0.69 3 8049.9 4434.9 1.56 0.70 4 8045.0 4430.0 1.56 0.70 5 8051.3 4436.3 1.56 0.69 6 8049.2 4434.2 1.56 0.70 average 1 58 0 69 The minimum void ratio was obtained by placing the dry sand inside the mold. It was leveled and a surcharge plate was placed on top, making sure the surface of the plate is in full contact with the sand. The mold was then attached to a vibrating table, with a guide sleeve on the mold in order to apply the surcharge, and vibrated at 60 Hz for 8 min. Finally the surcharge and guide sleeve were removed and 10 height measurements were taken of the surcharge plate. Knowing the volume and mass of the sand inside the mold it is possible to determine e m i n . The procedure was repeated 4 times yielding an average value of 0.49 (see Table 3.3). 60 Table 3.3: e m j n calculations for Badger sand Test Mass of mold + soil + cap (g) Mass of soil (g) H (cm) H, (cm) V (cnr) Pmax (g/cm ) e m i n M H H H I 1 9319 359 2.078 12.10 2224.06 1.78 0.49 2 9146 3786 2.700 11.47 2109.61 1.79 0.48 3 9350 3990 1.934 12.24 2250.52 1.77 0.49 4 9368 4008 1.944 12.23 2248.62 1.78 0.49 average 1 78 0 49 Values obtained for the maximum and minimum void ratio were very repeatable. It is important to note that the range of le=em a x-em i n is only 0.2, which is a somewhat narrow and unusual value for sands. For example, Ottawa sand is also a rounded sand with a C u = 1.1 but e m i n = 0.5 and e m a x = 0.8, hence le = 0.3. It appears that the minimum void ratio obtained for Badger sand is within the expected values, although the maximum void ratio is perhaps slightly lower than expected. Cubrinovski and Ishihara (2002) measured the minimum and maximum void ratios of about 300 soils including clean sands, sands with fines and silty soils. They found a correlation between e m i n and e m a x for clean sands (as Badger sand) given by: e m a x =0.072+ 1.53em i n 3.1 Similarly, from tests on 17 crushed sands from Georgia, 16 natural sands, and materials such as glass beads, granite powder and Syncrude tailings, Cho et al (2005) found a correlation between e m a x and e m i n taking into consideration the coefficient of uniformity C u , given by: e m a x =1 .35e m i n + 0 .15 (C u - 1 ) 3.2 Also, Shimobe and Moroto (1995), established a relation, based on 182 data points on uniform samples ( C u < 2), where /e_ = 0.574e m a x -0 .198 3.3 and le is the difference between the maximum and the minimum void ratios (le = e m a x - e m i n ] 61 Each of these three relations is reproduced in Figure 3.4, together with the data for Badger sand. The generally good agreement with Cho et al (2005) and Shimobe and Moroto (1995) is attributed to their consideration of grain size, as it is known that the minimum and maximum void ratios increase as the particle size decreases. Figure 3.3: e m a x and e m i n determinations Figure 3.4: Correlations between minimum and maximum void ratio 62 3.3.2. Angle of repose The angle of repose was determined according to ASTM C1444-00. As defined in Chapter II, the angle of repose is the angle of a heap formed by dry sand deposited as loose as possible before the heap starts flowing. A funnel with an internal diameter of discharge spout of 9.54 mm was used to pour the sand onto a metal plate, using a constant drop height of 38.1 mm. A stopper was initially placed in the tip of the funnel nozzle, and the funnel filled with sand. The stopper was then removed, allowing the sand to drop. When the tip of the sand cone formed on the base plate entered the funnel nozzle, the test was stopped. The diameter of the resulting cone was measured at 4 different locations and the angle of repose calculated using Equation 3.4 (see Figure 3.5). <j>rep = tan where: H is the height of the cone (38.1 mm); D A is the average diameter of the test determinations; and, d is the internal diameter of the funnel nozzle (9.54 mm). The procedure was repeated 10 times obtaining an average value of the diameter of the cone of 136.9 mm. The angle of repose obtained was ^ r e p = 30 .9 ° . 3.3.3. Roundness and sphericity The roundness of Badger sand grains was obtained using the definition given in Equation 2.29 proposed by Youd (1973), Shimobe and Moroto (1995), and Santamarina and Cho (2004). The method described by Santamarina and Cho (2004) was used to calculate the sphericity of Badger sand. It involves analysis of a two-dimensional image, from microscopic enlarged pictures of the individual grains, and was considered entirely suitable for these small rounded particles. After the law of large numbers, if a sample is selected with simple random sampling, the mean of the sample will approach the mean of the population as the sample size grows. When calculating the mean value of a sample, it is important to know the confidence interval and the sampling error to indicate the accuracy of the sample mean as an estimate of the true population mean (Lemay, 2005). 2H DA-d 3.4 63 Figure 3.5: Angle of repose of Badger Sand As a rule of thumb, before taking a sample, it is commonly accepted that the strong law of large numbers applies with a number of elements equal to 30 or more (Moore and McCabe, 2003). Hence, as the sand grains are not expected to vary much, 30 particles of Badger sand were taken randomly and the sphericity and roundness calculated using Equations 2.28 and 2.29 (see Figure 3.6). s = r m a x a n d ft = N (Eqs. 2.28 and 2.29) vmin Values of 0.77 and 0.81 respectively (see Tables 3.5 and 3.6), classify the Badger sand as a well-rounded sand (after Powers (1953) and Table 2.2) The variance of a sample is calculated as: n-1 3.5 6 4 were x is the mean value of the sample, n -1 is taken to calculate the variance given that this is the variance of a sample of size "n" and it is not the variance of the entire population. Hence, the variance obtained for the 30 particles of Badger sand was 0.006 and 0.011 for roundness and sphericity respectively. The standard deviation (cr) is obtained by taking the square root of the variance. For Badger sand, the values are 0.078 and 0.1 for roundness and sphericity respectively. The confidence interval is the maximum difference between the observed sample mean x and the true value of the population, and it is given by: x±t-^= 3.6 where t is taken from Table 3.4: Table 3.4: Relation between t for a 95% confidence and size "n" of a sample n 10 20 30 t 2.228 2.086 2.042 Replacing the values obtained for Badger Sand in Equations 3.5 and 3.6, the sampling error obtained for a 95% confidence interval for 30 particles of Badger sand is ± 0.028 and ± 0.039 for roundness and sphericity, respectively. Hence, it is possible to report, with 95% confidence that the mean values of roundness (R=0.81) and sphericity (S=0.77) obtained from the 30 particles are between (0.782, 0.838) and (0.731, 0.809). The Badger Mining Corporation reported values of R=0.8 and S=0.75 following the method of Krumbein/Sloss, which is in very good agreement with the values obtained with the Santamarina and Cho approach. Table 3.5: Sphericity of Badger sand Number of | Mean Value ] Variance j Standard Particles ' : Deviation Confidence interval 10 0.75 0.007 0.084 0.059 20 0.77 0.011 0.105 0.050 30 0.77 0.011 0.105 0.039 65 Table 3.6: Roundness of Badger sand Number of Particles Mean Value Variance Standard Deviation Confidence interval 10 0.80 0.005 0.071 0.053 20 0.81 0.007 0.084 0.040 30 0.81 0.006 0.075 0.028 Figure 3.6: Sphericity and roundness after Santamarina and Cho, 2004 66 3.2. Triaxial testing program 3.2.1. Description of the triaxial equipment The triaxial apparatus used at UBC is shown in Figure 3.7. A schematic diagram is shown in Figure 3.8. The cell pressure (P1) and back pressure (P2) are applied to the specimen through air hoses from the pressure regulator. The cell pressure and pore water pressure acting on the specimen are measured with pressure transducers (T1 and T2 respectively). An LVDT (Linear Variable Differential Transformer) measures axial deformation of the test specimen. Change in volume is deduced from the change in water level in a pipette using a differential pressure transducer (D). An external load cell and a frictionless (continually air leaking) seal on the loading ram ensure an accurate measurement of the deviator stress. 3.2.2. Data acquisition system The triaxial apparatus uses a "National Instrument AT-MI016x" 16-bit high speed data acquisition card for signal input. Five A/D channels are used; one for each transducer, one for axial load, one for volume change and one for axial displacement. One D/A channel is used for stress controlled tests to control the electro-pneumatic system. The channels in this card are connected to a computer and software is used to monitor axial displacement (from the LVDT), volume change (from DPT), effective confining pressure (from pressure transducer), deviator stress (from load cell) and pore water pressure (from pressure transducer). 3.2.3. Calibration of the system and measurement resolutions All readings on the computer are in mV. Hence it is necessary to apply a known pressure, force or displacement, in order to obtain the appropriate calibration factors, and thereby transform the mVA/ excitation signal into engineering units. The calibration of the pressure transducers was done through a DPI (Differential Pressure Indicator) which reads the pressure applied in kPa, connected to the transducer. The LVDT was calibrated through a micrometer, reading the displacement in mm and reading the associated voltage on the computer. The calibration of the cell pressure was calculated by reading the load.applied to the load cell through a proving ring (in 67 Figure 3.7: Triaxial apparatus Load cell L V D T Lr"n-Data acquisition system Figure 3.8: Schematic diagram of UBC triaxial apparatus 68 kgf) and on the computer (on mV). Finally the calibration of the volumetric deformation pipette and differential pressure transducer was done by letting water drain from the pipette and recording the volume of water in mL, reading the mV in the computer and the head of water in the pipette. 3.2.4. Preparation and testing of the specimen Specimens were reconstituted using a water pluviation technique; a known dry mass of sand is boiled with de-aired water and then deposited in the membrane lined cavity of a split mold filled with de-aired water. This technique ensures complete saturation of the sand as the grains are not in contact with air. Prior to the preparation of the sample, it is important to saturate all the lines in the triaxial apparatus. For that, a pressure of around 120 kPa was applied for 24 hours from the pressure regulator to the reservoirs of the triaxial frame (P1 and P2). A step by step preparation of the sample is explained below: • Boil a known dry mass of sand and water. Let it cool, and apply vacuum in order to dissolve the air bubbles trapped in the sand. • Boil the porous stones for 10 minutes and let them cool to room temperature. • Grease the O-rings. • Build up a dummy specimen with the aluminum sample and the loading cap used in the test. Take reference height to determine later the height of the specimen. • Saturate drainage line from the sample and place bottom porous stone. • Put the membrane and an O-ring around the pedestal. Place split mold steel jacket around the pedestal and stretch the membrane over its top. • Apply a small vacuum to the jacket (-20 kPa) and fill it with de-aired water. • Prepare the picnometer, fill it with de-aired water then put the stopper with glass stem and fill it with water. Invert the flask and fill the jacket with the sand. When the jacket is filled with sand to the desired height and touches the exit tip, the flow of sand will stop. The nominal dimensions of the samples are about 125 mm in height and 63 mm in diameter. • Carefully remove all the excess sand. This must be dried and weighed to compute the relative density of the sample from subtraction. • Carefully place the top cap with the top porous stone into the jacket in contact with the sand surface. Measure height. Snap the membrane and fit the O-ring, being careful to not 69 have air bubbles between the membrane and the top cap. Make sure that the top cap is leveled. Take height measurement again. • Apply vacuum of -20 kPa through the drainage line (the sample now stands because of the vacuum and not the jacket). Take measurement of height. Remove the steel jacket and measure the diameter (top, middle, and bottom) and height of the specimen. Figure 3.9 shows the shape of the sample before placing in the triaxial chamber. • Carefully assemble triaxial chamber and fill the chamber with de-aired water. • Close the vacuum valve and take the final height measurement and place the specimen in the triaxial apparatus. • On the computer, take offset values, being sure that the cell pressure and pore pressure transducers are opened to air. • Connect a line from the specimen to the cell pressure transducer and connect the air hose to the specimen through the reservoir at its top. • Apply a small pressure to the sample (30 kPa); this action will allow water to flow through the lines to get rid of the air bubbles and connect the drainage line to the pore pressure transducer. • Check the Skempton B value by opening the drainage valve from the sample and increasing the cell pressure step by step until a target pressure is reached checking the B value at each time. When a B value greater than 98% is obtained, apply the back pressure in order to impose the effective stress desired for the test, carry out the consolidation phase and then finally perform the triaxial test. A rate of displacement of 0.02 inch/min (about 0.5 %/minute) was used for every test allowing drainage of the specimen at the base only. A picture of the final shape of one test specimen is shown in Figure 3.10. 3.3. Triaxial test results The objective of the triaxial test program is to obtain properties and thereby establish parameters of Badger sand for the critical state soil model, NorSand. As explained in Chapter II, the parameters needed are critical state parameters (r and X), elastic parameters (G and v) and plastic parameters (M, H and x). 70 3.3.1. Test description Five triaxial compression tests were performed at UBC: two consolidated undrained (CU) tests at confining pressures of 100 kPa and 150 kPa, and three consolidated drained (CD) tests at confining pressures of 50 kPa, 100 kPa and 150 kPa. Test codes and characteristics are summarized in Table 3.7. Table 3.7: Codes and characteristics of tests Code Cell Pressure Void Ratio Drainage After Consolidation e.: (DR%) CUP100 100 0 624 (33%) Undrained CUP150 150 0.620 (35%) Undrained CDP50 50 0.623 (34%) Drained CDP100 100 0.624 (33%) Drained CDP150 150 0.620 (35%) Drained The initial void ratio of all five tests is very similar, which shows the reconstitution method was very repeatable. It was not possible to obtain specimens looser than 33% relative density after consolidation, considered indicative of a loose state, yet one that is close to that of a medium dense state (DR > 35%), which is attributed to the shape of the sand particles. As they are well-rounded particles, any minor disturbance yielded a rearrangement of the grains and densification of the sample. Therefore, although it was intended to test loose specimens, all the specimens showed an initial dilative behaviour of the sand. Hence, the steady state or critical state line was very difficult to achieve. Studies have shown that even after 25% to 30% strain, dilative specimens have not reached the steady state (Sivathayalan, 2000). 3.3.2. Void ratio determination The determination of void ratio for each test was made by measuring the height and cross-sectional area of the specimen. The height was obtained by comparison to a dummy sample of known height. The cross-sectional area was obtained by filling the steel jacket (which had the membrane snapped and vacuumed), with de-aired water. By calculating the mass of water inside 71 Figure 3.9: Sand sample before testing (dummy sample) the jacket, corrected for temperature, and the height of water, the average diameter of the specimen obtained was 63.38 mm and was taken as a constant for every test. 3.3.3. Triaxial test results The variation of deviator stress and volumetric strain, with axial strain, for the three drained tests is shown in Figure 3.11. The specimens are very stiff at strains less than 0.5%, and the maximum strength is reached at about 10% axial strain. With reference to the volumetric strain, the sand contracts until the maximum volumetric strain is reached (or maximum contraction), at about 1% axial strain, and then dilates until the test was terminated prior to reaching the critical state. Note the final rate of dilation in all tests is very similar, which is attributed to the very similar initial void ratio of the specimens. Interestingly, CDP100 contracts more than CDP150. As CDP100 is slightly looser than CDP150, it is possible that the void ratio-confining pressure combination made CDP100 more susceptible to contraction. Figure 3.13 shows the void ratio observed in all five tests. The open circle indicates the void ratio at the end of the test. The drained tests clearly show a little contraction before the sample starts dilating through to the end of the test. Given the density of the specimens and the low confining pressures, all the samples dilated, hence obtaining the critical state line (CSL) was not straightforward. Figure 3.12 shows the deviator stress and the pore water pressure for the two undrained tests. Again, the specimens are very stiff at strains less than 0.5%, then the slope reduces after 1% axial strain, and the higher the confining pressure, the higher the strength of the specimen. The tests initially tend to contract, generating a positive value of excess pore water pressure, until they reach the maximum pore pressure (or phase transformation) at about 1% axial strain, which is consistent with the maximum contraction points found for the drained tests. The sand then tends to dilate. Hence the excess pore water pressure decreases until the tests are stopped. A steady state condition was not reached as the sand continued to dilate until the end of the test. Figure 3.14 shows the stress path for the five triaxial tests in q-p' space. Both undrained tests show an initial contraction, where the pore pressure increases, until they reach the phase transformation point (defined by Ishihara et al (1975) as the maximum excess pore water pressure), and then dilate until the end of the test. The drained tests follow the expected 1:3 slope, reaching the peak angle of friction and then softening for the rest of the test. 73 a) ro Q. co CO CD CO o ro > Q 400 300 200 100 0 CD P150 DP 100 ^ CDP5 0 b) 4 6 8 10 Axial strain, s a (%) 12 14 Axial strain, s a (%) Figure 3.11: Triaxial (CD) test results: a) deviator stress and b) volumetric strain 74 Figure 3.12: Triaxial (CU) test results: a) deviator stress and b) excess pore pressure 75 0.64 0) o '•*-> ro i _ o 0.63 0.62 0.61 10 100 Mean stress, p' (kPa) 1000 Figure 3.13: Triaxial test results: void ratio 400 — 300 ro C L V) 2 200 o +-> TO > at Q 100 CDPIOO \ y / CUP 150 CUP10C n CDP50 / ( y ) I \ 0 DP 150 100 200 300 Mean stress, p' (kPa) 400 Figure 3.14: Triaxial test results: stress path 76 3.3.4. Shear modulus Additional triaxial tests on specimens of Badger sand, that were reconstituted by moist tamping, were reported by Golder Associates (2004). By attaching bender elements to the specimen and recording the shear wave velocities as the specimen was tested, it was possible to obtain the value of the shear modulus as a function of the confining pressure (see Figure 3.15). The relation obtained between shear modulus and mean stress is: G m a x = 9.35 (p')° 4 7x 1000 (MPa) 3.7 It is well recognised that the specimen reconstitution method determines the fabric of a sand specimen, hence the value of G m a x from a water pluviated specimen will be different to the G m a x obtained from moist tamped specimens. However, as elasticity parameters are needed later in this study (see Chapter V), this finding is reported, in order to establish a value for the same well-rounded sand rather than assuming a value believed appropriate to Badger sand. CHAPTER IV: ANALYSIS OF RESULTS The parameters derived from the UBC triaxial testing program described in Chapter III, are somewhat unusual for a silica sand, given the relatively low friction angles and a difference between extreme void ratios equal to 0.2. Also, in contrast to many sands, the grains are very rounded. In order to determine if the approaches used to describe typical sands are able to capture the behaviour of rounded sands such as Badger sand, results obtained from the triaxial tests are analyzed and discussed. Properties for Badger sand are determined: some of the properties are measured, and others are curve fitted with the help of the NorSand model. 4.1. Friction angles results from triaxial test program The angles were obtained following the definitions given in Chapter II. In summary, the angle of repose, (<|>rep), is calculated by measuring the angle of the heap formed when sand is deposited as loose as possible. The peak friction angle, (§ p ) , is obtained from drained tests on dense sand samples. The maximum contraction friction angle, ((j>Mc), is obtained from drained tests and corresponds to the point where the incremental volumetric strain changes from contractive to dilative. The phase transformation friction angle, (<|)PT), is measured in undrained tests and corresponds to the maximum pore water pressure. Finally, the maximum obliquity, ( § M 0 ) , is the state of maximum effective stress ratio, ie, ( o y / o y ) in undrained tests. 4.1.1. Constant volume friction angle (<j)Cv) In order to obtain the critical state friction angle, loose specimens must reach the critical state within the strain limitations of the test. As explained in Chapter III, although loose tests were planned, it was not possible to obtain specimens looser than the critical state line by the water pluviation method: the relative densities were 34%, 33% and 35%, for tests CDP50, CDP100 and CDP150, respectively (see Table 3.7). The same phenomenon was found by Vaid and Chern (1985) with Ottawa sand (which is also a rounded sand). The minimum relative density they obtained by the water pluviation method was DR=30%. In addition, Golder Associates used the moist tamped specimen reconstitution method, which is supposed to give the loosest possible state with no success; the sand specimens were always denser than the critical state. Dense 78 samples dilate and the critical state is reached at very large strains. Studies have shown that even after 25% to 30% strain, dilative specimens have not reached the critical state (Sivathayalan, 2000). The tests on Badger sand were carried to a maximum strain of 15%, so, the critical state was not reached by the end of the tests. Hence, another approach is needed to determine the critical state friction angle for Badger sand. It is possible to establish the critical stress ratio, M, from the D m i n versus r |m a x plot from several drained triaxial tests. As D=0 at the critical state, taking the best fit through the points as a basis for extrapolation, and taking r|max at Dm i n=0, M t c can be obtained (Bishop, 1966). The advantage of the method is that it does not require triaxial test data at large strain. Values for CDP50, CDP100 and CDP150 are plotted in Figure 4.1. It may be seen that the best fit through these three drained tests gives a straight line. Projecting this line to Dmin=0 the value of M c r i t was found to be equal to 1.13, which corresponds to a constant volume friction angle of 28.4° Figure 4.2 shows the plot r| versus D p , for CDP100. M c r i t can also be estimated by plotting the stress ratio versus the dilatancy for the drained tests and projecting the post peak r\ line to the critical state Dp=0. This procedure is most accurate if the tests approach the critical state. The values obtained from the three drained tests on Badger sand, were about 1.125, 1.13 and 1.133 for CDP50, CDP100 and CDP150, respectively. Hence from the two methods described above a good estimate of the critical state friction angle for Badger Sand is <j>cv = 28.4°. Following the ASTM method described in Chapter III, the angle of repose obtained for Badger sand was 30,9°. This value agrees with Fannin et al (2005), who reported an angle of repose for Badger sand of 31° following the method described by Bolton (1986). This method consists in excavating the toe of a heap formed by loose dry sand. It is generally assumed that the angle of repose corresponds to the internal friction angle of a material. Cornforth (1973) suggests that (j)Cv can be predicted by performing a static angle of repose test on dry sand. However, for Badger sand the angle of repose is 2.5° higher than the critical state friction angle (see Table 4.5). Many factors such as amount of material, rate of pouring of the material and surface roughness influence the value of the angle of repose (Miura et al, 1997). As well, properties such as grain size and angularity influence the value of <j)rep. 79 1.20 1.19 1.18 1.17 1.16 1.15 1.14 1.13 1.12 1.11 1.10 CDP100 CDP150 CDP50 r)max=0.37Dmin+1.13 0.00 -0.05 -0.10 -0.15 -0.20 Figure 4.1: D m i n versus r|max for CDP50, CDP100 and CDP150 -0.2 -0.1 0.1 0.2 0.3 0.4 Figure 4.2: Stress dilatancy plot for CDP100 80 4.1.2. Peak friction angle (fyp) The peak friction angles for the three drained tests are reported in Table 4.1; the peak friction angles for Badger sand are very similar, with an average value of (j>p=29.4°. Bolton (1986) proposed a relative dilatancy index, lR, given by Eq. 2.23. IR =DR(Q-\np')-R (Eq. 2.23) This relative dilatancy index was obtained considering 17 different sands for confining pressures between 150 kPa and 600 kPa. Table 4.1 shows the values obtained for Badger sand using Q=10andR=1. Table 4.1: Relative dilatancy index for Badger sand Test DR p (kPa) ; L 3IF (|)peak ~" (bp.. ,..-<|>rv CDP50 0.34 50 1.07 3.21 29.7 28.4 1.0 CDP100 0.33 100 0.78 2.34 29.4 28.4 1.0 CDP150 0.35 150 0.75 2.25 29.1 28.4 0.7 As well, Bolton (1986) suggested a relation between the peak friction angle, the constant volume friction angle and the relative dilatancy index, given by (Eq. 2.24). I'p rev <t>cv ~ 3 / P (Eq. 2.24) From Table 4.1 the difference between Bolton's prediction and the values of Badger sand is approximately 1.5°, which suggests that the peak friction angle might be lower than expected. Similarly, Bolton (1986) suggested a relation between l R and the rate of dilatancy given by: de = 0.3/ R 4.1 1 ; 81 Table 4.2 shows the predicted values with Equation 4.1 and the values obtained for Badger sand. Again, the predicted values are larger than the ones obtained with the triaxial tests. Table 4.2: Rate of dilatancy for Badger sand Test MH| |mHfl i p ! (kPa) (.- drJ { de\ -J '• CDP50 0.34 50 0.19 U.J2 CDP100 0.33 100 0.17 0.23 CDP150 0.35 150 0.19 0.23 It is known that maximum strength is related to maximum dilatancy and that maximum dilatancy is affected by initial density and effective stress. From Table 4.2 the rate of dilatancy obtained for the three triaxial tests on Badger sand is very similar, which is consistent with the very similar values of peak friction angle. It will be shown later in this Chapter that the dilatancy of Badger sand is unusual, which may contribute to the iow values of peak friction angle obtained with the triaxial tests. 4.1.3. Maximum contraction friction angle (<J>Mc) Table 4.3 shows the values obtained for the maximum contraction friction angle for Badger sand found with the triaxial tests, which are very similar. Miura et al (1998) found that the angle of maximum contraction and the friction angle at failure depend on the void ratio extent (le = e m a x - em i n) and the angularity of the grains; as l e and angularity increase (or roundness decreases), <j>MC and (j>cv increase. Table 4.3: Maximum contraction friction angle for Badger sand Test D R (%) it ) 1 v /max (""i''0y) (., C D P 5 0 34% 0.12 2.51 25.4 C D P 1 0 0 3 3 % 0.25 2.63 26.7 C D P 1 5 0 3 5 % 0.21 2.50 25.4 (average) 25.8 82 They reported the values of <j>MC and 0rep as shown in Figure 4.3. The values for Badger sand are added to that database. The maximum contraction angle plots between the proposed boundaries and links sands with glass beads. o c re c o 45 ifc 35 c o o re i _ +^  c o <J 25 E E •>< re 15 15 o Natural Sample A Uniform Sample o Graded Sample • G lass Beads • L W A & Badger Sand o . - - - A o •A A " S B O D D A , IA A • « . . ^ A^O.A--25 35 Angle of repose, (t>rep, (deg.) 45 Figure 4.3: Relation between angle of repose and maximum contraction friction angle (after Miura etal, 1998)3 4.1.4. Phase transformation friction angle ((j)PT) Table 4.4 shows the values of the phase transformation friction angles obtained at the point where the pore water pressure is maximum for CUP100 and CUP150. 3 After Miura et al (1998), Natural s a m p l e s are se lected from natural s a n d , Uniform s a m p l e s and G r a d e d s a m p l e s are reconstituted s a n d s a m p l e s , and L .W.A . (Light Weight Aggregates ) and G l a s s b e a d s are artificial granular materials. 83 Table 4.4: Phase transformation friction angle for Badger sand Test DR (%) (.u;)m 3 X (kPa) CUP100 3% 35.3 25.5 CUP150 35% 55.7 25.6 (average) 25 6 There has been considerable discussion of the uniqueness of (J)Cv, §PT and § M C in the literature. Further debate has addressed whether or not the critical state friction angle, steady state friction angle, maximum contraction friction angle and phase transformation friction angle are the same. The following is a summary, by authors, in chronologic order: • Kirkpatrick (1961) suggested that (bMC is a unique parameter for each sand. He also suggested that (bMc and t|)cv are the same, and independent of the specimen behaviour (contractive or dilative). Luong (1980) performed triaxial compression and extension tests on various cohesionless materials. He found that the maximum contraction friction angle is independent of stress path, initial porosity, grain size and anisotropy. He also found that, for contractive specimens, <j)Cv and (bMc are the same. • Vaid and Chern (1985) performed monotonic and cyclic triaxial tests on Ottawa sand and a mine tailings sand. They found that the phase transformation and the steady state line are the same for contractive specimens, and that <bPT is a unique property. • Negussey et al (1988) performed ring shear tests on medium Ottawa sand and fine and coarse copper particles. They found that <j)Cv is independent of normal stress, density, and grain size, and hence a unique parameter. They suggested that for loose samples, §Cv and (bMc are the same and that (bMc decreases with initial relative density and increases with increasing confining pressure. As well, they found that <bpT and (bcv are the same and that (j)PT is a unique property of sands, independent of density and effective stress. 84 Oyenuga and Tisot (1989) performed triaxial tests and direct shear box tests to measure (j)PT and t|)Mc, and ring torsion tests to measure <bCv with Fountainebleau sand and three Moselle sands. They found that <|)cv is independent of grain size distribution, effective stress and density. As well they suggest that (bCv and (bMc are the same. • Vaid et al (1990) performed triaxial tests Ottawa sand. They found that the phase transformation friction angle and the steady state friction angle are the same and a unique property for contractive sand specimens but are different for dilative sand specimens. They found that <j)PT is a unique property of a sand for contractive specimens, independent of initial void ratio, consolidated state and mode of loading. • Chu (1995) performed a different triaxial test in order to reach the critical state with dense sand specimens of Sydney sand. The procedure is to isotropically consolidate a dense sand sample and shear it to failure at constant confining pressure. When failure is reached, a constant volume condition is imposed. A full description of the procedure is found in Chu and Lo (1993). With this procedure they ensured homogeneous deformations and that the critical state was reached in the homogeneous zones. They observed that the critical state friction angle is stress level dependent, the higher the confining pressure, the smaller the friction angle, but independent of the initial density. The same conclusion was found for the phase transformation and maximum contraction friction angle but these angles are also dependent on the initial density of the specimen. As well, Chu (1995) found that for contractive soils, tj)cv, <t>PT and (bMc are the same. However for dilative specimens (b P T and §MC are the same but different from (bCv-Table 4.5 provides a summary of the different friction angles obtained during the triaxial testing program for Badger Sand. The values of M,c and their corresponding angle of friction are reported for different states; repose, failure envelope, maximum contraction, phase transformation and constant volume. From Table 4.5 the two numbers for Badger sand are very close and indistinguishable within the experimental resolution of the triaxial tests. 85 Table 4.5: Different friction angles for Badger sand State M,, '< .b (••) itiiiiliS Repose 1.24 30.9 Maximum obliquity (MO) 1.13 28.4 Maximum contraction (MC) 1.02 25.8 Phase transformation (PT) 1.01 25.6 Critical state (CV) 1.13 28.4 From Table 4.5 the maximum contraction friction angle is practically the same than the phase transformation friction angle. However, the constant volume friction angle is higher than the phase transformation friction angle (or maximum contraction friction angle) and is equal in magnitude to the maximum obliquity friction angle. These results are consistent with the findings of Chu (1995). Additionally, the maximum obliquity friction angle is equal to the phase transformation friction angle plus 2.8°. This finding agrees with Sivathalayan (2000). As Badger sand always had a dilative behaviour for the void ratio - mean stress combination presented in the triaxial testing (see Figs 3.11 and 3.12), it seems reasonable to have a different value for the angle of friction for the phase transformation state and the constant volume state. Typical values of critical state friction angles for sands stated in chapter II range between 30° and 39° (see Table 2.1). Hence, the values found for Badger sand are lower than commonly found. Also, the angle of repose and the constant volume friction angle measured for Badger sand are different. Similarly, the values of the peak friction angle were found to be lower than expected. Additionally, as explained in Chapter III, the values of the minimum and maximum void ratio were somewhat unusual. In order to be confident that the values found for Badger sand in the triaxial tests are correct, these atypical results are now compared with values of other sands found in the literature. 4.2. Comparison of Badger sand properties with literature As reported in Chapter III, the minimum and maximum void ratios found for Badger sand are unusual: e m i n = 0.49 and e m a x = 0.69, yielding l e = e m a x - e m i n = 0.2, which is a very low value for sands. Table 4.6 summarizes values of roundness, minimum and maximum void ratio and l e for different materials, with roundness ranging from 0.2 to 0.6. As suggested by Shimobe and Moroto (1995) and Cubrinovski and Ishihara (2002), the values of e m i n and e m a x increase with decreasing the roundness of the particles, and le = e m a x - e m i n increases with decreasing roundness. 86 Table 4.6: Values of roundness and void ratio for various sands (from Youd, 1973) Material R e n ,A 1 Ottawa sand 0.60 0.704 0.408 0.296 Lapis Lustre sand 0.44 0.754 0.460 0.294 Monterey sand 0.39 0.772 0.469 0.303 Del Monte sand 0.27 0.971 0.503 0.468 Crushed basalt 0.20 1.190 0.700 0.490 In order to determine if the values of the maximum and minimum void ratios found for Badger sand are reasonable, 30 particles were taken randomly, and following the procedure explained in Chapter III and Eqs. 2.28 and 2.29 of Santamarina and Cho (2004), the sphericity and roundness were determined. Average values of S = 0.77 and R = 0.81 were obtained, as shown in Table 3.5 and Table 3.6. This value of roundness classifies Badger sand as a well rounded sand (Powers, 1953, Table 2.2). Figure 4.4 shows the relations for the minimum and maximum void ratios defined by Santamarina and Cho (2004), Shimobe and Moroto (1995), and Youd (1973). A clear tendency of decreasing void ratios with increasing roundness is appreciable, converging to a null slope as roundness approaches a value of 1. As well, the difference between e m a x - e m i n decreases with increasing roundness. Youd (1973) reported materials with roundness between 0.17 and 0.60. Shimobe and Moroto (1995) correlated e m a x and the roundness from 76 different materials; they reported values of roundness between 0.1 and 0.7 for sands and a roundness of 1 for glass beads. As well, Santamarina and Cho (2004) reported values of roundness between 0.2 and 1 from natural and crushed sands. The values of R, e m i n and e m a x obtained for Badger sand are included, and lend confidence to the general trend in the data. Santamarina and Cho (2004) found that the constant volume friction angle of granular materials depends on the shape of the particles. Based on observations of 54 specimens they measured (bcv as the angle of repose for materials with roundness between 0.1 and 1 and proposed: rcv=42-^7R 4.2 The values of (j>Cv and (J)repose are added to the database, showing an excellent agreement with the proposed relation (Figure 4.5). 87 Generally, the values of mobilized friction angles given in Table 4.5 indicate the range of friction angles for Badger sand is very narrow: the difference between the maximum obliquity and the phase transformation friction angle is i j > M 0 - < j > P J = 2.8° . Table 4.7 shows the values of <j>M0 and 4>PT for Fraser River sand (Vaid and Eliadorani, 1998), which is an angular sand, and Ottawa sand which is a rounded sand (Vaid and Chern, 1985). Data for the rounded Badger sand appear in excellent agreement with the findings for Ottawa sand. Table 4.7: Friction angles for Ottawa sand and Fraser River sand Material 0MO <I>PT 0MO ~-<PPT Ottawa sand 31.5° 29.4= 2.1° Fraser River sand 37° 32° 5.0° This interpretation of the data for Badger sand and comparison with the literature, confirms the shape of the grain plays an important role in governing intrinsic parameters of granular materials. As the roundness of the particles increases, the maximum and minimum void ratios decrease, as does the difference l e =e m a x -emin- Also, the constant volume friction angle decreases as the roundness of the particle increases. In addition, it seems that for rounded sands the difference between the mobilized friction angles is smaller than for angular sands. 1.6 1.4 1.2 | 1.0 o O 0.8 £ 0.6 0.4 0.2 0.0 Youd Shimobe and Moroto Santamarina and Cho A Badger Sand 0.2 0.4 0.6 Roundness, R 0.8 Figure 4.4: Relation between roundness and extreme void ratios 88 50 40 c 10 c o •« 30 cu E o > c CO co c o o 20 10 1 <j>cv=42-17R A Badger sand: (bc v A Badger sand: Repose 0.2 0.4 0.6 Roundness, R 0.8 Figure 4.5: Relation between roundness and constant volume friction angle (from Santamarina and Cho, 2004) 4.3. Analysis of NorSand results As explained in chapter II, the NorSand constitutive model has seven input parameters; two critical state parameters (r and X), three plastic parameters (M t C i H and xtc) and two elastic parameters (Ir and v). The critical state parameters, r and X, are ideally obtained from drained and undrained tests on very loose reconstituted sand samples at different confining pressures. M t c is obtained from the end of test condition of very loose triaxial drained tests or Bishop's r\-D plot for dense specimens. The plastic hardening parameter, H, is a fabric dependent parameter and is obtained by fitting the curves and iterating to have the best match. It is found that H is directly related to the state parameter, vy. Xtc is found by plotting the state parameter at image condition, v|/|, versus the minimum dilatancy D m i n . It is believed that x is also a fabric dependent parameter. 89 The elastic property Ir is obtained from bender elements during the triaxial tests. If bender elements are not included an unload-reload cycle can be performed. Poisson's ratio, v, is measured in laboratory tests and is about 0.2 and is typically assumed between 0.1 and 0.2. Generally a value of v close to 0.1 is used for modelling purposes. 4.3.1. Comparison between triaxial data and NorSand model for Badger sand Figure 4.6, Figure 4.7 and Figure 4.8 show the comparison between the three drained triaxial tests (CDP50, CDP100 and CDP150, respectively) and the predictions with NorSand. The parameters used for the fittings are described later and a summary table is presented at the end of this chapter. The peak deviator stresses predicted by NorSand are larger than the ones obtained with the tests (for CDP50 and CDP150), and they are very similar for the CDP100 test. Nevertheless the initial stiffness and final trends show on average a good fit, capturing the drained behaviour of the Badger sand. The main purpose of the comparison between the triaxial data and the model is to obtain the parameters needed to describe the behaviour of Badger sand by having a reasonable fit to all the tests and not one really good fit and the others poor. Similarly, for the volumetric strain, the model predicts more contraction than the real data, but the rate of dilation of the sand is captured until the 10% axial strain. A better fit for the initial stiffness and the contraction of the sand can be found by increasing the hardening parameter, H, as shown in Figure 4.9. However, the rate of dilation is not captured, the model showing a lower rate than the triaxial data. Also, Figure 4.10 and Figure 4.11 show the fitting for the undrained tests with the same parameters used to fit the drained tests. In this case, the deviatoric stress predicted by NorSand is smaller than the one obtained by the triaxial tests. Also, the test data show less pore water pressure than the prediction with NorSand, hence the difference in strength. However, for both, deviatoric stress and pore water pressure NorSand is able to capture the behaviour of Badger sand for the strains up to 10%. 90 Figure 4.6: Comparison of deviator stress and volumetric strain between triaxial data and NorSand model for CDP50 91 Figure 4.7: Comparison of deviator stress and volumetric strain between triaxial data and NorSand model for C D P 1 0 0 92 6 9 Axial strain, £a (%) • Triaxial Test Fitted Curve 12 15 0.4 > _1.2 J 1 1 1 Axial strain, s a (%) Figure 4.8: Comparison of deviator stress and volumetric strain between triaxial data and NorSand model for CDP150 93 250 re Q) Q 0 j 1 1 1 , 0 3 6 9 12 15 Axial strain, s a (%) 0.4 o > -1.2 J 1 — J I Axial strain, e a (%) Figure 4.9: Comparison between CDP100 and NorSand model for a greater hardening 94 100 .300 J 1 1 1 • • 1 • Axial strain, s a (%) Figure 4.10: Comparison of deviator stress pore water pressure between triaxial data and NorSand model for CUP100 95 700 Axial strain, E A (%) Figure 4.11: Comparison of deviator stress pore water pressure between triaxial data and NorSand model for CUP150 96 Figure 4.12: Comparison of deviator stress pore water pressure between triaxial data and NorSand model for CUP100, for R =2 97 The preceding figures show a good fit to dilation rate but a poor fit for the initial contraction. A better fit for the undrained tests can be obtained by raising the overconsolidation ratio to 2. The comparison between the predicted values and the triaxial test for CUP100 is shown in Figure 4.12. This change makes the predicted deviatoric stress increase and fit the triaxial data almost perfectly. In the case of the excess pore water pressure, the prediction with NorSand shows more contraction (less than the case where R=1), but the final trend shows a perfect fit. 4.3.2. Determination of NorSand parameters Critical state parameters The average value of M t c equal to 1.13 obtained using the method described by Bishop (1966) was used to fit the three drained and the two undrained tests the best. Badger sand only dilated for the void ratios - confining pressure used in these tests, the critical state line in the e-lnp' space has been estimated with the help of NorSand. r and X were found to be equal to 0.697 and 0.0105 respectively, giving the equation for the Critical State Line in the e-Inp' space equal to e = 0.697 - 0.0105 x ln(p'). This equation ensures that the tests start and end at the left of the CSL. Even if the critical state friction angle associated to Mlc=1.13 is smaller than typical values for quartz sands, it is consistent with the values proposed by Santamarina and Cho, as explained before. Figure 4.13 shows the place of the Critical State Line (CSL) estimated with the help of NorSand, in relation to the void ratios obtained during the lab tests. As said before, the critical state line remains above the tests which confirms that the samples were not loose, hence, after a short contraction they started dilating. Tests CDP100 and CDP150 are very close to the estimated CSL at the end of the tests. However, CDP50 is far away from the CSL as the two undrained tests are. Plastic parameters The value of the hardening parameter, H, used to fit the triaxial tests on Badger sand was taken constant and equal to 200. As the state parameter varies very little for the tests performed on Badger sand it is difficult to determine whether H should vary with Higher values of H could have matched better the triaxial results but in that case the rate of dilation would have been off as shown in Figure 4.9. 98 From the drained tests, a relation between the plastic dilation initial state parameter was determined and is equal to: ( • / ' ^ si j spq at peak strength and the V J D M I N = 3.6^ 4.3 Theoretically, Dmm = 0 at <// = 0. The best fit line in Figure 4.14 has a (9e"s) intercept which is practically 0, which reconfirms the approach. From Figure 4.14 the value of Xtc can be determined as the slope of the line, hence: Z,c = 3 - 6 4.4 On the other hand, as explained in Chapter II, the slope of the line of Figure 4.1 (r|m a x vs -Dm i n) is (l - N), hence for Badger sand the value of N is found to be equal to 0.63. Typical values of N and x for sands are about 0.3 and 3.5, respectively. Given those values, the later version of NorSand described in Chapter II assumes that N><x=1.0. Hence, the flow rule described in Equation 2.91. However, given the values obtained for Badger Sand (N=0.63 and % i c = 3.6), the product between N and % is equal to 2.3. In order to capture the behaviour of Badger sand, and given the unusual values found for N the flow rule has been modified to fit these tests. Instead of the conventional flow rule described in Chapter II, the flow rule used is Mj - M(\-y//Mlc) for i/s positive (or loose state) and M, =M ( l + 2.3 </y/M,c) for if/ negative (or dense state). The fits in Figure 4.6 to Figure 4.11 show that this modified flow rule is able to capture the stress-strain behaviour of Badger sand. 99 0.68 0.67 0) 0.66 6 += 0.65 (0 i _ •D 0.64 O > 0.63 0.62 0.61 9 I c 1 I) o . . . .—j 10 100 Mean stress, p' (kPa) 1000 Figure 4.13: Critical State Line and void ratios -0.045 -0.040 -0.035 State parameter at D m i n -0.030 -0.025 Figure 4.14: Dilation at peak strength, D m i n , as a function of the state parameter 100 Figure 4.15 shows the stress-dilatancy behaviour for CDP100 compared with the stress-dilatancy predicted using the modified flow-rule for NorSand. The figure shows a very good agreement between the predicted and the values measured for this test. As found by Been and Jefferies (2004) for Erksak sand, the graph is approximately linear, showing a natural reverse as the stress ratio reaches the maximum, as expected for dense sand samples. Again, this plot confirms that the value of <j)Cv obtained by Bishop's method is reasonable. If we assume that § P J in undrained tests is actually the constant volume friction angle, the rebound line of Figure 4.15 should end up at r p 1.06, which seems extremely unlikely. Finally the properties determined for the Badger Sand are summarized in Table 4.8. As stated previously, the critical state parameters, r and A, were deduced from the fits with the NorSand model and the lab test data. M t c was obtained with the methods of Bishop and stress-dilatancy. The elastic parameters were obtained from a test with bender elements and assuming a Poisson's ratio of 0.15. Finally the plastic parameters were deduced by iterating and finding the best fit in order to be consistent with all the other parameters. With this parameter combination, the average error between the initial void ratio obtained from the triaxial tests (e0r) and the one obtained by fitting (eot) is 0.01, which is with the accuracy expected from the lab testing. Most of the parameters obtained to fit Badger sand fall all between the expected values (Table 2.7). M is lower than expected and the parameter N is higher than normal values, hence the change in flow rule described above. As well, the dilatancy of Badger sand is unusual (Figure 4.6 to Figure 4.11) which explains the low values of peak friction angles obtained in the drained tests. As a general conclusion it is possible to say that Badger sand is a very unusual sand; the values of e m i n , e m a x , <|>cv , <!>PT, <|>MC and peak friction angles are lower than expected values for sands. However, these values were confirmed and explained by the shape of the sand grains, which, after Powers (1953) classifies Badger sand as a well-rounded sand with a roundness of R=0.81. As R increases, e m i n , e m a x , <|)cv , §PT and § M C decrease. Peak strength is related to maximum dilatancy. The dilatancy for Badger sand is also unusual which explains the low peak friction angles obtained in the drained tests. Finally, the NorSand constitutive model is able to capture the behaviour of Badger sand and confirm that the value of <|>cv obtained by Bishop's method is correct and it is different from the phase transformation friction angle. 101 102 Table 4.8: Parameters summary of NorSand 4 for Badger sand MEP50 CDP100 CDP150 wmmm- Comments 0.697 0.697 0.697 0.697 0.697 at 1 kPa X 0.0105 0.0105 0.0105 0.0105 0.0105 base e M,. 1.13 1.13 1.13 1.13 1.13 H 200 200 200 200 200 7. 3.6 3.6 3.6 3.6 3.6 Ir 1184.6 822.0 1200.0 831.8 664.7 Ir = G/p' G (MPa) 59 82 99 81 . 99 G = 9.35(p')0 4 7 V 0.15 0.15 0.15 0.15 0.15 E (MPa) 136 189 228 187 228 R 1 1 1 1 1 M' -0.043 -0.033 -0.036 -0.037 -0.034 e 0 r -e 0 l 0.011 0.008 0.012 0.012 0.010 4 Details of the implementation of NorSand in FLAC are shown in Appendix A. 103 C H A P T E R V : S U M M A R Y A N D C O N C L U S I O N S This research examines the properties of a well-rounded sand, in order to characterize it for purposes of numerical modelling. The silica sand is produced by the Badger Mining Corporation. The Badger sand is a uniformly graded coarse sand, for which C u = 1.3 and C c = 1.1 and D50=0.87 mm. The general intent is to contrast properties of Badger sand with those of other sands, and determine if the different available tools are able to capture the behaviour of well-rounded grains such as Badger sand. For this,, the main objectives of this study were: • ' Measure the physical properties of Badger sand grains, such as roundness, and the grain assembly, such as minimum and maximum void ratio. • Determine the strength parameters of Badger sand, including the angle of repose, failure envelope friction angle, phase transformation friction angle, maximum contraction friction angle and constant volume friction angle. • Correlate the roundness of Badger sand with its extreme void ratios, and constant volume friction angle, and contrast those findings with relations for sands published in the literature. • Compare the relative magnitude of the different friction angles obtained for Badger sand, and examine the implications for characterization of well-rounded sands. Determine the elastic, plastic and critical state properties of Badger sand for modelling purposes. • Capture the behaviour of Badger sand with the NorSand constitutive model. A series of three drained and two undrained triaxial tests was performed at the graduate geotechnical laboratory of UBC, with complementary studies using the NorSand constitutive model and FLAC. A summary of the main results and findings is presented below. 104 Image analysis of the sand grains yielded a roundness R=0.81 and S=0.77. Values of the minimum and maximum void ratio obtained by following the ASTM method are e m i n = 0.49 and em a x= 0.69. Friction angles established for Badger sand tests are as follow: Angle of repose: <|>rep = 30.9° Maximum obliquity: <|>MO = 28.4° Maximum contraction: <t>MC = 25.8° Phase transformation: <)>PT = 25.6° Constant volume : <)>CV = 28.4° The angle of repose was obtained following the ASTM method for a loose heap of sand. Triaxial tests were performed on specimens reconstituted by water pluviation. All specimens had a relative density less than DR = 35%, indicative of a loose state, yet one close to that of a medium dense state (DR>35%). All triaxial specimens exhibited considerable dilation. Accordingly, the constant volume friction angle was determined using the method proposed by Bishop (1966): it requires plotting the values of the maximum stress ratio, ^ m a x , and the dilatancy, D m i n , for drained tests, and taking the best fit through the points. It is possible to obtain the critical stress ratio, M, as the value of the maximum stress ratio, T i m a x , at Dmin=0. A second method was used to confirm the value of the critical state friction angle: for each drained test, M was obtained by plotting the stress ratio versus the dilatancy, D p , and projecting the post peak r\ line to Dp=0. The maximum contraction friction angle was obtained from the stresses related to the point where the behaviour of the sand passed from contractive to dilative in the three drained tests. The phase transformation friction angle was obtained from the stresses corresponding to the maximum pore water pressure in the undrained tests. The maximum obliquity friction angle was measured from the maximum effective stress ratio in the two undrained tests. The NorSand constitutive model was used to fit the triaxial tests on Badger sand. A summary of the parameters used is presented in Table 5.1. 105 Table 5.1: Summary of Badger sand parameters Parameter Badger sand Remark r 0 697 At 1 kPa X 0.0105 Defined in base e M 1.13 In triaxialcompression H 200 X 3.6 G 9.35(p')u"'xl000 MPa V . 0.15 Assumed R 1 Assumed As all the reconstituted test specimens were dilatant, the critical state parameters in the e-ln(p') space, r and X were obtained with the help of NorSand and ensuring that the CSL stays to the right of the end of test condition. H was found by iteration, x was found as the slope of the line in the v|/-Dm i n plot of the three drained tests. The relation of G with the mean effective stress was found using bender elements attached to the specimen and provided by Golder Associates. From the results presented above, the following conclusions can be stated: 1. For Badger sand it was not possible to reconstitute specimens looser than 33% relative density, with the method of water pluviation, which is consistent with the experience of Vaid and Chern (1958) for Ottawa sand. 2. For Badger sand the maximum contraction friction angle (25.8°) appears equal to the phase transformation friction angle (25.6°), and occurs at the same axial strain of about 1% to 1.5%. 3. For Badger sand the maximum contraction friction angle (25.8°) appears equal to the phase transformation friction angle (25.6°), and they are both different from the constant volume friction angle (28.4°). This finding is in agreement with that of Chu (1995). 1 0 6 4. For Badger sand the maximum obliquity friction angle (28.4°) is greater than the phase transformation friction angle (25.6°), by approximately 3°. This finding is consistent with the observation of Sivathayalan (2000). 5. The constant volume friction angle (28.4°) was found to be lower than the angle of repose (30.9°). 6. For Badger sand the maximum obliquity friction angle (28.4°) was found equal in magnitude to the constant volume friction angle (28.4°). 7. Notwithstanding the limited number of triaxial tests, an excellent agreement was found between the roundness (0.81) and constant volume friction angle (28.4°) of Badger sand, and a corresponding relation between R and <bCv proposed by Santamarina and Cho (2004). The latter relation is based on the assumption that the critical state friction angle is equal to the angle of repose. 8. For Badger sand, em a x=0.69 and emin=0.49, which is a very narrow range. A very good agreement was found between the roundness (0.81) and these extreme void ratios, and the relations proposed by Youd (1973), Shimobe and Moroto (1995), and Santamarina and Cho (2004). 9. Most sands exhibit a roundness of 0.2 < R < 0.7. In contrast, glass beads are nearly perfect spheres (R = 1.0). There is little information on well-rounded sands 0.7 < R < 1.0 in the literature, and the experimental data suggest these well-rounded sands exhibit somewhat unusual values for the angle of friction which can be attributed to grain shape. Accordingly, the Badger sand data provide a valuable link between data for commonly found sands and glass beads; they also serve to build confidence in empirical relations based on a quantification of grain shape. 10. The parameters used to fit Badger sand with NorSand fall between the expected values. 107 11. The volumetric coupling coefficient from Nova's flow rule, N, has an unusual value for Badger sand, equal to 0.63. As well % has a value of 3.6. As the product Nx% is different from 1, a modification of the flow rule is proposed for Badger sand, given by: M, = M(l + 23y//Mlc) for ¥ < 0 12. Peak strength is related to maximum dilatancy, and Badger sand was found to exhibit unusual dilatancy, which is believed to explain the unusually low friction angles found in the triaxial tests. 13. 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Zhang, H. and Garga, V., (1997), "Quasi-steady state: a real behaviour?", Canadian Geotechnical Journal, Vol 34, pp 749-761. 116 APPENDIX A: NORSAND MODEL IMPLEMENTATION In order to capture the behaviour of the Badger Sand and use it later for modelling purposes, a new constitutive model will be implemented within FLAC (Fast Lagrangian Analysis of Continua); NorSand. As explained in Chapter II, NorSand is able to realistically capture the dilatancy of sands through use of the state parameter. NorSand, unlike Original or Modified Cam Clay, is also able to simulate strain softening which is very important for dense sands. As discussed in Chapter II, the Critical State Soil Models, Modified Cam Clay (MCC), Original Cam Clay (OCC) and NorSand are very similar in concept. Modified Cam Clay is already a built-in constitutive model in FLAC, written as a FISH function. Hence the approach used here is to take the MCC FISH function, adapt this function to OCC and verify the approach used. Then modify the function to NorSand. The only difference between MCC and OCC is the shape (equation) of the yield surface. Hence, the Modified Cam Clay yield surface is modified in order to obtain the Original Cam Clay model. Because of the different yield surface shape, different methodologies are used. Finally, as OCC and NorSand have the same yield surface but different hardening rules, the next step is direct. This chapter explains the process followed to create the NorSand constitutive model function in FLAC, starting from Modified Cam Clay, continuing with Original Cam Clay and ending with NorSand (Figure A.1). The FLAC results are then compared with the results obtained with an Excel Visual Basic for Applications (VBA) (Jefferies and Shuttle, 2005) to verify that the FLAC function is working properly. A.1. Fast Lagrangian Analysis of Continua (FLAC) A. 1.1. General description of FLAC FLAC is a two-dimensional computational program that analyses the stress-strain response of solids subjected to static and dynamic states of load. FLAC's explicit finite difference formulation has been widely applied to model geotechnical and mining problems, and it is one of the most widely used programs world-wide for geotechnical stress-strain analysis. The FLAC program is able to simulate large deformations of continuous media, groundwater flow and pore water pressure dissipation. Interface elements are also available in FLAC, allowing simulation of the 117 separation of materials or slipping between them. FLAC also includes structural elements, such as cables, beams and reinforcing strips, among others, that behave in conjunction with the continuous media. Five optional features are also available to analyze specific problem types; dynamic analysis, thermal analysis, two-phase flow analysis, creep material behaviour and the ability to code user defined models (UDM) in C++ and compile them as dynamic link libraries (dll's). Well executed numerical models allow us to improve our understanding of the behaviour of soil and structures under the effect of different types of loading (static or dynamic). FLAC (Version 5.0) has 11 built-in constitutive models for continuous elements. The constitutive models are: Null, Isotropic Elastic, Mohr-Coulomb, Modified Cam Clay, Transversely Isotropic Elastic, Drucker-Prager, Ubiquitous-joint, Strain Hardening/Softening, Bilinear Strain-Hardening/Softening Ubiquitous-joint, Double-yield, Hoek-Brown. However, none of these models are able to realistically capture the behaviour of sands under general paths; Mohr-Coulomb predicts the dilatancy wrong and the strain hardening/softening model only allows cohesion, dilation and friction angle to change with strain level. A.1.2. FLAC nomenclature Figure A.2 is an example of a FLAC model showing all the commonly used terms. The soil body is formed by a grid (or mesh) of quadrilateral elements (or zones) that behave depending on the constitutive model, forces and boundary conditions assigned to them. A zone, or element, is the smallest quadrilateral domain where changes in stresses, fluid flow or heat transfer are calculated. Each zone is divided into four triangular subzones which are superimposed, as shown in Figure A.3. Each zone has four grid points (or nodes) associated with the four corners of the quadrilateral and each grid point is defined by a pair of x- and y-coordinates. Each zone of the entire grid behaves depending on the constitutive model assigned to it. Constitutive model means a model which describes the deformation and strength behaviour of the simulated material. Some examples of constitutive models already included in FLAC are Elastic-Plastic, Mohr-Coulomb and Modified Cam Clay. Constitutive models and material properties can be assigned independently to every zone of the model. The periphery of the grid is called the model boundary. Boundary conditions to the model can be applied as fixed boundaries (displacement restraint), stress boundaries or excavations. Excavations have the Null constitutive model assigned to them where no properties are assigned. 118 Modified Cam Clay Yield Surface Hardening Law Flow Rule Solve F = 0 Original Cam Clay Yield Surface Hardening Law Flow Rule Solve F = 0 NorSand Yield Surface Hardening Law Flow Rule Solve F = 0 F = q 2 + M2p'(p'-pcn') }-1 + e Pen' .P £P = AsM2(2p'-pcn') £p =Xs2q F = In + - S L - 1 Mp' Ei Pc 1 + e *„ A - K bq - As P / ' J M ; P ' P , ' : H exp M , (P/'/P'), max 1 r P ' B1 £p =As{Mj Figure A.1: General flow chart 119 ^^^T^^ ^^ ^^ ^^  ^d^^^ fixed bottom bouadarv Figure A.2: Example of FLAC model (from FLAC Version 5.0 Manual) Figure A.3: FLAC zone divided in 4 triangles; triangular element with velocity vectors; nodal force vector (from FLAC Version 5.0 Manual) 120 Structural elements, such as cables, beams or interfaces are used to represent the interaction of structures. Interfaces are used to simulate separation or sliding between elements and beams and cables are used to represent, for example, reinforcement in a wall. A.1.3. Sign convention and units in FLAC In FLAC, positive normal stresses and strains indicate tension and extension respectively, and negative normal stresses and strains indicate compression. Conversely, the sign convention for pore pressure is: positive pore pressure indicates compression and negative pore pressure indicates tension. For gravitational force, positive gravity will make an element move downward and negative gravity makes an element move upward. Figure A.4 shows the sign convention for positive shear stress components in FLAC and their associated shear strains. Positive shear stresses and strains are clockwise The system of units for mechanical parameters used by FLAC is shown in Table A.1. Table A.1: System of units used by FLAC Length m m m cm ft in Density Kg/m J 10 Kg/m" 1 0 b K g / m J . 10 b g/cm J slugs/ftJ snails/in'3 Force N KN MN Mdynes lbf lbf Stress Pa kPa MPa bar lbf/ff psi Gravity m/sec2 m/sec^ m/sec^ cm/sec^ ft/sec2 in/sec2 Stiffness Pa/m kPa/m MPa/m bar/cm lbf/ftJ lb/in" . , . . /ft sec 2 , . . . . /Dsei where, islug = 1 and 1sna;/ = 1 ft in 121 y 'yx •xy 'xy 'yx -> x Figure A.4: Positive shear stress (or strain) convention and associated distortion (from FLAC Version 5.0 Manual) 122 4.1.4. FISH in FLAC FISH is a programming language created by Itasca which allows FLAC users to code and define new functions and variables (the User Defined Model (UDM) option is not required to use FISH). An example of a FISH function which initializes the mean effective stress for the 'camclay' constitutive model is presented below. def camclay_ini _p loop i (1,izones) loop j (1 Jzones) if z_model(i,j)='camclay' then mean_p=-(sxx(ij)+syy(i,j)+szz(i,j))/3.-pp(i,j) z_prop(i,j,'cam_cp')=mean_p end_if endjoop endjoop end FISH functions start with the command DEFINE and end with the command END. Within the FISH function other functions can be called and created. In the previous example, the FISH function loops through all the elements and initializes the mean effective stress to each zone. FISH functions can be embedded in a normal FLAC data file or entered directly from the keyboard. The FISH language can also be used to implement new constitutive models within FLAC. A constitutive model is defined by the statement CONSTITUTIVEMODEL <n> where n is the number associated to the constitutive model. It will be executed as a built-in constitutive model, and will be called four times for each FLAC zone (1 time for each triangle shown in Figure A.3). Inside the constitutive model, new variables can be created and other FISH functions can be called. However, one of the disadvantages of the constitutive models implemented with FISH is that it takes a long time to run. For larger problems, where simulation speed is important, the UDM in C++, which runs at the same speed as FLAC, is a better option. 123 A.2. FLAC finite difference formulation5 The basic FLAC formulation consists of three conditions; Newton's law of motion, a stress-strain equation and a general finite difference formula.Taking a simple mechanical example, where a force (F) is applied to a mass (m) , causing a displacement (u), velocity f ^ l and acceleration ydt2 , , as shown in Figure A.5, from Newton's law of motion, dt For a continuous body, Equation A.1 can be written as: P — - = t j - + pg, A.2 dt dXj where, p is the mass density, x, is the coordinate vector (x,y) er,y are the components of the stress tensor, and gi is gravitation. In addition to the law of motion, a continuous solid must obey a stress-strain relation. For an elastic material this is: 2 <J,J ^o-ij +\Sij\K--G\ekk + 2Geij\At A.3 where, 5^ is the Kronecker delta, e,y is the strain rate component (Equation A.5), K and G are the bulk and shear modulus, 5 This is a summary of the FLAC User's Manual, for the full explanation, please refer to this manual. 124 At is the time step, and := means "replaced by". Equation A.3 can be generally written as: ' ay := M v j A.4 where, M () is the functional form of the constitutive law, K is a history parameter depending on the constitutive law, and en = -diij duj + dXj dx. A.5 where, ui are the velocity component. In the finite difference method, each derivative in the equations of motion and strain are replaced by an algebraic expression relating variables at specific locations of the grid. From the Gauss divergence theorem, the finite difference equation for all elements is given by (Figure A.3): \n,fdS= \—dA A .6 where: £ is the integral around the surface, S; n, is a unit normal to the surface; f is a scalar, vector or tensor; dS is an incremental arc length; A is the surface area of the element; I is the integral over the surface area, A; and •w dx, is a position vector. 125 Taking the average value of the gradient of f as: df \ 1 r df dX; / A JA dX: dA A.7 Substituting in Equation A.6 and approximating the integral through a sum over three sides of a triangular zone, the gradient can be written as: df A.8 where AS is the length of the side of the triangle. Substituting the average velocity vector in each side of Equation A.8, the strain rates can be written as nodal velocities for the triangular sub-zone as (Figure A.3): ~ d x ~ ^ 2 A ^ .(a) .(*>) ^  Ui + Ui |n yAS A . 9 dUj dUi + dx j dXj A.10 As for volumetric strains, they are averaged over each pair of triangle and the deviatoric stresses remain unchanged. Area weighting is used to obtain the new stress tensor from the strain-rate tensor. A ( e ) + A W A.11 where: o-£a) is the isotropic stress in triangle (a); and A ( a > is the area of triangle (a). 1 2 6 Once all the stresses are calculated, the nodal forces are obtained from the traction acting on the sides of each triangle; each node receives two forces (one from each side of the triangle as shown in Figure A.3). Hence, F, = la^ <1>S(1)+n<2)S(2>) A.12 At each node of the grid, the forces (from applied loads and gravity) of all the quadrilaterals are summed in order to obtain the net force vector ^ F, . If the node is not in equilibrium, it will be accelerated according Newton's second law of motion: , (f+Af/2) . (r-Af/2) = U i +yF(»M A.I 3 ^ 1 m Finally, Equation A.13 is integrated to determine the new coordinate of the grid point, given by: . (f+Af/2) X , ( ( + A f ) = x ( " + U , - At ^ 14 In order to have a stable solution, the time-step chosen to solve the problem must be smaller than the critical time-step given by: k i Ax A.15 At < — C where: Ax is the element size, and C is the maximum speed at which information can propagate, and is given by: C - ^ L l A , 6 For a triangular zone of area A, considering the shortest propagation distance as/4 /x m a x , and taking At = 1, C2p = K + 4G/3 and the zone mass a s m z = pA, the mass of a zone is given by: (K + 4G/3)Ax 2 a x A 1 7 A 127 Finally, as the grid point mass is one third of the zone mass, and the node mass is the sum of the masses of the grid points connected to the node, the mass of a node is given by: Figure A.5: Force acting on a mass A.3. Summary of standard plasticity implementation All elasto-plasticity models have four components, elasticity, a yield function which defines the stress combination for which plastic flow takes place, a hardening/softening function which controls the size of the yield surface and a flow rule which specifies the direction of the plastic strain increment vector. Plasticity models considered in this thesis are formulated in terms of effective stresses. For the triaxial conditions used in the rest of this analysis all stresses and strains are principal values and 128 the shear component is not considered, the vertical direction is the direction of maximum principal stresses and strains. The two horizontal directions are both the direction of minimum stresses and strains. For triaxial conditions, the stress components used are the mean effective stress (/>') and the deviator stress [q), defined as 6: A.19 The volumetric and shear strains are: 2s3 A.20 The volumetric and shear strain increments, sv a n d ^ must be work-conjugate (Equation 2.10), and are defined as: s, + 2s3 A.21 The specific volume (equal to 1 + void ratio) is defined as: A.22 where V is the total volume of the soil and V s is the volume of solid particles. The incremental relation between volumetric strain and specific volume is: A.23 1 Using the FLAC sign convention explained in A. 1,3. 129 From basic plasticity theory, the total strain increment can be decomposed into elastic and plastic components and that only the elastic part contributes to the stress increment. Hence, •* V V V . • • A.24 £ = £ e + P q q q Changes in stress are associated with the elastic strain only. Therefore from Hooke's law, the increment of mean effective stress and deviator stress is given by: A.25 q = 3Gs q where K is the bulk modulus and G is the shear modulus, which can be written in terms of the Young's modulus, E, and the Poisson's ratio, v, as: K = E 3(1 -2v) E A.26 2(1 + ") Rearranging Equation A.24, and substituting in Equation A.25: p =KSV-K£P . . • q = 3Geq-3Gsp The plastic flow is written as: dQ A.27 A.28 130 where Q is the potential surface of the constitutive model and As is the plastic volumetric multiplier. For associated flow, the yield surface, F, and potential surface, Q, are identical and Equation A.28 may be written as: dp' ~p -A.29 A.4. Modified and Original Cam Clay in F L A C Modified Cam Clay is a built-in constitutive model in FLAC. It is an incremental hardening/softening elastoplastic model including a particular form of non/linear elasticity and the behaviour is governed by volumetric plastic strain. Given that Modified Cam Clay already exists, the NorSand function development in FLAC will start from MCC. An intermediate step will be coding Original Cam Clay. Given that only the yield surface changes between both Cam Clays, O C C is a test that the proposed methodology works with FLAC. A.4.1. Modified Cam Clay formulation The Modified Cam Clay formulation is based on a combined shear and volumetric yield function (a closed yield surface) and an associated flow rule. In the FLAC version of Modified Cam Clay an elastic guess for the stress increment is first computed from the total strain increment. If the stresses violate the yield criteria, plastic deformation takes place and only the elastic part of the strain increment contributes to the stress increment. The stress increment is evaluated and corrected using the plastic flow rule to ensure that any stresses initially outside the yield function are end on the yield surface. The shape of the yield surface of Modified Cam Clay is shown in Figure 2.16 and is defined by: 131 F = q2+M2p'(p'-pcn') A.30 where M is the ratio q/p' at the critical state line and pcn' is the consolidation pressure of the current yield surface. For Modified Cam Clay, the yield condition F = 0, is represented by an ellipse passing through the origin, with horizontal axis pcn', and vertical axis M Pen' Differentiating F with respect to p' and q: ^ = M 2 (2p ' -p O T ' ) = C a dp 8F_ dq = 2q = Cb A.31 Replacing Equation A.31 in Equation A.29, the flow rule for Modified Cam Clay is given by: sp =AsM2(2p'-Pcn') ep=As2q A.32 Replacing Equation A.32 into Equation A.27, the incremental mean pressure and deviator stress are: p' = K ( • sv — AsCa q = 3G\ A.33 For each step FLAC substitutes the new stresses into the yield surface by adding to the old stresses the stress increment, as shown in Equation A.34. p'w = p'° + p' qN=q°+q A.34 132 where the subscript N refers to the new converged stresses and O to the old ones. Substituting Equation A.33 into Equation A.34: p'N = p " - 4 K C a q" =.q'-Zs3GCb A.35 where p'1 and q' are the elastic guesses defined by: A.36 q' =q°+3G£a Substituting p'N and qN in F(qN,p'N)= (qNf + M2p'N {p'N-pcn'N)= 0 , the equation to find As is given by: aX\ +bAs+c = 0 A - 3 7 where: a = (/WKCa)2 +(3GC 6)2 ft = - (KC a C a +3GC b Ci) A.38 c = ^ ' . P " ' ) Equation A.37 is a polynomial of degree 2which is easy to solve. From the two roots obtained the smaller must be retained. At each step, the new specific volume is updated by: v = v V J A.39 and preconsolidation pressure is updated by: 133 Pcn'N = P, A — K V N A.40 J where X is the slope of the normal consolidation line in e-lnp' space, and K is the slope of the elastic swelling line in the e-lnp' space. The tangent bulk modulus is then updated using: One of the idealizations of Cam Clay is that it is rigid in elastic shear. However Finite Element and Finite Difference programs do not allow such a condition. Hence, the shear modulus is calculated from K and Poisson's ratio, as: and the new elastic trials are calculated. Figure A.6 shows the flow chart for the model. Note that in Figure A.6 the yield surface is hardened after the new stresses are calculated. Hence, the yield surface is one step in arrears. A.4.2. Original Cam Clay formulation The shape of the yield surface for Original Cam Clay is shown in Figure 2.15 and is given by: A.41 K G = 3(1 - 2v)K 2(1 + v) A.42 A.43 where pc' is the critical mean effective pressure given by: Pc' = exp(1.0) A.44 134 Elastic guess <7/(-Compute elastic guesses for p'1 and q by adding the total strain increment to the old stresses (Eq. A.36) flp'.p'')>o Is yield exceeded? yes Yielding in shear Solve F = 0 with Eq. A.37 to find As (choose smaller root) Calculate sp and sP (Eq. A.29) and calculate p'N and q N by subtracting the plastic correction (Eq. A.35) from the elastic guess. Update zone parameters, and obtain zone volumetric strain and zone specific volume (Eq. A.39). Obtain new preconsolidation pressure (Eq. A.40) and update the tangential bulk modulus (Eq. A.41) Calculate the new shear modulus; if v =t 0 update usinn Fn A 4? otherwise G = nnnst Calculate p' and q using the new elastic stresses Figure A .6 : Flow chart model Modif ied C a m Clay fish function 135 The yield surface for Original Cam Clay is non-linear, so a solution to Equation A.37 is not possible. Hence a different approach was used to derive parameter Xs for OCC. In this case, instead of solving F = 0, the more usual plasticity approach of using the consistency condition is applied by solving F = 0 . Differentiating Equation A.43: 8F \ 8F * 8F • , SF = — P'+— <7+ — Pc A.45 dp' dq dp, c F = J L _ l P . + £ _ £ £ _ A.46 Mp' Mp' p' pc' Differentiating Equation A.46 with respect to p': dF 1 dp' p' Mp2 Multiplying Equation A.47 by Mp': dp' As well, from Equation A.46: A.47 dF- = M-rj = Dp A.48 ^ = 1 A.49 dq It is possible to swap between Equation A.47 and A.48 because F = 0 (if F ^ 0 this will not work). Replacing in the plastic flow (Equation A.29), the volumetric and shear plastic strains for O C C are: 136 P p A.50 £q ~ As~ - As dq Substituting A.50 in Equation A.27, the mean stress increment and deviator stress increment are: p' = K sv-AsD q = 3G\ f . ^ V J A.51 For Original Cam Clay, the consistency condition is applied by setting F = 0 instead of using F = 0, and pc' is allowed to harden during each load step and so convergence is obtained with the current yield surface. The hardening rule is given by: Pc' = pc'l±^£} A.52 A - K Hence, using Equation A.50, ^ = H£P =H(M-TJ)AS A.53 Pc' where H is the hardening parameter given by H = ^1 + E^. A - K P ' ' Substituting —— from A.53, and p'and q , from A.51, in Equation A.46 and applying the Pc' consistency condition the solution for A is: 137 The flow chart for the Original Cam Clay fish function is presented in Figure A.7. In Figure A.7 the terms in bold font have changed from Figure A.6 (MCC). Figure A.8 shows the comparison between the VBA Code and the fish function for Original Cam Clay, for the input parameters presented. It is possible to observe that the results show an excellent agreement. Hence, the formulation proposed is able to model Original Cam Clay correctly. A.5. NorSand Formulation in F L A C NorSand is similar to Original Cam Clay in that it has the same shape of yield surface. The original version of the NorSand constitutive model (Jefferies, 1993) is described in Chapter II. The changes to NorSand in this work are described below. The yield surface for Norsand is shown in Figure 2.22 and is given by: F = ln Pi' + 1 M,p' A.55 The size of the yield surface is controlled by p;, which varies with hardening and Mj is related to the state parameter as: M, = Mte f 1- ¥ V J A.56 where M t c is the critical stress ratio under triaxial compression condition. In the case of NorSand used in this study, Mj is not constant so will become a variable, hence when differentiating Equation A.55 the term associated with M, must be added as: 138 Compute elastic guesses for p'0|d and qQid f(<7„; ./ ./ ' 'rtjl - 0 (Eq A 43) Is yield exceeded'? yes Yielding in shear Solve F 0 with Eq A 54 to find Calculate s p and ep (Eq. A.50) and calculate p'N and q N by subtracting to the elastic guess the plastic correction (Eq. A.35) Update zone parameters, and obtain zone volumetric strain and zone specific volume (Eq. A.39) Obtain new preconsolidation pressure (Eq. A.40) and update the tangential bulk modulus (Eq. A.41) Calculate the new shear modulus; if v * 0 update using Eq. A.42, otherwise G = const. Calculate p' and q using the new elastic stresses Figure A.7: Original Cam Clay fish function, flow chart 139 200 0 5 10 15 20 Axial strain, e a (%) 10 0 5 10 15 20 Axial strain, sa (%) Figure A.8: Comparison of deviator stress and volumetric strain for Original Cam Clay, between VBA code and FLAC results 140 • DF • dF dF • dF • , F = Mi+—-p +—q+ -p, A.57 8M, dp' dq dp; 1 However, M, is a second order term that can be eventually neglected to come back to: • 8F \ dF • dF ' , F = —p + —q+ — Pi A.58 dp dq dp. Hence F can be written as: •F=^_VPl+PL_Pl A.59 Mp' Mp' p' Pi' Differentiating Equation A.59 with respect to p': dF 1 q dp' p' MjP2 Multiplying Equation A.60 by M,p': dp' Differentiating Equation A.59 with respect to q A.60 - = M,-ij A.61 ^ = 1 A.62 dq Replacing in Equation A.29, the plastic volumetric strain increment and the plastic shear volumetric increment are: £P =As(Mi-,j) 141 A.63 Re-writing the yield surface equation (Eq. A.55) as: exp v M , , P_ Pi' A.64 the hardening rule is given by: H exp v Mi j (P/'/P')ma> iPi'lP') A.65 where: XV i M, A.66 and: y/j = y/ — A A.67 H is a constant for each sand and y/ is the state parameter. Replacing in Equation A.59, As is calculated as: ZGsq+Ks„Dh 36 + Hexp { Pi'lp' ^•Mj+KiMj-rjf A.68 A flow chart for NorSand is presented in Figure A.9. Finally, the results obtained for Norsand with FLAC and the VBA Code are compared. The input parameters used are presented in the following table and are taken from the parameters found for 142 Ticino sand (Shuttle and Jefferies, 1998). Two extreme states are presented, an extremely loose state, where the state parameter is 0.15, and a dense state where the state parameter is -0.2. The results are presented in Figure A. 10 and Figure A.11, respectively, showing an excellent agreement. The NorSand fish function is presented in Appendix B. Table A.2: Parameter summary for NorSand model Parameter Dense Loose Remark r 0.962 0.962 At 1 kPa 0.0248 0.0248 Defined in base e M 1.23 1.23 In compression H 115-420M/ 115-420v|/ X 3.5 3.5 G 50 50 MPa V 0.2 0.2 V. -0.2 0.15 100 100 kPa Ko 1 1 143 Compute elastic guesses for p'0id and q0id Calculate pi'/p' (Eq. A . 6 4 ) , \Vi (Eq. A . 6 7 ) and (pi7p')max (Eq. A . 6 6 ) rfo»d P'M ) - 0 (Eq A 5b) Is yield exceeded? yes < , > ' Yielding in shear", ' „' • Solve F 0 with Eq A 6 8 to find Calculate sp and EP (Eq. A.63) and calculate p'N and q N by subtracting to the elastic guess the plastic correction (Eq. A.35) Update zone parameters, and obtain zone volumetric strain and zone specific volume (Eq. A.39). Update the tangencial bulk modulus (Eq. A.41) Calculate the new shear modulus; if v * 0 update using Eq. A.42, otherwise G = const Calculate p' and q using the new elastic stresses Figure A.9: NorSand fish function flow chart 144 Figure A.10: Comparison of deviator stress and volumetric strain for NorSand, between VBA Code and FLAC results for a loose state 145 146 A P P E N D I X B: N O R S A N D FISH FUNCTION FISH version of NorSand model September, 2005 Pascale Rouse Master's Thesis set echo off def nor_sand constitutive_model 110 f_prop m_g m_k m_m m_kappa mjambda mjDi m_p1 m_v0 m_poiss f_prop m_e m_ep m_v m j m_q m_ind m_kc f_prop m_psi m_hard m_chi m_gamma float $sum_p $sum_q $sum_e $sum_ep $p $q $tk $tg $mm $pc $p1 float $e1 $e2 $sh2 $s11i $s22i $s33i $s12i $ds11 $ds22 $ds33 $ds12 float $fs $sa $sb $sc $ba $bb $bc $boa $sqr float $alam $alam1 $qn $pn $add $v $vk $maxg $ming float $hard $pold $qold $dP $dQ $eta $fs2 $pi2 $eta2 $pi3 $pidot float $pt1 $pt2 $pt3 $pt4 $pt5 $pcrit float $Mp $pi $pip $psi $psii $chi $ec $gamma $mvp int $pind Case of mode Initialisation section Case 1 ; — data check — if m_poiss < 0.0 then nerr = 126 error = 1 endjf if m_p1 = 0.0 then m_p1 = 1.0 endjf if m_p1 < 0.0 then nerr = 126 error = 1 end_if if m_pi < 0.0 then nerr =126 error = 1 endjf if m_kappa = mjambda then . nerr =126 error = 1 endjf if m_kappa = 0.0 then nerr= 126 error = 1 endjf ; — initialize current bulk modulus — if m kc = 0.0 then 147 m_kc = m_k end_if ; — initialize current shear modulus — if m_g = 0.0 then if m_poiss # 0.0 then m_g = 3.0*m_kc*(1.0-2.0*m_poiss)/(2.0*(1.0+m_poiss)) endjf endjf Case 2 Running section zvisc = 1.0 if m jnd # 0.0 then mjnd = 2.0 end_if $pind = 0 ; — define constants locally — $tk = m_kc $tg = m_g $mm = m_m $p1 = m_p1 $e1 = $tk + 4.0 * $tg / 3.0 $e2 = $tk - 2.0 * $tg / 3.0 $sh2= 2.0 * $tg $hard=m_hard $chi=m_chi ;— calculate old stress invariants, etc — $pold = -(zs11 + zs22 + zs33) / 3.0 ; — OLD deviatoric stresses — $ds11 = zs11 + $pold $ds22 = zs22 + $pold $ds33 = zs33 + $pold $ds12 = zs12 ; — Qold sqrt(3.0 * J2) — $qold = sqrt(1.5*($ds11*$ds11+$ds22*$ds22+$ds33*$ds33)+3.0*$ds12*$ds12) ;— get new trial stresses from old, assuming elastic increments — $s11 i = zs11 + (zde22 + zde33) * $e2 + zdel 1 * $e1 $s22i = zs22 + (zdel 1 + zde33) * $e2 + zde22 * $e1 $s33i = zs33 + (zdel 1 + zde22) * $e2 + zde33 * $e1 $s12i = zs12 + zde12*$sh2 ;— zone volumetric strain accumulation $sum_e = $sum_e + zdel 1 + zde22 + zde33 ; — mean pressure — $p = - ($s11 i + $s22i + $s33i) / 3.0 ; — deviatoric stresses — $ds11 =$s11i + $p $ds22 = $s22i + $p $ds33 = $s33i + $p $ds12 = $s12i ; — sqrt(3.0 * J2) — $q = Sqrt(1.5*($ds11*$ds11+$ds22*$ds22+$ds33*$ds33)+3.0*$ds12*$ds12) $eta=$qold/$pold ;— Critical Void Ratio and State Parameter from old stresses — $ec=m_gamma-m_lambda*ln($pold) 148 $psi=($v-1.0-$ec) Definition of Mp — if $psi>0.0 then $Mp=$mm*(1.0-$psi/$mm) else $Mp=$mm*(1.0+$psi/$mm) end_if Definition of NorSand parameters — $pi=exp(($eta/$Mp)-1.0) $psii=$psi-m_lambda*(1.0-$eta/$Mp) $pip=exp(-m_chi*$psii/$Mp) - check for yielding — $fs =ln(1.0/$pi)+$eta/$Mp-1.0 if $fs > 0.0 then - yielding in shear — $dP = $p - $pold $dQ = $q - $qold $pt1 = $dQ $pt2 = $dP * ($Mp-$eta) $pt3 = 3.0 * $tg $pt4 = $tk * ($Mp-$eta)*($Mp-$eta) $pt5 = m_hard * $Mp*$pold*($pip/$pi-1.0)*exp(1.0-$eta/$Mp) $alam = ($pt1 + $pt2) / ($pt3 + $pt4 + $pt5) $sa = 3.0 * $tg $sc = $Mp -$eta $sb = $tk * $sc $pind = 1 if $alam < 0.0 then $alam = 0.0 endjf $qn = $q - $sa * $alam $pn = $p - $sb * $alam - new stresses — if $q = 0.0 then $add = $pn - $p zs11 = $s11i - $add zs22 = $s22i - $add zs33 = $s33i - $add zs12 = $s12i else zs11 = ($ds11 / $q) * $qn - $pn zs22 = ($ds22 / $q) * $qn - $pn zs33 = ($ds33 / $q) * $qn - $pn • zs12 = ($ds12/$q)*$qn endjf $sum_p = $sum_p + $pn $sum_q = $sum_q + $qn $sum_ep = $sum_ep + $alam * $sc zvisc = 0.0 mj'nd = 1.0 else - no failure at all — zs11 = $s11i zs22 = $s22i zs33 = $s33i zs12 = $s12i 149 $sum_p = $sum_p + $p $sum_q = $sum_q + $q endjf — update zone parameters — if zsub > 0.0 then m_p = $sum_p / zsub m_q = $sum_q / zsub m_e = m_e + $sum_e I zsub m_ep = m jap + $sum_ep / zsub $v = m_vO * (1.0+m_e) — data check — if $v < 1. then nerr= 167 error = 1 endjf if m_p > 0.0 then error = 0 else nerr = 168 error = 1 endjf if m_kc > m_k then nerr=169 error = 1 endjf — update shear modulus — if mjDoiss # 0.0 then constant Poisson's ratio — m_g = 3.0*m_kc*(1.0-2.0*m_poiss)/(2.0*(1.0+mjpoiss)) else constant G, check upper and lower bounds (nu=0 or 0.5) $maxg = 1.5*m_kc $ming = 0.0 if m_g > $maxg then m_g = $maxg else if m_g < $ming then m_g = $ming endjf endjf endjf ; — reset for next zone $sumj3 = 0.0 $sum_q =0.0 $sum_ep =0.0 $sum_e = 0.0 endjf Case 3 Return maximum modulus cmjnax = m_k + 4.0 * mjg / 3.0 smjnax = m_g Case 4 150 ; Add thermal stresses ztsa = ztea * m_kc ztsb = zteb * m_kc ztsc = ztec * m_kc ztsd = zted * m_kc End_case end opt nor_sand FISH function to initialize initial specific volume and tangent bulk modulus def set_vO loop i (1,izones) loop j (Ijzones) if model(ij) = 110 then pO = -(sxx(i,j)+syy(i,j)+szz(i,j))/3.-pp(i,j) if pO > 0.0 then ec = m_gamma(i,j) - m_lambda(i,j)*ln(pO) m_v0(i,j) = m_psi(i,j) + ec + 1.0 else nerr =170 error =.1 endjf endjf endjoop endjoop end opt nor_sand set echo on The fish function for NorSand will be updated and available for download from the UBC Geotechnical web site [ http://www.civil.ubc.ca/research/qeotech/analytical.htm ]. 151 

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