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Determination of microparameters for discrete element modelling of granular materials with varying particle… Mei, Chen-Jung Judy 2017

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DETERMINATION OF MICROPARAMETERS FOR DISCRETE ELEMENT MODELLING OF GRANULAR MATERIALS WITH VARYING PARTICLE SIZE USING ONE-DIMENSIONAL COMPRESSION TESTING  by  Chen-Jung Judy Mei  B.Sc. (Hons), New York University Abu Dhabi, 2015  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  December 2017  ©  Chen-Jung Judy Mei, 2017      ii Abstract A research program was undertaken to study the effect of particle size on the mechanical response of granular materials, with particular emphasis on supporting the study of the effect of backfill particle size on the soil-pipe interaction of buried pipelines. In this regard, laboratory one-dimensional compression tests of different-sized glass beads and crushed granite were conducted. One-dimensional compression tests on glass beads were simulated in a numerically equivalent discrete element model (DEM), in order to identify suitable DEM particle stiffness microparameters able to reproduce the corresponding laboratory results. Effects of particle size on bulk material behavior were first studied through the analysis of experimental one-dimensional compression test results of both glass beads and crushed granite. Axial stress-strain response of both materials revealed that an increase in particle size of a granular material matrix increased the stiffness of the overall granular matrix. Results also revealed that smaller particles resulted in higher side wall friction values than larger particles of the same material type. The dependence of constrained modulus and shear modulus on effective confining stress determined experimentally from all laboratory tests were in general agreement with relationships previously proposed by other researchers. Numerical simulations of the laboratory specimens were conducted using DEM; i.e. the numerical models were calibrated against experimental results obtained from one-dimensional compression tests of different-sized glass beads to determine suitable particle stiffness microparameters for granular materials of differing particle sizes. The findings indicated that the numerical value of particle stiffness microparameters increased with increasing particle size. In agreement with the experimental results, DEM results also showed that an increase in particle size resulted in increased stiffness of the overall granular matrix under one-dimensional compression. Through evaluation of numerical results, it was proposed that a preliminary relationship between “average” constrained modulus and particle stiffness could be established.  Results indicate that DEM simulations of one-dimensional compression tests can be successfully used to calibrate DEM particle stiffness microparameters. The findings suggest that particle stiffness microparameters should be carefully selected for DEM   iii simulations of granular materials of different-sized particles and, in turn, be utilized in quantitative analysis of geotechnical engineering problems.      iv Lay Summary Pipeline integrity may be achieved through the reduction of soil forces acting on the pipe due to ground movements. One strategy is through careful selection of pipe trench backfill materials. Previous research conducted at the UBC APSIRe™ facility has noted the potential of backfill particle size as an important parameter in controlling lateral soil restraints of buried pipelines. Further studies are needed to understand the effects of particle size – however, conducting full-scale experiments is difficult and expensive. Numerical methods provide an alternative method in the analysis of soil-pipe behavior. In this thesis, discrete element modelling (DEM) is utilized as a tool to study particle-scale interactions and their effect on overall material behavior. By calibrating an equivalent DEM model with one-dimensional compression laboratory test results, this thesis aims to provide further insight into the effect of particle size on DEM input microparameters and their influence on overall material behavior.    v Preface This thesis contains details of the research program conducted at UBC (Vancouver) during the period of September 2015 to December 2017. I was involved in the laboratory equipment development and conducted all the tests in the laboratory experimental program in this research project. I also conducted all the DEM simulations for the numerical program in this thesis. Dr. Dharma Wijewickreme was the supervisory author and was involved in the formation of concept during the conductance of experiments, organizing the content and final editing of the manuscript. All experimental and numerical work presented in this thesis was carried out by the author under the supervision of Dr. Dharma Wijewickreme. The author also acknowledges the guidance and detailed review work of Dr. Sadana Dilrukshi in the formation of the DEM model used for the numerical simulations in this research project, and her help in revising of initial drafts of the manuscript. It is noteworthy to acknowledge the presentation of previously published data and results extracted from other sources in the following figures:  A figure from Zhou et al. (2001) has been reproduced and presented in Figure 2-1 and Figure 5-5 to express details of the effect of particle size on DEM microparameters. Similarly, to illustrate the DEM model calibration process, a figure from Chung and Ooi (2006), has been reproduced and presented in Figure 2-2. Additionally, Figure 2-3 is featured with permission from Dabeet (2014).  For the illustration purpose of the effect of backfill particle size, figures from Wijewickreme et al. (2014) and Dilrukshi and Wijewickreme (2017) have been presented in Figure 2-5 and Figure 2-6, respectively, with permission from the authors. All the figures were prepared by the current author with the results obtained from publication from aforementioned researches reproduced accordingly, or presented directly in the manuscript with permission from the authors.     vi Table of Contents Abstract ......................................................................................................................... ii Lay Summary ................................................................................................................ iv Preface ........................................................................................................................... v Table of Contents ......................................................................................................... vi List of Tables ................................................................................................................ ix List of Figures ............................................................................................................... x List of Symbols .......................................................................................................... xiii Acknowledgements .................................................................................................... xiv Dedication .................................................................................................................... xv 1 Introduction ......................................................................................................... 1 1.1 Thesis Objectives .................................................................................................. 2 1.2 Thesis Organization ............................................................................................... 3 2 Literature Review ................................................................................................. 4 2.1 Discrete Element Modelling (DEM) ........................................................................ 4 2.1.1 Introduction to discrete element modelling ..................................................... 5 2.1.2 Advantages of discrete element modelling ..................................................... 6 2.1.3 General applications and limitations of DEM .................................................. 6 2.2 Utilization of DEM in Geomechanics ...................................................................... 8 2.2.1 DEM microparameter sensitivity studies ........................................................ 9 2.2.2 DEM studies on the effect of particle size .................................................... 11 2.2.3 Model calibration through laboratory element testing ................................... 14 2.2.4 Calibration of DEM models using one-dimensional compression testing...... 17 2.2.5 Application of calibrated DEM microparameters to large-scale studies ........ 21 2.3 DEM Modelling of Soil-Pipe Interaction ................................................................ 23 2.3.1 General overview of soil-pipe interaction modelling ..................................... 23 2.3.2 Numerical and experimental studies ............................................................ 25 2.3.3 Effect of backfill particle size ........................................................................ 27 2.4 Closure ................................................................................................................ 30 3 Experimental Aspects ....................................................................................... 33 3.1 Test Apparatus .................................................................................................... 33 3.1.1 One-dimensional compression chamber ...................................................... 33 3.1.2 Compression frame ..................................................................................... 35 3.1.3 Data acquisition system ............................................................................... 37   vii 3.2 Tested Materials .................................................................................................. 38 3.3 Specimen Preparation and Testing Program ....................................................... 41 3.3.1 Specimen preparation for one-dimensional compression testing of 1-mm diameter particles ................................................................................................... 41 3.3.2 Specimen preparation for one-dimensional compression testing of 9-mm diameter particles ................................................................................................... 43 3.3.3 Testing program .......................................................................................... 44 3.4 Experimental Results ........................................................................................... 46 3.4.1 One-dimensional compression of glass beads ............................................. 46 3.4.2 One-dimensional compression of crushed granite ....................................... 52 3.4.3 Further comments ....................................................................................... 58 3.5 Evaluation of Data ............................................................................................... 59 3.5.1 Stress dependency of the tangent constrained modulus, Mt ........................ 61 3.5.2 Stress dependency of the tangent shear modulus, Gt and tangent Young’s modulus, Et ............................................................................................................ 67 3.5.3 Summary ..................................................................................................... 69 4 Development of the Discrete Element Model .................................................. 71 4.1 DEM Contact Models ........................................................................................... 72 4.1.1 Linear contact model ................................................................................... 72 4.1.2 Hertz-Mindlin contact model ........................................................................ 74 4.2 Description of the Numerical Assembly ............................................................... 75 4.2.1 Numerical configuration for 9-mm diameter glass beads ............................. 75 4.2.2 Numerical configuration for 1-mm diameter glass beads ............................. 76 4.3 Input Parameters and Sensitivity Analyses .......................................................... 78 5 Numerical Results ............................................................................................. 80 5.1 Model Calibration with 9-mm Diameter Glass Beads ........................................... 81 5.1.1 Linear contact model: microparameter calibration of 9-mm diameter glass beads  .................................................................................................................... 81 5.1.2 Hertz-Mindlin contact model: microparameter calibration of 9-mm diameter glass beads ............................................................................................................ 86 5.2 Model Calibration with 1-mm Diameter Glass Beads ........................................... 91 5.2.1 Significance of particle rolling resistance ..................................................... 91 5.2.2 Linear contact model: microparameter calibration of 1-mm diameter glass beads  .................................................................................................................... 93 5.3 Evaluation of DEM Results .................................................................................. 98 5.3.1 Effect of particle size on particle stiffness microparameters ......................... 99 5.3.2 Further comments on the effect of particle size .......................................... 101 5.4 Summary ........................................................................................................... 102 5.4.1 DEM simulations of 9-mm diameter glass beads ....................................... 103 5.4.2 DEM simulations of 1-mm diameter glass beads ....................................... 103 5.4.3 Effect of particle size on DEM simulations with the linear contact model.... 104 6 Summary and Conclusions............................................................................. 106   viii 6.1 Laboratory One-Dimensional Compression Testing of Glass Beads and Crushed Granite ..................................................................................................................... 107 6.2 DEM Simulation and Calibration of One-Dimensional Compression Testing of Glass Beads ............................................................................................................ 108 6.3 Recommendations for Future Research ............................................................ 109 References ................................................................................................................. 111    ix List of Tables Table 2-1: List of references for general overview of DEM in geomechanics ................... 9 Table 2-2: List of DEM calibration studies of laboratory element tests with coarse, granular materials ......................................................................................................... 15 Table 3-1: Summary of one-dimensional compression testing program (with all tests conducted at a constant strain rate of 0.013 mm/s) ....................................................... 45 Table 3-2: Summary of material properties for soda lime glass beads and crushed granite rockfill, with average values taken from the five tests of each material type ....... 45 Table 3-3: Summary of legend labels for one-dimensional compression test results of glass beads and crushed granite, as presented in Section 3.4 ...................................... 46 Table 3-4: Young’s modulus exponents for sand, based on back analysis of observed settlements and laboratory tests, from Byrne and Eldridge (1982)................................. 60 Table 3-5: Determined tangent constrained modulus numbers, km, for all materials tested, obtained from experimental data........................................................................ 65 Table 3-6: Table of all calculated kM, kG and kE values for glass beads and crushed granite, assuming  ν = 0.3 ............................................................................................. 68 Table 4-1: Known microparameter inputs for DEM simulations with glass beads, for both the linear and Hertz-Mindlin contact models .................................................................. 79 Table 4-2: Microparameter inputs for stiffness sensitivity study, conducted with the linear contact model, using 9-mm diameter glass beads ......................................................... 79 Table 5-1: Microparameter inputs and “average” constrained modulus estimates for DEM stiffness sensitivity study with the linear contact model – 9 mm diameter glass beads .. 82 Table 5-2: Final calibrated DEM input parameters for one-dimensional compression testing of 9-mm diameter beads, using the linear contact model ................................... 86 Table 5-3: Microparameter inputs and “average” constrained modulus results for DEM simulations with the Hertz-Mindlin contact model and 9-mm diameter glass beads ....... 87 Table 5-4: Final calibrated parameters for one-dimensional compression testing of 9-mm diameter glass beads, using the Hertz-Mindlin contact model ....................................... 89 Table 5-5: Microparameter inputs and “average” constrained modulus estimates for DEM particle stiffness calibration with the linear contact model – 1 mm diameter glass beads ...................................................................................................................................... 94 Table 5-6: Final calibrated DEM input parameters for one-dimensional compression testing of 1-mm diameter beads, using the linear contact model ................................... 96 Table 5-7: Comparison of PFC3D results for the linear contact model, showing particle stiffness, k and “average” constrained modulus, Mt results from this thesis ................... 99      x List of Figures Figure 2-1: Dimensionless average rolling friction torque for different particle sizes, from DEM simulation of sandpile tests with glass beads (reproduced after Zhou et al. 2001) 12 Figure 2-2: Load-displacement response of corn grains under one-dimensional compression, comparison of DEM and experimental results (reproduced after Chung and Ooi 2006) ............................................................................................................... 17 Figure 2-3: Comparison between oedometer test results and the results of a simulated oedometer test performed using the non-linear Hertz-Mindlin model with various shear modulus values, with permission from ©  Dabeet (2014) ................................................ 19 Figure 2-4: Idealized 3-D soil spring model of soil restraint conditions (reproduced from ALA 2001) ..................................................................................................................... 24 Figure 2-5: Experimental observations - Nqh vs Y’ for Fraser River sand (MS) and crushed limestone (LS) backfill, for NPS 16 pipe buried at H/D=1.6, with permission from ©  Wijewickreme et al. (2014) ......................................................................................... 28 Figure 2-6: Effect of particle size - Nqh versus Yh′ relationship, with normal and shear stiffness of 15 x 106 N/m and friction coefficient of  0.5, and circular particles, with permission from ©  Dilrukshi and Wijewickreme (2017) .................................................. 30 Figure 3-1: Details of one-dimensional compression chamber setup with dimensions in mm ................................................................................................................................ 34 Figure 3-2: Photo of one-dimensional compression chamber setup with connected bottom load cell, top cap and loading ball ...................................................................... 35 Figure 3-3: Photo of Wykeham Farrance Eng. Ltd. mechanical compression frame ...... 36 Figure 3-4: Schematic of overall one-dimensional compression experimental setup ..... 37 Figure 3-5: Photo of 9-mm diameter glass beads (top) and 1-mm diameter glass beads (bottom) ......................................................................................................................... 39 Figure 3-6: Photo of 9-mm crushed granite (top) and 1-mm crushed granite (bottom) ... 40 Figure 3-7: Funnel deposition of 1-mm diameter particles into the steel one-dimensional compression chamber ................................................................................................... 42 Figure 3-8: Photo of siphon setup for the preparation of specimens with 1-mm diameter particles, with the one-dimensional compression chamber and siphon setup  and vacuum flask ................................................................................................................. 43 Figure 3-9: One-dimensional compression test results at 0.013 mm/s for 1-mm diameter glass beads with D50 = 1 mm, γd= 15.2 kN/m,3 showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction (refer to Table 3-3 for legend labels) ..................................................................................................... 48 Figure 3-10: One-dimensional compression test results at 0.013 mm/s for 9-mm diameter glass beads with D50 = 9 mm, γd= 15.0 kN/m3, showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction (refer to Table 3-3 for legend labels) ........................................................................................... 49 Figure 3-11: Comparison of average axial stress and axial strain for a) 1-mm diameter glass beads with D50 = 1 mm, γd= 15.2 kN/m3 and b) 9-mm diameter glass beads with D50 = 9 mm, γd= 15.0 kN/m3 (refer to Table 3-3 for legend labels) ................................. 50   xi Figure 3-12: Void ratio and vertical effective stress for a) 1-mm diameter glass beads with D50 = 1 mm, γd= 15.2 kN/m3 and b) 9-mm diameter glass beads with D50 = 9 mm, γd= 15.0 kN/m3 (refer to Table 3-3 for legend labels) ..................................................... 51 Figure 3-13: One-dimensional compression test results at 0.013 mm/s for 1-mm crushed granite with D50 = 1 mm, γd= 13.6 kN/m3 showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction (refer to Table 3-3 for legend labels) ........................................................................................................................... 54 Figure 3-14: One-dimensional compression test results at 0.013 mm/s for 9-mm crushed granite with D50 = 9 mm, γd= 14.7 kN/m3 showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction (refer to Table 3-3 for legend labels) ........................................................................................................................... 55 Figure 3-15: Comparison of average axial stress and axial strain for a) 1-mm crushed granite with D50 = 1 mm, γd= 13.6 kN/m3 and b) 9-mm crushed granite with D50 = 9 mm, γd= 14.7 kN/m3 (refer to Table 3-3 for legend labels) .................................................... 56 Figure 3-16: Test results at 0.013 mm/s for void ratio and vertical stress for a) 1-mm crushed granite with D50 = 1 mm, γd= 13.6 kN/m3 and b) 9-mm crushed granite with D50 = 9 mm, γd= 14.7 kN/m3 (refer to Table 3-3 for legend labels) .......................................... 57 Figure 3-17: Stress dependency of the tangent constrained modulus, Mt  for 1-mm diameter glass beads, obtained from experimental results ............................................ 62 Figure 3-18: Stress dependency of the tangent constrained modulus, Mt for 1-mm crushed granite, obtained from experimental results ..................................................... 62 Figure 3-19: Stress dependency of the tangent constrained modulus, Mt  for 9-mm diameter glass beads, obtained from experimental results ............................................ 63 Figure 3-20: Stress dependency of the tangent constrained modulus, Mt for 9-mm crushed granite, obtained from experimental results ..................................................... 63 Figure 3-21: Determination of constrained modulus number, km, for 1-mm diameter glass beads (left) and 1-mm crushed granite (right), for σ’v up to approximately 500 kPa ....... 64 Figure 3-22: Determination of constrained modulus number, km, for 9-mm diameter glass beads (left) and 9-mm crushed granite (right), for σ’v up to approximately 500 kPa ....... 64 Figure 3-23: Stress dependency of the constrained modulus for 1-mm diameter glass beads, showing comparison between experimental data and calculated Mt results ....... 66 Figure 3-24: Stress dependency of the constrained modulus for 1-mm crushed granite, showing comparison between experimental data and calculated Mt results ................... 66 Figure 3-25: Stress dependency of the constrained modulus for 9-mm diameter glass beads, showing comparison between experimental data and calculated Mt results ....... 67 Figure 3-26: Stress dependency of the constrained modulus for 9-mm crushed granite, showing comparison between experimental data and calculated Mt results ................... 67 Figure 4-1: Schematic of linear contact model parameters of stiffness and friction coefficients .................................................................................................................... 73 Figure 4-2: Numerical assembly of 9-mm diameter glass beads created in PFC3D ........ 76 Figure 4-3: Numerical assembly of 1-mm diameter glass beads created in PFC3D ........ 77 Figure 5-1: Comparison of one-dimensional compression DEM and experimental results (Test D) for 9-mm diameter glass beads, with the linear contact model for varying   xii particle stiffnesses showing a) top axial stress, b) bottom axial stress and c) side wall friction (refer to Table 5-1 for legend labels) .................................................................. 83 Figure 5-2: Comparison of one-dimensional compression DEM calibration results for 9-mm diameter glass beads with experimental results, for the linear contact model, showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction ............................................................................................................. 85 Figure 5-3: Comparison of one-dimensional compression DEM and experimental results (Test D) for 9-mm diameter glass beads, with the Hertz-Mindlin contact model, for varying particle stiffnesses showing a) top axial stress, b) bottom axial stress and c) side wall friction (refer to Table 5-3 for legend labels) ........................................................... 88 Figure 5-4: Comparison of one-dimensional compression DEM calibration results for 9-mm diameter glass beads with experimental results, for the Hertz-Mindlin contact model, showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction ............................................................................................................. 90 Figure 5-5: Effect of particle size on the dimensionless average rolling friction torque, obtained from sandpile tests with glass beads, reproduced from Zhou et al. (2001) ...... 92 Figure 5-6: Comparison of one-dimensional compression DEM and experimental results (Test D) for 1-mm diameter glass beads, with the linear contact model, for varying particle stiffnesses showing a) top axial stress, b) bottom axial stress and c) side wall friction (refer to Table 5-5 for legend labels) .................................................................. 95 Figure 5-7: Comparison of one-dimensional compression DEM calibration results for 1-mm diameter glass beads with experimental results, for the linear contact model, showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction ............................................................................................................. 97 Figure 5-8: Relationship between “average” constrained modulus and particle stiffness for one-dimensional compression simulations of glass beads, with superimposed hyperbolic curve, labelled the “hyperbolic function”. .................................................... 101     xiii List of Symbols  𝜎𝑣′                                       𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑠𝑡𝑟𝑒𝑠𝑠  𝑃𝑢                                      𝑃𝑒𝑎𝑘 𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑒 𝑓𝑜𝑟𝑐𝑒 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑝𝑖𝑝𝑒𝑙𝑖𝑛𝑒  𝛾                                        𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑜𝑖𝑙  𝐻                                       𝐷𝑒𝑝𝑡ℎ 𝑡𝑜 𝑐𝑒𝑛𝑡𝑒𝑟𝑙𝑖𝑛𝑒 𝑜𝑓 𝑝𝑖𝑝𝑒𝑙𝑖𝑛𝑒  𝑁𝑞ℎ                                    𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑏𝑒𝑎𝑟𝑖𝑛𝑔 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟  𝐷                                       𝐸𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝑝𝑖𝑝𝑒𝑙𝑖𝑛𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟  𝐷𝑟                                     𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦  𝐸𝑡                                       𝑇𝑎𝑛𝑔𝑒𝑛𝑡 𝑌𝑜𝑢𝑛𝑔′𝑠 𝑚𝑜𝑑𝑢𝑙𝑢𝑠  𝑀𝑡                                     𝑇𝑎𝑛𝑔𝑒𝑛𝑡 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑒𝑑 𝑚𝑜𝑑𝑢𝑙𝑢𝑠  𝐺𝑡                                       𝑇𝑎𝑛𝑔𝑒𝑛𝑡 𝑠ℎ𝑒𝑎𝑟 𝑚𝑜𝑑𝑢𝑙𝑢𝑠  𝑘𝐸                                      𝑌𝑜𝑢𝑛𝑔′𝑠 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡  𝑃𝑎                                      𝐴𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒  𝜎𝑚′                                     𝑀𝑒𝑎𝑛 𝑛𝑜𝑟𝑚𝑎𝑙 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑠𝑡𝑟𝑒𝑠𝑠  𝑛                                       𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡   𝜈                                       𝑃𝑜𝑖𝑠𝑠𝑜𝑛′𝑠 𝑟𝑎𝑡𝑖𝑜  𝑒                                       𝑉𝑜𝑖𝑑 𝑟𝑎𝑡𝑖𝑜  𝑘𝑛                                    𝑁𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠  𝑘𝑠                                    𝑆ℎ𝑒𝑎𝑟 𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠  𝜇                                      𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡   𝛿𝑛                                   𝑂𝑣𝑒𝑟𝑙𝑎𝑝 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝑖𝑛𝑔 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠  |𝐹|                                 𝑀𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑛𝑜𝑟𝑚𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝑎𝑡 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑐𝑜𝑛𝑡𝑎𝑐𝑡  𝑅                                    𝑆𝑝ℎ𝑒𝑟𝑒 𝑟𝑎𝑑𝑖𝑢𝑠  𝑑                                    𝑃𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟  𝐾0                                  𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑎𝑡 𝑟𝑒𝑠𝑡 𝑒𝑎𝑟𝑡ℎ 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡     xiv Acknowledgements The completion of this research project was the product of the efforts of many individuals. I would like to express my gratitude to those who supported me throughout this process, as without their assistance, this thesis would not have been possible.  First, I wish to express my most sincere thanks to my research supervisor, Dr. Dharma Wijewickreme, for his steadfast guidance and encouragement throughout the duration of this project. His wealth of laboratory experience and geotechnical knowledge made the culmination of this work a possibility. I would also like to extend my gratitude to Dr. Sadana Dilrukshi for her generous input and help with the setup and success of my numerical modelling work, as well as her countless hours spent revising and helping me during the thesis writing process. Her support during the preparation of my thesis and during my years at UBC are deeply appreciated. The technical assistance of Harald Schrempp, Bill Leung, John Wong and Scott Jackson of the Department of Civil Engineering Workshop during the development of the laboratory testing equipment and setup is also acknowledged with deep regard. I am also grateful to my peers and colleagues for all their support during my time at UBC, particularly their enthusiastic help and invaluable pieces of advice gleaned from their own experiences within the geotechnical engineering department. Finally, the financial support provided by the UBC Pipeline Integrity Institute (PII) and Natural Sciences and Engineering Research Council of Canada (NSERC) to supporting certain components of this research work is gratefully acknowledged.       xv Dedication       This thesis is dedicated to Dr. Roula Maloof, for all your encouragement and support during my journey to becoming a civil engineer, and inspiring me as a woman in engineering 1  1 Introduction Geotechnical hazards arising from permanent ground deformations can be a major cause of damage to buried pipelines.  Some of the common causes of permanent ground displacements are slope instability, landslides, earthquake-induced flow slides or lateral spreading, and ground movements induced by permafrost and thawing.  The key concerns are the unacceptable strains induced in the pipeline due to significant soil loads arising from permanent ground deformations. Thorough understanding of soil-pipe interaction is an important factor in the assessment of the development of soil restraints on pipelines, and in turn, the integrity of buried pipelines against geotechnical hazards.  Current understanding of pipeline soil restraint has been derived mainly from experimental and analytical research representing pipelines buried in sand backfill. In the majority of cases, excavated soil is used as trench backfill material.  Alternate granular materials may be needed when ideal backfill materials are unavailable.  Previous research conducted at the UBC Advanced Soil Pipe Interaction Research (ASPIRe™) facility has shown the potential of backfill particle size as an important parameter in controlling lateral soil restraints of buried pipelines (Wijewickreme et al. 2014).  Steel pipes were subjected to lateral soil restraint testing, buried in three different backfill materials: uniformly-graded Fraser River sand, well-graded crushed sand and gravel, and uniformly-graded crushed limestone. The coarser-grained soil backfill seemed to be able to “flow as individual particles”, as opposed to movement in the form of “blocks” for fine-grained backfill, during ground movement.  The observations of the study suggested that lateral soil loads on pipelines may reduce after reaching a peak load for pipes buried in coarse-grained backfill. In contrast, soil restraint did not exhibit a post-peak “load-shedding” response for pipes buried in fine-grained trench backfill. This noted load shedding (after the peak load) and or higher rates of load reduction may be potentially advantageous for the design integrity of buried pipelines subject to ground deformations.  Further studies are needed to clearly understand the effects of particle size – however, conducting full-scale experiments is difficult and expensive. Numerical methods, 2  when supported by experimental validations, provide an alternative method in the analysis of soil-pipe behavior. Discrete element modeling (DEM) has been shown to be a powerful tool in characterizing granular interparticle interactions, as it is able to capture effects of particle size on material response.  As such, DEM was considered as a useful numerical tool to study the noted pipeline interactions with the soil at particle-scale, and their subsequent effect on overall material behavior.  In spite of the attractiveness of the technique, the selection of suitable material model microparameters for DEM still presents a significant challenge.  Because the DEM model simplifies the complexity of the real physical system, it is important to acknowledge these simplifications, particularly those of particle geometry, contact and deformation characteristics. With this knowledge, it was considered prudent to calibrate the numerical model against data obtained from real-life testing.  In this regard, high quality data acquired from controlled laboratory element tests are particularly suited to the calibration of DEM models.  The intent is to use such calibrated model microparameters and apply them to the study of more complex engineering problems.  1.1 Thesis Objectives With the above background, a study was undertaken to investigate the effect of particle size on the behavior of the overall granular material matrix, as well as the relationship of particle size to DEM particle stiffness microparameters. The following methodology was used to achieve the thesis objectives:  Conducting one-dimensional compression tests with different-sized glass beads and crushed granite rockfill to experimentally investigate the effect of particle size on the mechanical response of a granular material matrix.  This included the assessment of the effect of particle size on the constrained modulus and confining stress.  The use of experimental data from one-dimensional compression tests with glass beads to determine suitable particle stiffness microparameters for granular 3  materials of differing particle sizes, and in turn, derive relationships and attributes with respect to particle stiffness microparameters useful for other DEM simulations. It is intended that these results will form the basis for the selection of suitable DEM material microparameters in future large-scale DEM studies of soil-pipe interactions, and therefore potential developments in our understanding of the effect of particle size in pipe trench backfill selection.  1.2 Thesis Organization The thesis is organized into six chapters. Chapter 1 introduces impetus and background along with the thesis objectives. A survey of literature on the application and calibration of discrete element modelling and soil-pipe applications is summarized in Chapter 2, thus presenting the rationale for this thesis. A detailed description of the test apparatus, test material, testing procedures and an outline of the test program is presented in Chapter 3, along with results and analyses obtained from the experimental testing program.  Chapter 4 provides an overview of the DEM research program, with details of the numerical specimen, setup procedure and input parameters. The results obtained from DEM simulations conducted under this study are explained in Chapter 5, with analysis and discussion of numerical results. Chapter 6 presents the summary of findings and main conclusions arising from this study, followed by a list of references. 4  2 Literature Review The research work in this thesis was performed with the objective of providing input to a broader research program to understand the effect of backfill particle size on the soil restraints on buried pipes subjected to relative ground movements.  The literature review below is presented in order to provide background information for this topic as well as rationale and basis for the research work undertaken. This chapter consists of two main topics of discussion covering the above – the first part comprises a review of discrete element modelling, and the second part presents information on the experimental and numerical studies undertaken with respect to soil-pipe interaction. An overview of discrete element modelling is first presented, with details of the advantages, general applications, and limitations of the method. This is followed by a discussion of applications of DEM in geomechanics, including calibration and sensitivity studies with laboratory element tests, and some examples of applications of DEM to large-scale or field studies. DEM studies on the effects of particle size and one-dimensional compression testing are also discussed, to offer justification for the selected topic and coupling of laboratory testing and numerical method for this study. Finally, a brief overview of DEM studies of soil-pipe behavior is provided, with details of previously published works which have expressed the significance of backfill particle size of buried pipelines, to provide a rationale for a numerical study into the effect of particle size on DEM microparameters and overall material bulk behavior.  2.1 Discrete Element Modelling (DEM) Discrete element modelling (DEM) is a way of modeling a real-life particulate system as an assembly of singular discrete and interacting particles. In comparison with continuum modelling methods, its considerable computing requirements created difficulties in the use of DEM for research and industrial applications. In particular, computing power has restricted the number, size and shape of particles that may be realistically modelled. Recent advances in computing speed and power (as well as improvements in programming) have opened new possibilities in modelling complex 5  granular flow using DEM. This section provides a brief introduction to the Discrete Element Method, with an overview of the method, its advantages (as compared to other continuum methods), and concluding with some general applications and limitations of DEM. 2.1.1 Introduction to discrete element modelling DEM considers a material as a collection of separate particles, which interact at their contact points. As defined in the review by Cundall and Hart (1992), the method allows for finite displacements and rotations of discrete bodies, where new particle contacts are identified automatically as the model progresses. Interactions of the particles are modelled as a dynamic process, with equilibrium states established whenever internal forces are balanced. Computation cycles alternate between applications of Newton’s 2nd Law of Motion to individual particles, and the selected force-displacement law at the inter-particle contacts. Newton’s Law of Motion is utilized to provide the relationship between unbalanced interparticle forces and particle velocity. A force-displacement law is used to calculate forces acting at contacts. The resultant force acting on each particle is then calculated from contact forces (while taking into account body forces and externally applied forces), and particle accelerations are calculated from forces using Newton’s 2nd Law of Motion. Particle positions are then updated, and this process repeats within the designated computational timestep (Δt). The timestep chosen is small enough that during a single timestep, disturbances cannot propagate further than a particle and its immediate neighbors (Itasca, 2016).  Contacts between two objects (i.e. Ball-Ball or Ball-Wall) are commonly simulated using the “soft contacts” approach, which allows a very small overlap distance between particle contacts. The magnitude of the overlap is related to the forces at the contacts through normal and shear stiffness values, Kn and Ks, respectively, and considered equivalent to particle deformation. These stiffness values are dependent on the selected particle contact model, as detailed in Section 5.1. 6  2.1.2 Advantages of discrete element modelling DEM plays an important role when systems may not be represented as a continuum. The assumption of a continuous material is not appropriate for physical processes where individual particle and local discontinuities affect the behavior of a material. As stated in O’Connor (1996), continuum models do not consider the behavior of the individual particle, particle geometry, particle contacts, particle slip, or other dynamic changes in soil structure. Examples include models of granular materials, rock fracture mechanics and particle flow applications. The fundamental difference between discrete and continuum methods is that in DEM, individual particles have their own velocity and force vectors (Walsh 1998). Each particle can only influence another particle though a force generated when two particles come into contact with each other. Simpler methods utilize circular or spherical shapes, while more advanced methods are able to create arbitrary shapes, ranging from geometric to irregular particle shapes. This is of particular importance for applications where particle shape is significant to material behavior – for example, within geomechanics, where particle shape of angular soils affects the overall soil behavior.  The above factors affirm that DEM is the suitable choice of method for assessment planned in this thesis. In order to study the effects of particle size on the bulk behavior of granular soils, it is important must model interactions at the particle scale. Not only may the physical differences in particle size be represented in the numerical model, but detailed analyses of stress-strain behavior, packing density and stress distributions may be conducted within the simulation.  The Section 2.1.3 below provides an overview of general applications of DEM, and some of its limitations, to introduce the reader to applications of DEM to real-life systems, as a preparation for Section 2.2, which will delve into details of the reasoning behind decisions to select the final model which will be used in this thesis. 2.1.3 General applications and limitations of DEM First applied to geomechanics by Cundall and Strack (1979), the DEM method is able to provide insight into the mechanisms governing particle flow, as well as capturing 7  information and detailed interactions at the particle level. DEM numerical simulations are able to enhance our fundamental understanding of granular materials, and therefore, help to improve the design and operation of particulate systems (Cleary, 2000). The complexities of the behavior of particulate systems has also generated research from mathematicians and physicists, where its successful use in modelling geophysical applications include studies of natural processes such as landslides (Cleary 1993, Campbell et al. 1995) and ice flows (Hopkins et al. 1991). Granular materials are encountered in a range of research fields outside of soil mechanics and geotechnical engineering. As noted by O’Sullivan (2011), DEM has been widely used by chemical and process engineers, by researchers in food technology, mining engineering, geomechanics, and pharmaceutical sciences. Some industrial applications of DEM in granular processes include industries such as ball mills (Mishra and Rajamai 1994, Cleary 1998), silo filling (Holst et al. 1997, González-Montellano et al. 2011), die filling for tableting (Wu 2008, Guo et al. 2010), fracture of agglomerates (Thornton et al. 1996, Liu et al. 2010), flow in screw extruders and conveyers (Moysey and Thompson 2005, Owen and Cleary 2009), and granular mixing in blenders (Arratia et al. 2006, Marigo et al. 2013). Due to computing considerations, there are limitations in the use of DEM for industrial applications. The main constraint and difficulty for DEM in an industrial context is the calibration of model input parameters within a reasonable time scale. A DEM model is inevitably a simplification of a real-life physical system, particularly in regards to the particle contacts, geometry, deformation and the number of particles involved (O’Sullivan 2011). Marigo and Stitt (2015) note that real-life industrial systems typically consist of millions of small, irregular particles, often with varying particle size distributions. Hence justification and validation of simulation input parameters becomes increasingly difficult when dealing with large numbers of non-ideal particles. These issues, as O’Sullivan (2011) states, bring researchers to the important decision of selecting a 2D or 3D DEM numerical study. The advantage of 2D simulations is that they are computationally cheaper (with fewer degrees of freedom), and that 2D codes are significantly cheaper than 3D codes. A 2D DEM model is able to capture key complex mechanical responses of granular materials, and may be used to assess effect of particle-scale parameters on the overall material response. However, 8  O’Sullivan cautions, it may be inappropriate to draw qualitative conclusions from 2D models, as physical tests are three-dimensional in nature, and calibration of 2D DEM models against three-dimensional physical test data should always be done with care.  2.2 Utilization of DEM in Geomechanics DEM is commonly used in geomechanics simulations of both cohesionless soil and cemented sand/rock masses of bonded particles. DEM was first applied to rock mechanics and soils by Cundall and Stack (1979). Typically, DEM methods applied to particles larger than 0.1 mm in size, to ensure that the surface attraction forces are insignificant in comparison to the particle inertia. Consequently, DEM simulations of clay particles are less common due to the difficulties in modelling surface interaction forces and particle geometries (O’Sullivan 2011).  As summarized in O’Sullivan (2011), DEM within geomechanics literature may be broadly classified into several categories: modifications of DEM algorithms, DEM model validation and calibration, investigation of relationships between macroscale material response and microscale mechanics, development of interpretation techniques, or simulations of element tests and field-scale boundary value problems.  An overview on the geomechanics applications of DEM is summarized in Table 2-1. Of particular interest to this thesis are DEM studies of laboratory element tests, although some field tests are also included. It should be noted that laboratory simulations outnumber those of field testing within the DEM community, although notable studies such as those on penetration testing (Huang and Ma 1994, Butlanska et al. 2009, Daniel 2008) and pile installation (Lobo-Guerrero and Vallejo 2005) exist. Undoubtedly, there are far more studies than those listed within Table 2-1, however, to highlight the breadth of laboratory tests which have been simulated using discrete methods, only limited references have been provided. Both 2D and 3D studies have been included, although within this thesis, DEM studies utilized three-dimensional models only. The purpose of the list of studies in Table 2-1 are to show the extent of laboratory element tests (in addition to some field tests) which have been studied using DEM. 9  Clearly, the use of DEM in the study of laboratory element tests is well established within the research community, with much of the focus on 3D simulations to accurately represent real-life conditions. Section 2.2.1 will provide a deeper look into the purpose of such element test studies, more specifically, the study of model sensitivity to microparameters such as friction, stiffness or particle shape using the DEM method. Section 2.2.3 discusses DEM studies on the effect of particle size, Section 2.2.3 provides an overview of DEM calibration studies to identify suitable material microparameters, Section 2.2.4 presents a focused discussion of one-dimensional compression testing microparameter calibration, and Section 2.2.5 explores the application of calibrated DEM microparameters in large-scale engineering simulations.  Table 2-1: List of references for general overview of DEM in geomechanics Simulated Test Reference Material 2D/3D Biaxial compression test O’Sullivan et al. (2002) Steel and glass rods 2D Simple shear test Matsushima et al. (2003) Toyoura sand 3D Uniaxial compression test Wu et al. (2010) Asphalt mixture (sands) 3D Plane strain test Powrie et al. (2005) Sand (unspecified) 3D Direct simple shear test Dabeet (2014) Glass beads 3D Triaxial compression test Cui et al. (2007)  Steel spheres 3D Direct shear box test Cui and O’Sullivan (2006) Steel spheres 3D Biaxial and Brazilian tests Potyondy and Cundall (2004) Lac du Bonnet granite 2D and 3D 1D compression test Wiacek and Molenda (2014) Non-uniformly sized spheres 3D Cone penetration testing (CPT) Butlanska et al. (2009) Ticino sand 3D Pile installation Lobo-Guerrero and Vallejo (2005) Calcareous and quartz sands 2D Rock indentation and cutting Huang and Detournay (2008) Analog rock (unspecified) 2D Standard and Large Penetration Testing (SPT and LPT) Daniel (2008) Analog soil (unspecified) 2D  2.2.1 DEM microparameter sensitivity studies One distinct advantage of using DEM is the ability to study the sensitivity of model behavior to specific material microparameters. Some examples of important microparameters include particle stiffness, particle friction, and particle shape. This section will provide an overview of past DEM studies utilizing laboratory element tests 10  to study the effect of model microparameters on the bulk material behavior, with a wide range of different element tests, selected materials and sensitivity studies (as summarized in Table 2-1). A wide range of different laboratory tests have been utilized to study particle-scale effects on bulk material behavior. Simulations of biaxial tests of steel and borosilicate rods (O’Sullivan et al. 2002) allowed for study of the sensitivity of model response to particle geometry and surface friction. Matushima et al. (2002) used simple shear tests to validate the effectiveness of image-based grain shape modelling of sand, while Wu et al. (2010) used uniaxial compression tests on asphalt to investigate the effect of variations in internal sample geometry, distribution of bond strengths, and the coefficient of friction. Powrie et al. (2005) modelled plane strain tests on sand, and explored the sensitivity of the numerical model to variations in the porosity, particle shape factor, and interparticle friction.  Two studies involving steel spheres are listed, with Cui et al. (2007) conducting triaxial tests to analyze the sensitivity of the macro-scale response to the friction coefficient and the circumferential boundary conditions, and Cui and O’Sullivan (2006) simulating direct shear box tests to investigate the model sensitivity to factors such as surface friction (both interparticle friction and particle-boundary friction) and the specimen generation approach.  Many uses of DEM exist for cemented materials – however, this thesis focuses specifically on granular materials, and so only a few prominent studies have been reviewed here. A notable DEM study of cemented materials was conducted by Potyondy and Cundall (2004), who conducted 2D and 3D biaxial, triaxial and Brazilian tests on granite, and the sensitivity of results to particle size was investigated. Huang and Detournay (2008) provide another example of DEM modelling of rock, simulating rock indentation and cutting tests to investigate the effect of length scales on the failure mode in the tool-rock interaction process.  It is important to note that the studies listed in Table 2-1 are DEM analyses of microscale mechanics and macroscale material response, and do not focus on calibration of model microparameters, which are provided in Section 2.3.1. Certain studies selected ideal rod-shaped or spherical materials to avoid misrepresentation 11  of irregular soil grains (O’Sullivan et al. 2002, Cui et al. 2007, Cui and O’Sullivan 2007, Wiacek and Molenda 2014). However, many other listed studies conducted tests on various types of sand, which adds in the difficulty of accurately representing particle shape and particle size distribution.  A number of studies have investigated the effects of particle geometry (Matushima et al. 2002, Wu et al. 2010, Powrie et al. 2005) on macroscale behavior. Studies investigating the effect of particle-size polydispersity also exist, such as those by O’Sullivan et al. (2002) and Wiacek and Molenda (2014).  It seems prudent then, if one is not interested in particle shape effects, to strive to avoid complications of representing particle geometry by selecting rod-shaped or spherical particles for testing. By eliminating the need to investigate the effect of particle shape, more detailed analyses of desired microparameters such as particle friction, particle-size polydispersity and boundary conditions may be conducted without the uncertainty of particle shape effects. Manufactured materials provide the benefit of providing control over variables such as particle size distribution and surface roughness. Suitable materials include steel and glass beads or rods, where material properties may be supplied by the manufacturer, or determined through laboratory methods such as tilt tests, and bead or rod sizes may be selected with accuracy. 2.2.2 DEM studies on the effect of particle size It is rare to find publications on the effect of particle size of monodisperse particles on the material global response. No DEM numerical studies were found within the research community on the effect of particle size on the calibrated DEM microparameters of shear or normal stiffness. However, there are some notable studies on the effect of particle size on the angle of repose, as summarized in the following section, alongside studies of the effect of particle size distribution (whilst not directly applicable to this thesis, this is a related topic in relation to DEM studies of particle size, albeit for polydisperse mixtures). The effect of particle size on the angle of repose has been studied experimentally, such as experimental studies of monodisperse powders (Carstensen and Chan 1976), granular material in a rotatable-drum apparatus (Carrigy 1970) and glass beads (Zhou et al. 2002).The general conclusion drawn is that an increase in particle size will 12  decrease the angle of repose. Cartensen and Chan (1976) suggested that the particle size effect in powders is related to particle cohesive force and sliding friction coefficient – in particular, that the coefficient of sliding friction decreases with particle size, obtaining an equation to relate the angle of repose to particle size.  Zhou et al. (2002) extended the effects of particle size to coarse, cohesionless spheres, conducting experimental and DEM studies of glass spheres in understanding the angle of repose through the use of sandpile tests. Previously in Zhou et al. (2001), DEM results indicated that particle size effects on the observed angle of repose acted primarily through its effect on rolling friction, instead of sliding friction. The relationship between rolling friction torque and particle size is shown in Figure 2-1, and it can be seen that there is an inverse relationship between particle size and rolling friction torque.  Experimental sandpile tests were carried out by Zhou et al. (2002) to validate the DEM numerical simulation, and it was shown that the measured angle of repose could be represented by a power law for particle sizes between 0.05 to 10 mm diameter.    Figure 2-1: Dimensionless average rolling friction torque for different particle sizes, from DEM simulation of sandpile tests with glass beads (reproduced after Zhou et al. 2001)  Excluding the work of Zhou et al. (2002), no other DEM microparameter studies on the effect of particle size of granular and uniform spheres have been published in the 1001000100001000000 5 10 15 20 25Dimensionless average rolling friction torqueParticle size (mm)t = 1 st = 11 s (stable sandpile)13  research community. Some studies of grain size effects on field tests were found in literature, such as a DEM study of simple penetration testing (SPT) and large penetration testing (LPT) by Daniel (2008). A numerical investigation into the grain size effects of single and dual platen penetration tests was conducted, with the study of particle size effects related to platen dimensions and platen spacing. Daniel (2008) concluded that the required single and dual platen penetration energy increased with average particle size, and particle size effect trends were compared to experimental effects observed in SPT-CPT and SPT-LPT correlations. Some publications on the study of the effect of particle size distribution on bulk material response also exist, however, simulations are based on the study of mixtures of non-uniform particle sizes. An example is O’Sullivan et al. (2002), who investigated the effect of rod size distribution on the global response of steel rods under biaxial compression, with the conclusion that peak friction angle decreased significantly with increases in standard deviation of the rod size distribution. Studies on granular packings of spheres include those of Wiacek and Molenda (2014), who concluded that the degree of polydispersity had minor effect on the elastic properties of the system. Another study is that of Minh and Cheng (2013), who investigated the effect of particle size distribution on packing characteristics and compressibility of spherical particles under one-dimensional compression. Several studies on the effect of particle size may also be found for non-granular material applications. An example is a DEM study of cemented materials conducted by Potyondy and Cundall (2004), who conducted 2D and 3D biaxial, triaxial and Brazilian tests (Paterson, 1978) on granite, where the sensitivity of the stress-strain response and damage patterns to particle size effects was investigated. It was found that particle size affects the fracture toughness of cemented materials, which then affects damage processes in rocks. Particle size effects in particle-fluid mixtures have also been investigated, such as in Chu et al. (2017), who simulated flows in industrial cyclones, which are systems used to separate particles from fluids or classify particles by size or density. The authors concluded that both the particle and medium flows are sensitive to particle size, where spatial distributions of solid flow patterns would vary due to differences in particle size. 14  It can be seen from this attempt to review DEM studies on the effects of particle size, that studies on granular, mono-disperse particles are quite rare within the research community. Some studies of cemented materials (such as rock) and solid-fluid mixtures have been conducted, but are not relevant to this study, as they are not granular materials comparable to granular pipeline backfill material. DEM studies of poly-disperse particles are relatively common, with invested interest in the effects of particle size distribution on bulk material behavior. As soils have varying particle size distributions, it seems natural to have an interest in the effects of particle size gradation. However in this thesis, the focus is specifically on the effects of particle size; therefore, eliminating the variability of particle size in a given computer simulation is important since the objective is to understand the effects of particle size itself. The preceding review justifies the need for additional research to determine more refined microparameters as input to discrete element modeling – in this instance, it also serves to assess the effect of particle size on DEM microparameters and, in turn, overall material bulk behavior.  The outcomes of such research also has the potential to add significant value to the geomechanics DEM research community. Further literature related to the effect of particle size (of monodisperse, granular materials) on the global material behavior, as well as its effects on DEM model microparameters, are presented in the following Sections 2.2.3 and 2.2.4. 2.2.3 Model calibration through laboratory element testing DEM can be utilized as a tool to study particle-scale interactions, and their subsequent effect on overall material behavior. DEM is a powerful tool in characterizing granular interparticle interactions, as it is able to capture effects of particle size (as well as particle shape and interlocking effects). Selection of suitable material model parameters pose a significant challenge, and model calibrations with respect to high-quality laboratory element test data are essential in DEM, or any numerical analysis. Because the DEM model significantly simplifies the complexity of the real physical system, it is also important to acknowledge these simplifications, particularly those of particle geometry, contact and deformation characteristics.  15  As stated in O’Sullivan (2011), model calibration aims to take these simplifications into account. The calibration process allow for the simulation of an analog model using simplified particle geometries and contact models. The model parameters are then systematically changed to capture the observed laboratory macroscale response. Sensitivity studies may also be important, such as the sensitivity of observed material response to various microparameters such as particle shape, particle stiffness or particle friction. Of particular interest to this thesis are studies investigating the relationship between particle-scale (microscale) mechanics and the bulk (macroscale) material response, through simulation and calibration of physical tests. Numerical reproduction of laboratory element tests are an effective method to achieve this. As reported in literature, one-dimensional compression tests (Chung and Ooi 2006, Coetzee and Els 2009, Dabeet 2014, Coetzee et al. 2010), direct shear tests (Coetzee and Els 2009, Härtl and Ooi 2008, Coetzee et al. 2010), and triaxial compression tests (O’Sullivan et al. 2004, Belheine et al. 2008) have been successfully utilized to calibrate important material microparameters such as stiffness and friction coefficient. A selection of DEM calibration studies are summarized in Table 2-2 below, with details on the selected laboratory element test and calibrated DEM microparameters, to provide a brief overview of previous works in the DEM geomechanics community.  Table 2-2: List of DEM calibration studies of laboratory element tests with coarse, granular materials Reference Calibrated Test and Material Tested Calibrated Microparameters Chung and Ooi (2006) Single particle compression test with glass beads and corn grains Particle elastic modulus Coetzee and Els (2009) One-dimensional compression test and direct shear box test with corn grains Particle stiffness and friction coefficient O’Sullivan et al. (2004),  Drained triaxial compression test with steel spheres Particle stiffness Dabeet (2014) Oedometer test with glass beads Particle stiffness modulus Härtl and Ooi (2008) Direct shear box test with glass beads Particle friction coefficient Coetzee et al. (2010) One-dimensional compression test and direct shear box test with crushed rock Particle stiffness and friction coefficient Belheine et al. (2009) Drained triaxial compression test with Labenne sand Particle stiffness and friction coefficient,  16  All references indicated in Table 2-2 consist of 3D DEM studies, an important distinction because of the uncertainty in calibrating 2D simulations with three-dimensional laboratory results. Materials tested range from glass beads (Chung and Ooi 2006, Dabeet 2014, Härtl and Ooi 2008), grains (Chung and Ooi 2006, Coetzee and Els 2009), steel spheres (O’Sullivan et al. 2004), sand (Belheine et al. 2009) to crushed rock (Coetzee et al. 2010). Considering the focus of this thesis (based on the selection of backfill materials such as sands or crushed rock), all cited studies consist of drained laboratory tests on coarse, granular materials. Chung and Ooi (2006) calibrated particle elastic modulus of glass beads and corn grains with single particle compression tests to use in one-dimensional compression simulations, as shown in Figure 2-2. Coetzee and Els (2009) and Coetzee et al. (2010) calibrated particle stiffness through oedometer testing, combined with direct shear box tests to calibrate particle friction coefficient. Calibrated parameters were then used in further simulations of industrial processes.  Similarly, Härtl and Ooi (2008) also used direct shear box tests to calibrate particle friction coefficients of glass beads. An important conclusion from Coetzee and Els (2009) is that both particle stiffness and particle friction influence the resulting material internal friction angle in direct shear testing, and therefore a direct shear test cannot be used to determine a unique set of material parameters. The test must be used in conjunction with oedometer testing, in order to calibrate particle stiffness first, then to calibrate particle friction with the direct shear test (once particle stiffness is known). O’Sullivan et al. (2004) conducted triaxial compression tests on steel spheres to calibrate particle stiffness parameters. Belheine et al. (2009) also used triaxial compression tests to calibrate particle friction and stiffness parameters of sand, where stiffness parameters were calibrated against the stress-strain curve, and particle friction coefficient calibrated against the dilatancy curve. However, it seems prudent to maintain some caution with the results in Belheine et al. (2009), where calibration of two microparameters was done with only one laboratory test, instead of two separate laboratory tests, as recommended in Coetzee and Els (2009). It can be seen from this overview that laboratory element tests such as direct shear, triaxial and one-dimensional compression tests are an effective way to calibrate DEM material microparameters. Overall, it can be summarized that direct shear box tests 17  were selected for calibration of particle friction coefficient, whilst compression tests - including triaxial compression tests and one-dimensional compression testing - were used to calibrate particle stiffness properties. Further details on the calibration of stiffness parameters with one-dimensional compression tests is presented in Section 2.2.4, to justify its selection as the test method for this thesis.   Figure 2-2: Load-displacement response of corn grains under one-dimensional compression, comparison of DEM and experimental results (reproduced after Chung and Ooi 2006)  2.2.4 Calibration of DEM models using one-dimensional compression testing An important aspect of this thesis is to investigate the effects of particle size on the calibrated DEM stiffness microparameters of dry, granular materials. A review of past studies indicated that one-dimensional compression testing is particularly suited to calibration of stiffness microparameters in DEM simulations of dry, granular materials. This section will summarize past DEM calibration studies that have utilized one-dimensional compression testing, and explain the reasoning behind the eventual selection of one-dimensional compression as the laboratory element test for this study.  In the one-dimensional compression test, vertical or axial stress is applied to the specimen, while strains in the horizontal or radial direction is prohibited. Shear and 010020030040050060070080090010000 1 2 3 4Force (N)Displacement (mm)DEM ResultsTest 1Test 2Test 3Test 418  compressive stresses, shear strains and volume changes are observed during testing. However, since the specimen is unable to fail in shear, compression is the governing source of strain.  Walsh (1998) simulated one-dimensional compression tests to investigate the drained one-dimensional behavior of granular soils. Using a 2D DEM model, Walsh (1998) studied the effects of initial specimen density on the stress-strain response of a granular matrix under one-dimensional compression, including stiffness response and lateral earth pressure measurements. The author also noted that soil friction angles determined from the two-dimensional models were lower than those found in the three-dimensional soils.  Northcutt and Wijewickreme (2013) studied how the particle fabric would impact the one dimensional compression characteristics with particular reference to the coefficient of lateral earth pressure at rest. Coetzee and Els (2009) noted that one-dimensional compression is particularly suited to stiffness calibration, as the friction coefficient parameter did not significantly affect the stress-strain response of the DEM simulation. The authors performed a series of one-dimensional compression simulations, each with different particle stiffness and friction values. It was found that the friction coefficient had little effect on the resulting overall specimen stiffness. In addition to this, a linear relationship was observed between the numerical particle stiffness and the confined Young’s modulus. Based on these findings, the authors concluded that observed strain in one-dimensional compression testing is primarily caused by deformation of the individual particles and not due to particle rearrangement. In order to calibrate microparameters for corn grains (Coetzee and Els 2009) and crushed rock (Coetzee et al. 2010), one-dimensional compression was combined with direct shear testing, to determine the stiffness and friction coefficient of the selected material. Both studies utilized 2D DEM models. As stated in Coetzee and Els (2009), a combination of the two laboratory tests resulted in a unique set of calibrated parameters. One-dimensional compression allowed for the calibration of stiffness parameters – with the stiffness parameters known, direct shear results could be used to determine particle friction coefficient. 19  Dabeet (2014) also used one-dimensional compression tests to calibrate stiffness parameters for simulations of direct simple shear (DSS) testing of glass beads (Figure 2-3). Particle friction coefficients for glass beads were determined from literature, and a 3D DEM one-dimensional compression simulation used to calibrate the shear modulus (G) of the glass beads. Dabeet (2014) drew an important conclusion from simulations results, that the non-linear Hertz-Mindlin model was able simulate experimental conditions better than the alternative linear contact model.   Figure 2-3: Comparison between oedometer test results and the results of a simulated oedometer test performed using the non-linear Hertz-Mindlin model with various shear modulus values, with permission from ©  Dabeet (2014)  Chung and Ooi (2006) determined particle friction parameters through experimental three-particle sliding tests, and particle elastic modulus (E) was calibrated through single-particle compression tests. Chung and Ooi (2006) then simulated 3D DEM one-dimensional compression tests of corn grains and glass beads for comparison with laboratory results. These authors also investigated the processes of load transmission (from the top to the bottom plate), normal wall pressure distribution, mobilized bulk friction coefficient, and the effect of particle shape. 20  Experimental and numerical studies of powder compaction processes have also been conducted, such as those by Sheng et al. (2004) and He et al. (2015). Sheng et al. (2004) investigated the possibility of using DEM to integrate microscopic properties of particles into numerical continuum models. By using a 3D DEM simulation, coupled with experimental square die compaction tests, the authors were able to study the correlations between material properties (such as interparticle friction coefficient and limiting contact pressure) with the mechanical response of the powder compact. He et al. (2015) conducted a 3D DEM study on die compaction of iron ore fines. The DEM model was calibrated with the die compaction curves, unloading curves, and stress-strain response of the powder compact obtained from experimental uniaxial compression tests. The calibrated model was then used to investigate the effect of consolidation pressure, bond thickness and bond strength on compressive strength and failure modes of the formed compact. This overview suggests that the well-established one-dimensional compression also serves as a good element test providing input to the DEM research community. Some applications include simulations of powder compacts; whilst not directly relevant to this thesis’s study of granular materials, they showcase the usefulness of one-dimensional compression in model calibrations and microparameter studies. The most significant contributions have been those of Coetzee and Els (2009), who proved the effectiveness of using one-dimensional compression as a stiffness calibration method. Dabeet (2014) also successfully used one-dimensional compression to calibrate the stiffness parameters for glass beads, while Chung and Ooi (2006) studied the effect of particle shape under one-dimensional compression.  Based on the literature review above, drained one-dimensional compression of granular materials was selected as the calibration method for this thesis, as a way to study the effects of particle size on overall material behavior. The objective for this study is to determine the effects of particle size on the particle stiffness, as calibrated through comparison with one-dimensional compression tests of different-sized glass beads. In this manner, a study of the effect of particle size on DEM model microparameters may be conducted, through the specific study of particle stiffness in a coupled experimental and DEM study of a one-dimensional compression laboratory element test. 21  2.2.5 Application of calibrated DEM microparameters to large-scale studies DEM microparameters, once calibrated with respect to laboratory element tests, can then be applied to study complex engineering problems. After the calibration of important DEM input microparameters such as particle stiffness and particle friction, the microparameters may be used for DEM simulations of large-scale physical models. It should be noted that due to the large numbers of particles required, the majority of simulations on such studies use 2D DEM simulations. In studies by Coetzee and Els (2009), calibrated corn grain microparameters were used in large-scale 2D DEM simulations of silo discharge and bucket filling. A combination of one-dimensional compression and direct shear testing were used to calibrate particle stiffness and particle friction (using the 2D linear contact model). A model silo was then built to study the granular material outflow, where the depth to width ratio of the silo was large enough to assume 2D flow. Colored grains were used to create distinct layers within the material, to allow for observation of flow patterns and flow rates. For the bucket filling tests, observed drag forces, fill rates and flow zones were compared. The authors concluded that a close correlation with silo flow results could be achieved, and that DEM was able to accurately model the filling process of a bucket or scoop. Coetzee et al. (2010) used similar techniques to calibrate and simulate dragline bucket filling of crushed rock. Similar to Coetzee and Els (2009), a combination of one-dimensional compression and angle of repose tests were used to calibrate the particle stiffness and particle friction properties of the crushed rock. A random sample of 300 rocks was taken, and particles categorized into four distinct particle shapes. These were then represented through the use of clumps within the 3D DEM model. Comparisons between numerical and experimental results of pitch angle and drag force were made to validate the numerical model. A noteworthy DEM field study is that by Jenck et al. (2009), who investigated the arching mechanism observed in piles supporting embankments which overlie soft soils. The authors studied the problem by coupling small-scale laboratory model tests on steel rods with 2D DEM simulations. Biaxial compression tests of a rod assembly were used to calibrate the granular platform analog soil parameters. A 2D DEM 22  simulation of platform layer installation (and the subsequent load acting on the supporting piles) was created to compare with laboratory results. Validation of the DEM model was achieved with good agreement between load and displacement behavior at the platform base. The authors then further supplemented their study with continuum simulations of the same test, to compare the influence of the micromechanical properties for the DEM model and the macroscopic parameters for the continuum analysis. It is important to keep in mind that due to the large-size models required to simulate such industrial processes at full scale, most DEM studies are conducted with 2D simulations, due to restrictions in computer processing power and speed. As discussed in Section 2.1.3, calibration of three-dimensional real life tests with 2D models should always be approached with caution, and it may be inappropriate to draw qualitative conclusions from 2D models, as physical tests are three-dimensional in nature. As O’Sullivan (2011) states, the ability for DEM to provide insight into industrial or field-scale problems is still a challenging task and “work in progress” compared to DEM simulations of element tests that are relatively easier to conduct. Although not a full representation of material behavior in real life, consideration is often given to use 2D DEM simulations for large-scale tests, while remembering that calibration of a 2D model with three-dimensional laboratory results is an analog model.  In an overview sense, the above published information shows the potential of using standard laboratory element tests, conducted under controlled conditions and known stress states to calibrate particle parameters for large scale simulations – a faster and more accurate approach to model calibration, as material parameters are calibrated against standard laboratory element tests. With regard to this thesis, the significance of applying calibrated material parameters to large-scale simulations lies in the desire to represent soil-pipe behavior under different backfill conditions. Section 2.3 will link this together with numerical studies of soil-pipe interaction, in order to show the potential in using laboratory element tests to calibrate material parameters for a large-scale soil-pipe simulation.  23  2.3 DEM Modelling of Soil-Pipe Interaction As expressed in the introductory sections, the background of this thesis is based upon the interest in investigating the effect of trench backfill particle size on the development of soil forces on buried pipelines subject to lateral loading. Whilst the model calibration process herein focuses on utilizing laboratory element testing, the overall objective of this thesis is to contribute to the understanding of the soil-pipe interaction problem.  From a completion of information point of view, this section will provide a general overview of existing numerical studies of soil-pipe interaction, particularly those with combined experimental and numerical methods. Section 2.3.1 presents a general overview of numerical modelling of soil-pipe behavior, particularly the analysis aspects of lateral soil-pipe loading. Section 2.3.2 provides an overview of experimental and DEM studies on soil-pipe behavior. Section 2.3.3 explores the previous studies which have suggested the significance of backfill particle size for buried pipelines under lateral loading, which triggered the development of the backbone of this thesis topic. 2.3.1 General overview of soil-pipe interaction modelling The interaction between a unit cross section of a pipe and the surrounding soil (typically under monotonic load) can be modelled using perpendicular discrete springs in the axial, vertical and horizontal directions, as shown in Figure 2-4. In many cases, soil-spring behavior is assumed to be bilinear, although hyperbolic and multi-linear models may also be considered. Methods for evaluating these soil springs are covered in pipeline design guidelines such as ASCE (1984), ALA (2001) and PRCI (2009).  24   Figure 2-4: Idealized 3-D soil spring model of soil restraint conditions (reproduced from ALA 2001)   Current approaches in the determination of lateral soil restraint on buried pipelines are influenced by methods developed for retaining walls, laterally loaded piles, and vertical anchors. Estimation of lateral soil loads on buried pipes largely focuses on analytical methods presented in works by Hansen (1961) and Trautmann and O’Rourke (1983). As summarized in Monroy-Concha (2013), the ALA (2001) and PRCI (2009) guidelines recommend the use of charts based on works by Hansen (1961). ASCE (1984) guidelines suggest the use of charts by both Hansen (1961) and Trautmann and O’Rourke (1983), without particular recommendation of one method. The peak transverse force per unit length of pipeline may be determined using the following relationship: 𝑃𝑢 = 𝛾𝐻𝑁𝑞ℎ𝐷              [1] where:  𝛾 = effective unit weight of the soil H = depth to centerline of the pipeline kvertical kvertical kaxial khorizontal z y x 25  Nqh = horizontal bearing capacity factor D = external pipeline diameter Selection of the horizontal bearing capacity factor, Nqh, is crucial to the design process. The parameters for analysis may be obtained following PRCI (2009) guidelines. Whilst this thesis does not directly deal with calculations of soil-pipe loads, the holistic aim of this study is to investigate potential material parameter inputs for large-scale models of buried pipelines under lateral loading. Therefore a brief summary of soil-pipe modelling has been provided for context to the following sections, which will explore past numerical studies of soil-pipe modelling, as well as detailed discussion of the significance of backfill particle size, which forms the backbone of this thesis topic. 2.3.2 Numerical and experimental studies In addition to the applications of DEM summarized in Sections 2.1 and 2.2, DEM has also been used as a tool in numerical studies of soil-pipe interaction. Full-scale testing of buried pipelines is expensive, and numerical methods provide an alternative method in the analysis of soil-pipe behavior. Previous studies have presented DEM models coupled with experimental soil-pipe loading results, in order to study soil-pipe interaction of buried pipelines under permanent ground deformations. Some studies in literature have combined scaled-down laboratory tests with 2D models, such as Sakanoue (2008), where full-scale experiments were conducted with a 100-mm diameter pipe, which was subjected to lateral displacement with loose and dense backfill conditions. The selected backfill was dry Chiba sand, and the reaction force with displacement between the test pipe and fixed wall measured during testing. Experimental results were then coupled with a 2D numerical DEM study, of equal scale to the experimental setup. Model contact stiffness and friction coefficient were identified through numerical triaxial tests. Other studies have coupled scaled-down experimental studies with 3D DEM models (Prisco and Galli 2006, Calvetti et al. 2004). Prisco and Galli (2006) conducted small-scale plain strain experimental tests with a sample pipe of 50 mm diameter and 160 26  mm length. Ticino sand was used as backfill, and horizontal, uplift and oblique (45o inclined upward) tests were performed at three relative depths. A 3D DEM numerical specimen was then generated to study the coupling effect and failure modes of buried pipelines under various loading modes. Calvetti et al. (2004) also combined small-scale experimental tests with a 3D DEM model, to calibrate an equivalent numerical model, as well as utilizing their DEM model as a numerical substitute for further experimental tests under different loading conditions. Numerical triaxial tests were used to calibrate interparticle friction parameters. Load-displacement curves, as well as qualitative observations on patterns of displacement increments and contact forces were obtained through the use of 3D DEM modelling.  Some tests have coupled soil-pipe DEM analysis through more specialized laboratory tests (Marshall et al. 2010), who presented centrifuge test data of tunneling effects on buried pipelines and compared them to predictions made using DEM simulations. The study aimed to examine the capability of DEM to model the interaction problem of tunneling beneath buried pipelines, with a 3D DEM model created for direct comparison of bending moment results. The authors noted good agreement between the DEM and centrifuge results, and the success was partially attributed to the use of a non-linear Hertz-Mindlin contact model. Prior DEM simulations had shown that good agreement with experimental results could not be obtained with linear-elastic soil models (a significant conclusion as many soil-pipe simulations use linear-elastic soil models to minimize computational costs). Comparisons between DEM and FEM results have also been made, such as those by Yimsiri and Soga (2006). Soil-pipeline interaction under lateral and upward pipe movements in sand were investigated using DEM analysis. Simulations were conducted for medium and dense sands at different embedment ratios, to represent shallow and deep embankment conditions. The comparison of peak dimensionless forces from the DEM and earlier FEM analyses showed that for dense sand, the DEM analysis gave more gradual transition of shallow to deep failure mechanisms than the FEM analysis. It was also shown that peak dimensionless forces at deep depths were higher in the DEM analysis than in the FEM analysis. It was suggested by the authors that this is due to the differences in soil movements around the pipe due to its granular nature. Yimsiri and Soga (2006) concluded that DEM analysis is able to provide 27  additional information of soil-pipeline interaction in sand at deep embedment conditions (as a supplement to FEM methods). It can be seen that in the study of soil-pipe interaction, DEM provides a valuable tool to create numerical substitutes for experimental testing, or for detailed study of particle flow, pipe displacement and observed forces, or qualitative observations on contact force distributions. Previous soil-pipe studies have pointed out the potential significance of non-linear soil behavior, which warrants careful consideration, considering the popularity of using linear soil models due to limitations in computational power. Studies have also shown differences in DEM and FEM results for deep failure mechanisms of buried pipelines, associated with the particulate nature of soil.  It is clear from these examples that numerical simulations are able to enhance our fundamental understanding of the behavior of buried pipelines under permanent ground deformations, more so when coupled with experimental results for calibration and comparison. DEM can be utilized as a tool to study particle-scale interactions, and their subsequent effect on overall material behavior. DEM is a powerful tool in characterizing granular interparticle interactions, as it is able to capture particle-scale effects which are not possible with the traditional FEM methods. The following section will focus specifically on application of DEM on the effect of backfill particle size of buried pipelines, which forms the basis for the research aims of this thesis. 2.3.3 Effect of backfill particle size In the majority of pipeline installations, excavated soil is used as trench backfill material except in urban areas where sand backfill is preferred. Current understanding of soil restraints on buried pipelines has been derived mainly from experimental and analytical research representing pipelines buried in sand backfill.  Sometimes, alternate granular materials are sought when ideal backfill materials are unavailable in certain geographic regions; for example, there are instances where crushed rockfill, with coarse particles of significantly larger size than that of sand, has been considered as an alternative to sand backfill (Wijewickreme et al. 2017). 28  Previous research conducted at the UBC ASPIRe™ facility has noted the potential of backfill particle size as an important parameter in controlling lateral soil restraints of buried pipelines (Wijewickreme et.al. 2014).  Larger full scale soil-pipe loading tests such as those conducted at the UBC ASPIRe™ facility are much less common, with 406-mm and 457-mm diameter steel pipes loaded in the axial, horizontal and oblique directions (Wijewickreme et al. 2014). The UBC study consisted of a pipe buried in two different backfill materials: uniformly-graded Fraser River sand and uniformly-graded crushed limestone. The observations of the study suggested that lateral soil loads on pipelines may reduce after reaching a peak load for pipes buried in coarse-grained backfill. In contrast, soil restraint remains constant for pipes buried in fine-grained backfill, as shown in Figure 2-5.   Figure 2-5: Experimental observations - Nqh vs Y’ for Fraser River sand (MS) and crushed limestone (LS) backfill, for NPS 16 pipe buried at H/D=1.6, with permission from ©  Wijewickreme et al. (2014)  024681012140 0.2 0.4 0.6 0.8 1MS LSNormalized Lateral Soil Restraint (Nqh)Normalized Pipe Movement (Y')29  Dilrukshi and Wijewickereme (2014) then conducted a numerical investigation in conjunction with results obtained from lateral loading of the buried pipe. An equivalent 2D DEM model was created to simulate lateral loading conditions of the buried pipeline with crushed limestone backfill. DEM allowed for study of the flow of particles around the pipe, and investigations into the observed load-shedding (after peak load) behavior of the limestone backfill. The numerical results showed that coarser-grained soil was able to “flow as individual particles”, as opposed to “blocks” for fine-grained backfill, during ground movement. This “particle flow effect” was suggested as a term for the observed post-peak load reduction during vertical and vertical oblique movement of pipes.  In Dilrukshi and Wijewickreme (2017), this load-shedding behavior of the coarse backfill was explored in further detail, through DEM studies consisting of backfills of different particle sizes. Figure 2-6 presents a parametric study examining how the post-peak load reduction trend varied with backfill particle size. It was concluded that the two relatively smaller particle-sized backfills (of 10 mm and 5 mm diameter) exhibited a “plateauing” response after reaching a peak load. In contrast, the coarser backfill particles (of 20 mm diameter) displayed a noticeable “load-shedding” response. Parametric studies showed that post peak load reduction was not related to the model parameters of particle stiffness or friction. It was concluded that in addition to agreeing with experimental results obtained from the UBC ASPIRe™ facility, the results from the study support the notion that the development of soil restraint on buried pipes may be influenced by the trench backfill particle size. This noted “load-shedding” and higher rates of load reduction may be potentially beneficial for the design of buried pipelines subject to ground deformations. Further studies are needed to clearly quantify the effects of particle size – however, conducting full-scale experiments is difficult and expensive. DEM simulations provide a way to conduct parametric studies through a calibrated model, providing a potentially cheaper and faster way of conducting the many tests needed for such in-depth parametric studies. However, as discussed in Section 2.2, the calibration process for DEM simulations is a significant challenge, and much uncertainty lies in the determination of suitable soil microparameters in DEM numerical simulations. This uncertainty is the backbone for this thesis, where its purpose is to study the effect 30  of particle size on calibrated DEM microparameters, to aid in accurate determination of microparameters for future DEM studies on the effect backfill particle size.   Figure 2-6: Effect of particle size - Nqh versus Yh′ relationship, with normal and shear stiffness of 15 x 106 N/m and friction coefficient of  0.5, and circular particles, with permission from ©  Dilrukshi and Wijewickreme (2017)  2.4 Closure 1) Discrete element modelling plays an important role when systems may not be represented as a continuum. The assumption of a continuous material is not appropriate for physical systems where individual particle and local discontinuities affect the mechanical behavior of a material. 2) The main constraint and difficulty for implementation of DEM in an industrial context is the calibration of model input parameters within a reasonable time scale. A DEM model is inevitably a simplification of a real-life physical system, particularly in regards to the particle contacts, geometry, deformation and the number of particles involved 024681012140 0.1 0.2 0.3 0.4 0.5Normalized Lateral Soil Restraint (Nqh)Normalized Pipe Displacement (Yh')22 mm 10 mm 5 mm31  3) The use of DEM in the study of laboratory element tests is well established within the research community, with the distinct advantage of being able to study the sensitivity of model behavior to specific material microparameters, such as particle stiffness, friction or shape. 4) An attempt to review DEM studies on the effects of particle size showed that studies on granular, mono-disperse particles are rare within the research community. 5) One-dimensional compression testing is particularly suited to stiffness calibration, as the friction coefficient parameter does not significantly affect DEM model behavior. With the stiffness parameters known, other laboratory tests (such as direct shear) may be used to determine particle friction coefficient. 6) DEM microparameters, once calibrated with respect to laboratory element tests, can then be applied and validated with further studies of more complex engineering problems. 7) In the study of soil-pipe interactions, DEM provides a valuable tool to create numerical substitutes for experimental testing, or for detailed study of particle flow, pipe displacement and observed forces, or qualitative observations on contact force distributions. 8) Experimental and numerical DEM soil-pipe studies have suggested that lateral soil loads on pipelines may reduce after reaching a peak load for pipes buried in coarse-grained backfill. In contrast, soil restraint remains relatively constant for pipes buried in fine-grained backfill. In closure, to accurately determine DEM input parameters for soil-pipe studies on the effect of backfill particle size, a well-established technique of DEM model calibration with laboratory element tests was selected for this thesis. Past studies have shown that one-dimensional compression testing is particularly suited to stiffness calibration of granular materials, as particle friction did not have much effect on simulation results. Therefore, one-dimensional compression testing was selected as the laboratory element testing method for this thesis, firstly, for calibration purposes and secondly, for parametric studies of the effect of particle size on overall material response.  32  To negate the significant effects of particle shape, spherical glass beads were chosen for experimental and numerical calibration purposes. However, crushed granite rockfill was also chosen for experimental testing purposes, to provide some insight into the effect of particle size with real-life selections of backfill material (as glass beads do not realistically represent the angular shape of typical backfill soils or rockfill). The results of this thesis aim to supplement those from Wijewickreme et al. 2010, who first noticed the potential significance of load-shedding behavior in coarse backfill, for buried pipelines under lateral loading. By calibrating an equivalent DEM model with quality, accurate one-dimensional compression laboratory test results, this thesis aims to provide further insight into the effect of particle size on calibrated DEM particle stiffness microparameters, and their subsequent influence on overall material response. 33  3 Experimental Aspects As indicated in the introductory chapters, one-dimensional compression tests were conducted on two select granular materials to study the effect of particle size on overall specimen behavior.  In particular, laboratory data generated from these tests were specifically used in conjunction with further numerical studies, to determine the effect of particle size on particle stiffness microparameters employed in DEM modelling.  This chapter presents the experimental aspects related to this study. A detailed overview of all testing apparatus used is presented in Section 3.1, followed by details of the materials tested in this study in Section 3.2. Material parameters are provided, including details on specific gravity tests carried out on one of the materials. Details on specimen preparation methods and the testing program are then described in Section 3.3. Section 3.4 presents the experimental results from the one-dimensional compression tests, and the final section, Section 3.5, presents analysis and interpretation of the test results using a stress-strain model for sands, as proposed by previous studies in literature.  3.1 Test Apparatus The one-dimensional compression apparatus contains three main components – an instrumented one-dimensional compression chamber setup, mechanical compression frame, and data acquisition system. The overall setup is shown in Figure 3-4, and the details of the mechanics behind the one-dimensional compression test are presented in Section 2.3.2. 3.1.1 One-dimensional compression chamber Test specimens were prepared in a polished, rigid, stainless-steel compression chamber setup (Figure 3-1 and 3-2). The inner dimensions of the steel chamber measure 76 mm in diameter, and 100 mm in height. Two steel base plates of differing heights were used to alter the total height of the tested specimen; for example, by 34  switching the setup to have a thicker bottom plate, the inner height of the chamber could be reduced to create a smaller chamber size of 56 mm inner height. This ensured consistency between wall frictions of different testing programs and a particle diameter to ring height ratio of at least 1/8. A top cap of 7 mm thickness and a steel loading ball was placed on top of a reconstituted specimen, in order to complete the setup. Both the top cap and bottom plate were also made of polished stainless-steel, in order to reduce the friction between particles and chamber walls. A top cap with a steel loading ball was placed on top of a reconstituted specimen, in order to complete the setup. The entire chamber as described above was stationed upon a solid steel base, with a steel outer ring to prevent any lateral displacement of the various stacked sections.   Figure 3-1: Details of one-dimensional compression chamber setup with dimensions in mm  35   Figure 3-2: Photo of one-dimensional compression chamber setup with connected bottom load cell, top cap and loading ball  3.1.2 Compression frame A 44 kN (five-ton) compression machine (manufactured by Wykeham Farrance Eng. Ltd.) was used to apply normal loads to the test specimens at a constant strain rate of 0.013 mm/s (Figure 3-3). The loading ram was fitted with a 4 kN capacity load cell to measure the applied axial force. The precision of the load cell is ±3x10-5 kN normal force on the specimen. An identical bottom load cell was placed underneath the base plate of the chamber setup. The difference between the top and bottom load cell readings allowed was used to estimate the side wall friction. Loading ball Top cap One-dimensional compression chamber Outer ring Bottom load cell 36  A 10-mm range linear potentiometer was also magnetically attached to the loading platform to measure displacement during loading. The precision of the potentiometer is ±0.002 mm. A detailed view of the overall experimental setup is shown in Figure 3-4.   Figure 3-3: Photo of Wykeham Farrance Eng. Ltd. mechanical compression frame  Top load cell Loading ram Linear potentiometer with magnetic mount  Loading platform 37   Figure 3-4: Schematic of overall one-dimensional compression experimental setup  3.1.3 Data acquisition system A computerized data acquisition system was used to receive and record voltage signals from the top/bottom load cells and potentiometer. The system comprised of a signal conditioning unit, laptop, and data acquisition software. The signal conditioning unit was custom-designed with a 12-bit data acquisition system and three signal channels. The data acquisition software used was DASYLab®  (Measurement Computing Corp, USA) with sampling frequency set to 10 Hz.  LOAD DIRECTION 38  3.2 Tested Materials Two granular materials were employed in the testing program. The first set of tests were performed on solid soda-lime glass beads (supplied by Walter Stern, Inc., Port Washington, USA) of two different particle sizes, 1-mm and 9-mm diameter, with maximum diameter and sphericity variation of ±10% of the glass bead diameter. The glass bead material had a specific gravity of 2.49 and Poisson’s ratio of 0.22. All the listed soda-lime glass bead material properties in this section were provided by Walter Stern, Inc. Pictures taken on two representative samples of glass beads is shown in Figure 3-5.  All the materials were deposited dry during deposition for specimen preparation; moreover, all the compression tests were conducted under dry conditions. Granite rockfill (Figure 3-6) was used for the second series of tests. The specific gravity of the granite was determined to be 2.77 when tests were conducted as per ASTM Standard Test Method ASTM-C127.  Larger granite particles of 8.0 – 9.5 mm size were tested to represent a coarse material fraction in the 9 mm range, with a resulting D50 of 9 mm. Smaller granite particles of 0.83 – 1.18 mm size were tested to represent finer material in the 1 mm range, with a D50 of 1 mm. Both sets of particles originated from the same source of granite rockfill, as larger granite rocks were crushed down into smaller sizes for the finer material. This ensured consistency between materials tested for both larger and smaller particle sizes.   39     Figure 3-5: Photo of 9-mm diameter glass beads (top) and 1-mm diameter glass beads (bottom)   40     Figure 3-6: Photo of 9-mm crushed granite (top) and 1-mm crushed granite (bottom)   41  3.3 Specimen Preparation and Testing Program This section presents an overview of the specimen preparation methods for 1-mm and 9-mm diameter particles. Details of the laboratory testing program are also provided, for both glass beads and crushed granite materials.  3.3.1 Specimen preparation for one-dimensional compression testing of 1-mm diameter particles The specimens of both 1-mm diameter glass beads and 1-mm crushed granite were prepared using the method of air-pluviation (Wijewickreme et al. 2005).  A small funnel was used to deposit a concentrated flow of 1-mm particles directly into the specimen chamber (Figure 3-7). To produce a loose packing arrangement, the funnel was positioned to provide a near zero particle drop-height. As the top surface of pluviated particles in the chamber grew in height with increasing deposition, the funnel was also raised accordingly to maintain near-zero drop height. The funnel was traversed across the specimen footprint to provide a depositional process as even as possible until the top surface of the placed material was above the desired specimen height. A siphon connected to a vacuum was then used to level the specimen surface at the end of the pluviation process (Figure 3-8). The siphon was passed over the specimen surface at a fixed height with a vacuum suction of about 30 kPa, until no more particles could be removed, and the resulting specimen surface was level with a final specimen heights of about 53 mm. The top cap was then carefully placed on top of the specimen, and the horizontal alignment checked.   42   Figure 3-7: Funnel deposition of 1-mm diameter particles into the steel one-dimensional compression chamber  Plexiglass cylinder Funnel (with glass beads) 43   Figure 3-8: Photo of siphon setup for the preparation of specimens with 1-mm diameter particles, with the one-dimensional compression chamber, siphon setup  and vacuum flask  3.3.2 Specimen preparation for one-dimensional compression testing of 9-mm diameter particles Unlike for the 1-mm diameter particles, the method of air-pluviation was not practical for the deposition of relatively coarser 9-mm diameter particles tested in this study. Simply raining the large particles proved to create much too loose packing arrangements, with the tendency to result in non-uniform specimens. Therefore a method involving placement of particles in layers followed by tamping was used to prepare the specimens of 9-mm diameter particles.  Vacuum regulator Vacuum flask Siphoning tube One-dimensional compression chamber C c Plexiglass cylinder 44  For glass beads, the process involved the placement of 9-mm diameter glass beads in 12 layers. After the placement of each layer, the particles were manually shifted to fill in as many gaps as possible, then gently tamped with a plastic tool to make the top surface as flat as possible. After the placement of the top cap, the horizontal alignment was checked carefully – often, some gentle rearrangement of the particles was needed to ensure a horizontal top surface.  Because of the coarseness of the particles, it was possible count the number of particles to create a specimen; approximately 650 glass beads were required to form a specimen. For the 9-mm crushed granite, irregularly shaped rockfill meant that particles could not be laid down in exact layers. However a method similar to that used above for the glass beads was followed, with layers of granite being gently tamped and rearranged to pack efficiently. Granite particles were then placed individually by hand in the final (top) layer, to ensure a flat, horizontal top surface for testing. As with the 9-mm diameter glass beads, gentle rearrangement of particles was often needed to ensure a horizontal top surface. Final specimen heights were about 96 mm. 3.3.3 Testing program All the specimens prepared as per above were loaded in one-dimensional compression under a constant compression strain rate of 0.013 mm/s. The testing program undertaken is summarized in Table 3-1; as noted, five identical compression tests were performed on specimens of each of the materials tested.  The axial loads on the top and bottom plates were measured during the compression tests. As noted earlier, the side wall friction was estimated as the difference between the axial loads on the top and bottom load cell readings, divided by the area of the vertical chamber walls. Normal loads from the two load cells and axial displacement from a linear potentiometer were recorded during testing. Care was taken to check the horizontal alignment of the top cap before and after each test.  The tests on both 1-mm and 9-mm diameter glass beads were terminated as the vertical load approached the load cell capacity of 4 kN. Most tests on 9-mm crushed granite were stopped as displacement values reached a vertical load of 2 kN; it was noted that audible particle crushing would begin to occur around this load, and higher loads tended to result in “scratching” and “gouging” of the polished interior surface of 45  the stainless steel chamber.  As indicated in Table 3-1, two tests were conducted until a maximum vertical load of 3 kN, whilst the remaining three tests were performed to a maximum vertical load of 2 kN. Table 3-2 presents a summary of the material properties for both the soda lime glass beads and crushed granite rockfill, including the average initial void ratio (e0) and the average dry unit weight, (γd) of the specified granular matrix. Average values were taken from the five tests conducted for each material type. For e0 values of each individual test, refer to Figure 3-12 and Figure 3-16, as presented in Section 3.4  Table 3-1: Summary of one-dimensional compression testing program (with all tests conducted at a constant strain rate of 0.013 mm/s) Material tested Number of tests Maximum applied vertical load 1-mm diameter glass beads 5 4 kN 9-mm diameter glass beads 5 4 kN 1-mm crushed granite 5 3 kN 9-mm crushed granite 5 2 tests to 3 kN, 3 tests to 2 kN  Table 3-2: Summary of material properties for soda lime glass beads and crushed granite rockfill, with average values taken from the five tests of each material type  1-mm diameter glass beads 9-mm diameter glass beads 1-mm crushed granite 9-mm crushed granite Particle size 1 mm 9 mm 1 mm* 9 mm* Specific gravity 2.49 2.49 2.77 2.77 Poisson’s ratio, v 0.22 0.22 N/A N/A Average initial unit weight, γd 15.2 kN/m3 15.0 kN/m3 13.6 kN/m3 14.7 kN/m3 Average initial void ratio, e0 0.62 0.64 0.97 0.83 *D50 of crushed granite rockfill 46  Table 3-3 details the legend labels for the five successful tests run for each material type, as presented in the experimental results in Section 3.4.  Table 3-3: Summary of legend labels for one-dimensional compression test results of glass beads and crushed granite, as presented in Section 3.4 Test 1-mm diameter glass beads  9-mm diameter glass beads  1-mm crushed granite  9-mm crushed granite  1 Glass_1mm_A Glass_9mm_A Granite_1mm_C Granite_9mm_B 2 Glass_1mm_B Glass_9mm_C Granite_1mm_E Granite_9mm_C 3 Glass_1mm_C Glass_9mm_D Granite_1mm_F Granite_9mm_F 4 Glass_1mm_D Glass_9mm_E Granite_1mm_G Granite_9mm_G 5 Glass_1mm_E Glass_9mm_F Granite_1mm_H Granite_9mm_H  3.4 Experimental Results Experimental results from all one-dimensional compression laboratory tests are presented in this section. The variation of the axial stresses with applied axial strain is presented for each material type tested, along with side wall friction values with strain, and calculated changing void ratios with applied vertical stress. The side wall friction is obtained by deducting the bottom load cell response from the top load cell response, divided over the surface area of the vertical chamber wall. To allow for investigation into the effect of particle size on the overall material response, the average axial stress-strain response for a given solid material type, with differing particle sizes are also presented together, for ease of comparison. 3.4.1 One-dimensional compression of glass beads The axial stress-strain response during the testing of 1-mm glass beads is presented in Figure 3-9.  The computed stress-strain values of the 1-mm diameter glass bead 47  tests, from the top and bottom load cells, are presented in the Figures 3-9(a) and 3-9(b), respectively. Side wall friction is presented in Figure 3-9(c). The computed stress-strain values of the 9-mm diameter glass bead tests, from the top and bottom load cells, are presented in the Figures 3-10(a) and 3-10(b), respectively, and the side wall friction presented in Figure 3-10(c) An average of top and bottom axial stresses was computed, and a comparison of average axial stresses are shown in Figure 3-11. The void ratio (e) versus vertical effective stress (v) computed from the tests are presented in Figure 3-12. In regards to Figure 3-11, the average axial stresses were selected for comparison because side wall friction responses were determined to be above 1% of axial stress, for both 1-mm and 9-mm diameter glass beads. For 1-mm diameter glass beads, side wall friction is approximately 6% of overall axial stress, and for 9-mm diameter glass beads, side wall friction is approximately 3% of axial stress. Therefore, the side wall friction was determined to be substantial enough that an average of the top and bottom axial stresses was chosen to be most representative of the stresses in the overall specimen.  The variation of the average axial (vertical) effective stress with applied axial strain is presented in Figure 3-11(a) for 1-mm diameter glass beads and Figure 3-11(b) for 9-mm diameter glass beads.  Figures 3-9(c) and 3-10(c) present calculated side wall friction values with strain. Figure 3-12(a) Figure 3-12(b) and show calculated changing void ratios with applied vertical effective stress, where initial void ratios for both sets of tests showed good repeatability, with almost all tests lying within the 0.60 – 0.65 void ratio range. As noted from Figure 3-11(a) and Figure 3-11(b), both tests displayed relatively linear stress-strain response after load mobilization. The distinct points of load mobilization are in agreement with the Lambe and Whitman (1969), as the stress strain response may be categorized into two distinct sections. Initially, before load mobilization, little stress occurs as strain is governed mainly by particle rearrangement. Further strain past the load mobilization point displays the locking phenomena, where greater stress is needed as contacts become more stable with less voids for particles to move into, and deformation of the individual glass beads begin. 48     Figure 3-9: One-dimensional compression test results at 0.013 mm/s for 1-mm diameter glass beads with D50 = 1 mm, 𝛾𝑑= 15.2 kN/m,3 showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction (refer to Table 3-3 for legend labels)  010020030040050060070080090010000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Top axial stress (kPa)Axial strain (%)Glass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_E010020030040050060070080090010000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Bottom axial stress (kPa)Axial strain (%)Glass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_E0102030405060700 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Side wall friction (kPa)Axial strain (%)Glass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_Ea) b) c) 49     Figure 3-10: One-dimensional compression test results at 0.013 mm/s for 9-mm diameter glass beads with D50 = 9 mm, 𝛾𝑑= 15.0 kN/m3, showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction (refer to Table 3-3 for legend labels)  010020030040050060070080090010000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Top axial stress (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_F010020030040050060070080090010000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Bottom axial stress (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_F0102030405060700 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Side wall friction (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_Fa) b) c) 50    Figure 3-11: Comparison of average axial stress and axial strain for a) 1-mm diameter glass beads with D50 = 1 mm, 𝛾𝑑= 15.2 kN/m3 and b) 9-mm diameter glass beads with D50 = 9 mm, 𝛾𝑑= 15.0 kN/m3 (refer to Table 3-3 for legend labels)  010020030040050060070080090010000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Averaged axial stress (kPa)Strain (%)Glass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_E010020030040050060070080090010000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Averaged axial stress (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_Fa) b) 51    Figure 3-12: Void ratio and vertical effective stress for a) 1-mm diameter glass beads with D50 = 1 mm, 𝛾𝑑= 15.2 kN/m3 and b) 9-mm diameter glass beads with D50 = 9 mm, 𝛾𝑑= 15.0 kN/m3 (refer to Table 3-3 for legend labels)  0.50.550.60.650.70.750.80.850.90.9511 10 100 1000Void ratio, eVertical effective stress (kPa), log scaleGlass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_E0.50.550.60.650.70.750.80.850.90.9511 10 100 1000Void ratio, eVertical effective stress (kPa), log scaleGlass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_Fa) b) 52  Values from Figure 3-11 (a) and (b) were then used to estimate the constrained modulus of the specimen for 1-mm and 9-mm diameter glass beads, respectively, within an axial strain range of 0.8 – 1.2% strain. If the generally linear part of the stress-strain response was considered, an “average” constrained modulus of approximately 80 MPa was estimated for specimens of 1-mm diameter beads, and approximately 133 MPa for specimens of 9-mm diameter beads. Therefore, the observed results indicate that specimens of 9-mm diameter glass beads exhibit a stiffer response than specimens of 1-mm diameter beads under one-dimensional compression. The similarity of void ratios across all tests negates possibilities of the effect of void ratio on the test results. In other words, it may be concluded that an increase in particle size corresponds to an increase in the overall stiffness of a specimen under one-dimensional compression. For 1-mm diameter glass beads, side wall friction was determined to be approximately 6% of axial stress, and for 9-mm diameter glass beads, approximately 3% of axial strain. This indicates that observed side wall friction is higher for the 1-mm diameter glass beads.  The specimens of 1-mm diameter beads in particular show good repeatability in stress-strain results, with only slight deviations in load mobilization response. More variation in load mobilization response is seen in specimens of 9-mm diameter glass beads, which may be attributed to larger variations in particle packing and rearrangement during testing.  3.4.2 One-dimensional compression of crushed granite This section presents results obtained from one-dimensional compression tests of 1-mm and 9-mm crushed granite. The axial stress-strain response during the testing of 1-mm crushed granite is presented in Figure 3-13.  The computed stress-strain values of the 1-mm diameter crushed granite tests, from the top and bottom load cells, are presented in the Figures 3-13(a) and 3-13(b), respectively. Side wall friction is presented in Figure 3-13(c). The computed stress-strain values of the 9-mm crushed granite tests, from the top and bottom load cells, are presented in the Figures 3-14(a) and 3-14(b), respectively, and the side wall friction presented in Figure 3-14(c). An average of the axial stresses measured from the top and bottom load cells was 53  computed, and a comparison of average axial stresses are shown in Figure 3-15 (a) and (b). The void ratio (e) versus vertical effective stress (v) computed from the tests are presented in Figure 3-16 (a) and (b). Similar to the procedure followed for the one-dimensional compression tests of glass beads, average axial stresses were selected for comparison because side wall friction responses were determined to be above 1% of axial stress, for both 1-mm and 9-mm crushed granite. For 1-mm crushed granite, side wall friction is approximately 15% of axial stress, and for 9-mm crushed granite, side wall friction is approximately 11% of axial strain. Therefore, an average of the top and bottom axial stresses was chosen to represent the stresses in the overall specimen, as shown in Figure 3-15 (a) and (b). During testing, it became apparent that for 9-mm crushed granite, a gradual increase of axial load up to 3 kN resulted in damage to the steel chamber and plates. Surficial damage to the chamber walls introduced uncertainties into one-dimensional compression results, as variation of the side wall roughness could affect the repeatability of following tests. Therefore, only two tests were conducted until the axial load reached a maximum load value of 3 kN (Test G and Test H), and the remaining three tests were stopped at axial load values of 2.5 kN (Test B, Test C and Test F). All 1-mm crushed granite tests were performed to a maximum axial load of 3 kN. Figure 3-16 (a) and (b) shows calculated changing void ratios with applied vertical effective stress, and in contrast to the tests on glass beads, differences in initial void ratio can be seen with the 1-mm and 9-mm crushed granite tests. For 1-mm crushed granite, initial void ratio lies in the 0.95 – 1.00 range, while for 9-mm crushed granite, initial void ratio lies in the 0.80 – 0.85 range. Figures 3-13(c) and 3-14(c) present the calculated side wall friction values with axial strain. Observed side wall frictions for crushed granite are noticeably higher than those observed in glass beads. For 1-mm crushed granite, side wall friction is approximately 15% of axial stress, and for 9-mm crushed granite, side wall friction is approximately 11% of axial stress. 54     Figure 3-13: One-dimensional compression test results at 0.013 mm/s for 1-mm crushed granite with D50 = 1 mm, 𝛾𝑑= 13.6 kN/m3 showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction (refer to Table 3-3 for legend labels)  01002003004005006007000 0.5 1 1.5 2 2.5 3 3.5 4 4.5Top axial stress (kPa)Axial strain (%)Granite_1mm_CGranite_1mm_EGranite_1mm_FGranite_1mm_GGranite_1mm_H0501001502002503003504000 0.5 1 1.5 2 2.5 3 3.5 4 4.5Bottom axial stress (kPa)Axial stress (%)Granite_1mm_CGranite_1mm_EGranite_1mm_FGranite_1mm_GGranite_1mm_H0204060801001200 0.5 1 1.5 2 2.5 3 3.5 4 4.5Side wall friction (kPa)Axial strain (%)Granite_1mm_CGranite_1mm_EGranite_1mm_FGranite_1mm_GGranite_1mm_Ha) b) c) 55     Figure 3-14: One-dimensional compression test results at 0.013 mm/s for 9-mm crushed granite with D50 = 9 mm, 𝛾𝑑= 14.7 kN/m3 showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction (refer to Table 3-3 for legend labels)  01002003004005006007000 0.5 1 1.5 2 2.5 3Top axial stress (kPa)Axial strain (%)Granite_9mm_BGranite_9mm_CGranite_9mm_FGranite_9mm_GGranite_9mm_H0501001502002503003500 0.5 1 1.5 2 2.5 3Bottom axial stress (kPa)Axial strain (%)Granite_9mm_BGranite_9mm_CGranite_9mm_FGranite_9mm_GGranite_9mm_H01020304050607080900 0.5 1 1.5 2 2.5 3Side wall friction (kPa)Axial strain (%)Granite_9mm_BGranite_9mm_CGranite_9mm_FGranite_9mm_GGranite_9mm_Ha) b) c) 56    Figure 3-15: Comparison of average axial stress and axial strain for a) 1-mm crushed granite with D50 = 1 mm, 𝛾𝑑= 13.6 kN/m3 and b) 9-mm crushed granite with D50 = 9 mm, 𝛾𝑑= 14.7 kN/m3 (refer to Table 3-3 for legend labels)  01002003004005006007000 0.5 1 1.5 2 2.5 3 3.5 4 4.5Averaged axial stress (kPa)Axial strain (%)Granite_1mm_CGranite_1mm_EGranite_1mm_FGranite_1mm_GGranite_1mm_H01002003004005006000 0.5 1 1.5 2 2.5 3 3.5 4 4.5Averaged axial strain (kPa)Axial stress (%)Granite_9mm_BGranite_9mm_CGranite_9mm_FGranite_9mm_GGranite_9mm_Ha) b) 57    Figure 3-16: Test results at 0.013 mm/s for void ratio and vertical stress for a) 1-mm crushed granite with D50 = 1 mm, 𝛾𝑑= 13.6 kN/m3 and b) 9-mm crushed granite with D50 = 9 mm, 𝛾𝑑= 14.7 kN/m3 (refer to Table 3-3 for legend labels)   0.50.550.60.650.70.750.80.850.90.9511 10 100 1000Void ratio, eAxial effective stress (kPa), log scaleGranite_1mm_CGranite_1mm_EGranite_1mm_FGranite_1mm_GGranite_1mm_H0.50.550.60.650.70.750.80.850.90.9511 10 100 1000Void ratio, eAxial effective stress (kPa), log scaleGranite_9mm_BGranite_9mm_CGranite_9mm_FGranite_9mm_GGranite_9mm_Ha) b) 58  As noted from Figure 3-15, similar to the glass bead tests, both tests displayed a relatively linear stress-strain response after load mobilization. Values from Figure 3-15 were then used to estimate the constrained modulus for 1-mm and 9-mm crushed granite tests, respectively, within an axial strain range of 2.5 – 3.5% strain (1-mm crushed granite) and 1.5 – 2% strain (9-mm crushed granite). An “average” constrained modulus of approximately 26 MPa was estimated for crushed granite specimens of 1-mm particle size, and approximately 45 MPa for crushed granite specimens of 9-mm particle size. Similar to the results for glass beads, observed results indicate that 9-mm crushed granite exhibits a stiffer bulk response than 1-mm crushed granite under one-dimensional compression.  It is important to note the differences in initial void ratio for the 1-mm and 9-mm particle size tests, where 9-mm crushed granite tests consisted of a denser initial particle matrix than the 1-mm crushed granite tests. In other words, 9-mm crushed granite tests had a lower initial void ratio value range. As noted by Lambe and Whitman (1969), dense specimens are more dependent on elastic deformation of particles and the soil skeleton, causing a stiffer material response. Therefore, part of the stiffer response of 9-mm crushed granite can be attributed to its denser initial state. 3.4.3 Further comments Comparisons between stress-strain behavior of 9-mm diameter glass beads (Figure 3-11) and 9-mm crushed granite (Figure 3-15) show some of the difficulties experienced during tests of 9-mm crushed granite. The stress-strain response of the 9-mm diameter glass beads is a relatively smooth, linear response after load mobilization. In contrast, the 9-mm crushed granite has a more variable response, with some fluctuations in axial stress after load mobilization.  A potential reason for this difference is the difference in particle shape, more notably the angular, jagged edges of the larger crushed granite particles. It was much more difficult to achieve a horizontal top surface with 9-mm crushed granite specimens, and some tests could not be conducted to completion because the top cap would slip out of alignment during loading. This indicates the occurrence of more particle rearrangement during the testing of 9-mm crushed granite, as compared to 9-mm diameter glass beads. 59  A few other factors may have contributed to the fluctuations in stress-strain response. Some particle crushing was also evident during the 9-mm crushed granite tests, although there were negligible amounts of crushed material. Additional problems also occurred with scratching and minor damage to the steel chamber and top cap after testing 9-mm crushed granite particles. Due to this issue, not all tests were performed to a 3 kN vertical load, to reduce damage to the apparatus. Evidence of damage to the vertical chamber walls also suggests that care should be taken with estimated side wall frictions for 9-mm crushed granite tests, as visible scratching of the steel walls was noticed in tests with axial loads higher than 2 kN. It should be noted that none of the above points were an issue for tests of 1-mm crushed granite, and that the particles were small enough that top cap alignment did not pose any problems during testing, and no damage to the testing apparatus occurred.  3.5 Evaluation of Data The main objective of the experimental work is to provide soil stiffness values for consideration in developing numerical parameters for discrete element modeling (DEM) of soil-pipe interaction problems.  The data generated from the above testing also provided an opportunity to assess the stiffness of the materials tested and compare the results with other known stiffness values of other materials.  Such a comparison was considered prudent since it allows a valuable cross-check on the information derived in addition to generating the DEM parameters. The stress-strain behavior of soil is complex, being nonlinear, inelastic and stress level dependent. With this in mind, deformation characteristics of soils have been described using hyperbolic stress-strain models. The following section will present data evaluations conducted on the experimental results to obtain stiffness parameters and comparing the values with those proposed by other researchers. Numerous studies in literature have proposed stress-strain relationships for sand accounting for the dependency of material properties on confining stress, including Duncan and Chang (1970), Byrne and Eldridge (1982) and Yan and Byrne (1987). 60  The stress-strain relations are assumed to follow a power law, resulting in the determination of elastic modulus varying with both mean normal effective stress and relative density, as expressed by a relationship of the general form: 𝐸𝑡 = 𝑘𝐸𝑃𝑎 (𝜎𝑚′𝑃𝑎)𝑚                  [2] where Et = tangent bulk modulus; kE = Young’s modulus coefficient, Pa = atmospheric pressure (for normalizing); m = mean normal effective stress; and m = exponent constant.  Table 3-4 provides the kE and m values reported by Byrne and Eldridge (1982) based on back analysis of observed settlements as well as laboratory tests. The increase in kE in terms of increasing relative density (Dr) was noted, and the exponent, m, of 0.5 was found to be in reasonable agreement with experimental data.  Table 3-4: Young’s modulus exponents for sand, based on back analysis of observed settlements and laboratory tests, from Byrne and Eldridge (1982) Relative density, Dr (%) Young’s modulus coefficient, kE Exponent constant, m 25 250 0.5 50 500 0.5 75 750 0.5 100 1000 0.5  It is important to note that the above parameters depend on the grain size, shape, grading and composition of the sand, as well as its density and loading history. Byrne and Eldridge (1982) noted that the value of kE is strongly dependent on grain shape, where angular sands will have lower values of kE, and rounded well-graded sands will have higher values of kE. 61  3.5.1 Stress dependency of the tangent constrained modulus, Mt Using the above stress level dependency of soil stiffness as the basis, the tangential constrained modulus values were determined from the tangent slope of the axial stress-strain results at regular strain intervals; this allowed assessing the stress dependency of the constrained modulus across the entire range of axial stresses observed during testing. (Note: the stress-strain results from one-dimensional compression testing directly allow for direct computation of the constrained modulus since the radial strains in the specimen are virtually zero due to the presence of the stainless steel chamber). The derived experimental tangential constrained moduli (Mt) from this exercise are presented in Figure 3-17, Figure 3-18, Figure 3-19 and Figure 3-20. It can be seen that the general shape of all figures shows a non-linear variation of Mt with increasing vertical effective stress (v) for v < 500 kPa; the value of Mt seems to reach a plateau when v > 500 kPa. It was judged to prudent to assess the variation of constrained modulus with respect to mean normal stress using a formula similar to Equation 2. Replacing Young’s tangent modulus with constrained tangent modulus, the following relationship was considered for the data analysis: 𝑀𝑡 = 𝑘𝑚𝑃𝑎 (𝜎𝑚′𝑃𝑎)0.5                [3] where Mt = tangent constrained modulus and km = soil modulus coefficient.   Since the horizontal stress was not measured, it was decided to compute the value of 𝜎𝑚′  assuming a lateral at-rest coefficient (K0) value of 0.5 for both the tested glass beads and crushed granite, prevalent with the one-dimensional compression stress conditions.  This allowed for determination of 𝜎𝑚′ =23𝜎1′  . With this approach, the Mt values computed for the stress range of 𝜎𝑣′ < 500 kPa were plotted with respect to √𝜎𝑚′  knowing that the constrained modulus coefficient km may be determined from the gradient of the trend line. The resulting plots for the four materials are provided in Figure 3-21 and Figure 3-22. Linear trend lines and their equations drawn with the intent of determining km as well as the coefficient of correlation (R2) values are also shown in the figures. 62    Figure 3-17: Stress dependency of the tangent constrained modulus, Mt  for 1-mm diameter glass beads, obtained from experimental results   Figure 3-18: Stress dependency of the tangent constrained modulus, Mt for 1-mm crushed granite, obtained from experimental results  0204060801001201401601802000 200 400 600 800 1000Tangent constrained Modulus, Mt(MPa)Axial stress (kPa)Glass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_E010203040506070800 100 200 300 400 500 600 700Tangent constrained modulus, Mt(MPa)Axial stress (kPa)Granite_1mm_CGranite_1mm_EGranite_1mm_FGranite_1mm_GGranite_1mm_H63   Figure 3-19: Stress dependency of the tangent constrained modulus, Mt  for 9-mm diameter glass beads, obtained from experimental results   Figure 3-20: Stress dependency of the tangent constrained modulus, Mt for 9-mm crushed granite, obtained from experimental results  0501001502002503003500 200 400 600 800 1000Tangent constrained modulus, Mt(MPa)Axial stress (kPa)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_F0204060801001201400 100 200 300 400 500 600 700 800Tangent constrained modulus, Mt(MPa)Axial stress (kPa)Granite_9mm_BGranite_9mm_CGranite_9mm_FGranite_9mm_GGranite_9mm_H64   Figure 3-21: Determination of constrained modulus number, km, for 1-mm diameter glass beads (left) and 1-mm crushed granite (right), for σ’v up to approximately 500 kPa    Figure 3-22: Determination of constrained modulus number, km, for 9-mm diameter glass beads (left) and 9-mm crushed granite (right), for σ’v up to approximately 500 kPa   y = 5.3049xR² = 0.90380204060801001200 5 10 15 20Tangent constrained modulus, Mt(MPa)σm0.5 (√kPa)y = 7.8963xR² = 0.87330204060801001201401601802000 5 10 15 20Tahngent constrained modulus, Mt(MPa)σm0.5 (√kPa)y = 1.7853xR² = 0.882305101520253035400 5 10 15 20Tangent constrained modulus, Mt(MPa)σm0.5 (√kPa)y = 2.6282xR² = 0.769501020304050600 5 10 15 20Tangent constrained modulus, Mt(MPa)σm0.5 (√kPa)65  All figures show a reasonably good linear fit, with R2 values close to 90%, except for the 9-mm glass beads that had a R2 value close to 75%. In general, the 1-mm diameter glass beads and 1-mm crushed granite display a better linear correlation than 9-mm diameter glass beads and 9-mm crushed granite, with the best fit achieved with the 1-mm diameter glass beads (with an R2 value of above 90%). Table 3-5 presents all km values determined from the experimental data for one-dimensional compression testing of glass beads and crushed granite.  Table 3-5: Determined tangent constrained modulus numbers, km, for all materials tested, obtained from experimental data  1-mm Glass Beads 1-mm Crushed Granite 9-mm Glass Beads 9-mm Crushed Granite km 530 179 790 263  The Mt values could be back-calculated by substituting the above values km back into Equation 3 along with the corresponding axial stress. It is of interest to compare the back-calculated constrained modulus values with the experimental data. Constrained modulus curves developed using Equation 3 are superimposed on experimental results in Figure 3-23, Figure 3-24, Figure 3-25 and Figure 3-26, labelled as the “M Function”, to allow for comparison with experimental results, and it can be seen that there is a good fit of the constrained modulus function with experimental values at low stresses below 500 kPa, which is as expected, due to the range of selected data for the constrained modulus calculations. All four constrained modulus functions above illustrate a clear deviation of constrained modulus behavior at higher axial stresses. For 1-mm diameter glass beads and 1-mm crushed granite tests, (Figure 3-23 and Figure 3-24), the deviation from the constrained modulus function is evident after axial stresses of 500 kPa, as the experimental constrained modulus appear to flatten out to a constant Mt value, and the calculated values are notably higher than the experimental values.  For 9-mm diameter glass beads and 9-mm crushed granite, (Figure 3-25 and Figure 3-26), a good fit between the M function and experimental data can be seen up to 66  600 kPa, where experimental Mt values also appear to level off to a constant value, and the calculated values are notably higher than the experimental values.  These results indicate the suitability of using the sand deformation model proposed by previous researchers in representing the non-linear deformation characteristics of glass beads and crushed granite under low axial stresses in one-dimensional compression.  Figure 3-23: Stress dependency of the constrained modulus for 1-mm diameter glass beads, showing comparison between experimental data and calculated Mt results   Figure 3-24: Stress dependency of the constrained modulus for 1-mm crushed granite, showing comparison between experimental data and calculated Mt results  0204060801001201401601802000 200 400 600 800 1000Tangent constrained Modulus, Mt(MPa)Axial stress (kPa)Glass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_EM Function010203040506070800 100 200 300 400 500 600 700Tangent constrained modulus, Mt(MPa)Axial stress (kPa)Granite_1mm_CGranite_1mm_EGranite_1mm_FGranite_1mm_GGranite_1mm_HM Function67   Figure 3-25: Stress dependency of the constrained modulus for 9-mm diameter glass beads, showing comparison between experimental data and calculated Mt results   Figure 3-26: Stress dependency of the constrained modulus for 9-mm crushed granite, showing comparison between experimental data and calculated Mt results  3.5.2 Stress dependency of the tangent shear modulus, Gt and tangent Young’s modulus, Et Tangent shear modulus, Gt and tangent Young’s modulus, Et, may be expressed in terms of tangent constrained modulus, Mt, and Poisson’s ratio, ν with the following relationships: 0501001502002503003500 100 200 300 400 500 600 700 800 900 1000Tangent constrained modulus, Mt(MPa)Axial stress (kPa)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_FM Function0204060801001201400 100 200 300 400 500 600 700 800Tangent constrained modulus, Mt(MPa)Axial stress (kPa)Granite_9mm_BGranite_9mm_CGranite_9mm_FGranite_9mm_GGranite_9mm_HM Function68  𝐺𝑡 =𝑀𝑡(1−2𝜈)2(1−𝜈)                 [4] 𝐸𝑡 =𝑀𝑡(1+𝜈)(1−2𝜈)1−𝜈                     [5] For all glass bead and crushed granite tests, assuming the behavior of sand or gravelly sand, a value of ν = 0.3 was assumed for all following analyses (Bowles, 1996). This allowed for the conversion of the determined constrained modulus numbers (kM) to shear modulus numbers (kG) and Young’s modulus numbers (kE). Table 3-6 presents all kG and kE values calculated from the determined kM values.  Table 3-6: Table of all calculated kM, kG and kE values for glass beads and crushed granite, assuming  ν = 0.3  1-mm Glass Beads 1-mm Crushed Granite 9-mm Glass Beads 9-mm Crushed Granite kM 530 179 790 263 kG 151 51 226 75 kE 394 133 587 195  The computed kE values for the 1-mm diameter glass beads and 1-mm crushed granite were also compared with those in Table 3-4, as the particle size of these two tested materials are comparable to sands. For 1-mm diameter glass beads, the determined kE value of 394 falls within the values for sand with a relative density of 25 – 50%. For 1-mm crushed granite, the determined kE value of 133 is lower than the kE value of 250 for very loose sands with relative density of 25%. Byrne and Eldridge (1982) noted that the values of kE are strongly dependent on grain shape, where angular sands will have lower values of kE, and rounded well graded sands will have higher values of kE. The calculated tangential kE values fit with this observation, as values of kE for glass beads are significantly higher than values of kE for crushed granite. The authors also claimed that rounded well graded sands may 69  have values of kE twice of those of angular sands. This is echoed in these test results, with perfectly-rounded glass bead kE values more than double those of their angular crushed granite counterpart. It may also explain why the determined kE value for 1-mm crushed granite is lower than the minimum values for sand provided in Table 3-1. 3.5.3 Summary 1) One-dimensional compression tests were conducted on 1-mm and 9-mm diameter glass beads, and 1-mm and 9-mm crushed granite. Tests were conducted at a strain rate of 0.013 mm/s, and five successful tests conducted for each of the materials. 2) All observed stress-strain responses indicated two distinct sections in terms load development. Initially, before load mobilization, little stress occurs as strain is governed mainly by particle rearrangement. Further strain past the load mobilization point displays the locking phenomena, where greater stress is needed as contacts become more stable, and deformation of the individual particles begin. Similar observations have also been made by Lambe and Whitman (1969). 3) For both glass beads and crushed granite tests, 9-mm diameter particles showed a noticeably stiffer stress-strain response than 1-mm diameter particles of the same material type. 4) For both glass beads and crushed granite tests, 1-mm diameter particles resulted in higher side wall friction values than 9-mm diameter particles of the same material type. 5) In all the tests, side wall friction development was notable (i.e., above 1%) of the measured overall vertical stress.  This suggests that the use of top and bottom load cells to assess the side wall friction is meaningful. 6) Observed side wall friction values are noticeably higher for crushed granite tests, more than twice of the observed side wall friction values for tests on glass beads. 70  7) Care should be taken in assessing the observed side wall friction values during tests on 9-mm crushed granite, as visible damage occurred on the vertical walls of the one-dimensional compression chamber. 8) Some particle crushing was evident in the 9-mm crushed granite tests, as well as issues arising in alignment of the top cap during testing. 9) The dependence of tangent constrained modulus and tangent shear modulus on effective confining stress determined experimentally from the tests are in general agreement with the equations reported by previous studies in literature – in particular, for 1-mm diameter glass beads and 1-mm crushed granite, the relations seems to be suitable up to axial stresses of 500 kPa. For 9-mm diameter glass beads and 9-mm crushed granite, the equations seems to provide a good fit up to slightly higher axial stresses of 600 kPa. 10) Calculated values of tangent Young’s modulus coefficients kE also agree with the observations of Byrne and Eldridge (1982), of the dependency of kE values on particle shape. The authors noted that angular sands had lower kE values than rounded sands, which agree with the lower kE values of angular crushed granite, as compared to spherical glass beads.    71  4 Development of the Discrete Element Model The objective of this study is to determine the effect of particle size on calibrated DEM stiffness microparameters, through numerical simulation of laboratory one-dimensional compression tests conducted on different-sized glass beads. Demonstrating the ability of the discrete element modeling approach to simulate real-life experimental laboratory element tests is a prudent way of identifying suitable DEM microparameter inputs for geomechanics applications, as discussed in the Section 2.2. Assessing the effect of particle size on DEM microparameters is important since it contributes to the high-level objective of the present research program at UBC, which is to understand the effect of backfill particle size on the soil restraint of buried pipes subjected to relative ground movements using DEM (also see Chapter 1, and Section 2.3 for background details).   In brief, following the methods utilized by previous DEM calibration studies (Chung and Ooi 2006, Coetzee and Els 2009, Dabeet 2014, Coetzee 2016), numerical DEM simulations of one-dimensional compression tests of glass beads were used to identify suitable stiffness microparameters for calibration against laboratory results. The experimental stress-strain and side-wall friction responses obtained through one-dimensional compression testing of 1-mm diameter and 9-mm diameter glass beads, as presented in Section 3.4, were used to calibrate/validate the discrete element models. The calibrated DEM model parameters were then utilized to determine the effect of particle size on bulk material response, as well as its effect on the particle stiffness parameters used in the DEM simulation.  This chapter presents the development of the discrete element model. A detailed overview of DEM contact models used in the study are provided in Section 4.1. This is followed by descriptions of the numerical assembly in Section 4.2, as well as details on the final DEM input parameters and selected microparameter sensitivity studies in Section 4.3. For details of the DEM calibration results, please refer to Chapter 5.  72  4.1 DEM Contact Models All simulations reported in this study were conducted as three-dimensional (3D) numerical models, using commercially available Particle Flow Code software, PFC3D version 5.0 (Itasca, 2016). In PFC, contacts between two objects (i.e. Ball-Ball or Ball-Wall) are simulated using the “soft contacts” approach. This method allows a small overlap distance between the contacts, where the magnitude of the overlap distance is related to forces at the particle contacts through normal stiffness (Kn) and shear stiffness (Ks) values. The linear contact model (described in detail in Section 4.1.1) was mainly used for simulations in the identification of suitable stiffness microparameters, and to study the effect of particle size on the selected stiffness microparameter inputs. Similar to previous studies, the first set of simulations were performed using the linear contact model (O’Sullivan et al. 2002, O’Sullivan et al. 2004, Coetzee et al. 2009, Coetzee et al. 2010), and in accordance with these studies, it was assumed that normal stiffness was equal to the shear stiffness. The laboratory one-dimensional compression tests conducted on both 1-mm diameter glass beads and 9-mm diameter glass beads were numerically simulated with the linear contact model.  The Hertz-Mindlin contact model has also widely been used by many researchers in 3D DEM studies (Chung and Ooi 2006, Härtl and Ooi 2008, Dabeet 2014, Cui and O’Sullivan 2006). Some researchers have found that the Hertz-Mindlin model can simulate granular material behavior better than the linear contact model, since it can capture the non-linear response of granular assemblies (Dabeet 2014). Therefore, some numerical simulation of 9-mm diameter glass beads with the Hertz-Mindlin contact model allowed for comparison of the two different contact models, and their suitability on capturing bulk material behavior under one-dimensional compression. 4.1.1 Linear contact model The linear contact model utilizes linear and dashpot components, where the linear component provides linear elastic (compression) frictional behavior, while the dashpot component provides viscous behavior. Both components then transmit 73  forces at particle or wall contacts. In this contact model, the linear springs cannot sustain tension. Slip is controlled by the assigned friction coefficient, μ (Itasca, 2016).  The model is defined by the normal and shear stiffnesses, kn and ks. For the linear model, kn and ks are constants, and specified as a material property. It is suitable for use in both 2D and 3D models. Figure 4-1 provides a schematic of the contact model parameters. In this study, calibration of the linear contact model will refer to the calibration of the DEM stiffness microparameters of the one-dimensional compression model, assuming normal stiffness equal to shear stiffness.  It should be noted that the linear contact model is the simplest contact model, and therefore one of the least computationally expensive contact models to use. Because of its simplicity, it is often used in large-scale DEM simulations, to minimize computational time and associated costs.   Figure 4-1: Schematic of linear contact model parameters of stiffness and friction coefficients  Normal stiffness, kn Shear stiffness, ks Friction coefficient, μ 74  4.1.2 Hertz-Mindlin contact model Alternatively, the non-linear Hertz-Mindlin contact model updates the values of kn (Equation 6) and ks (Equation 7) by considering the particle interactions of the numerical assembly at each time step. It is defined by the Poisson’s ratio (ν), shear modulus (G) and surface friction coefficient (μ) of the particles where δn is the overlap distance between the two contacting entities, |F| is the magnitude of normal force at the contact, and R is sphere radius. The contact model is recommended for 3D simulations. In this study, calibration of the Hertz-Mindlin contact model will refer to calibration of the DEM shear modulus microparameter of the one-dimensional compression model.  𝑘𝑛 = (2𝐺√2𝑅3(1−𝜈)) √𝛿𝑛                   [6] 𝑘𝑠 = (2(𝐺23(1−𝜈𝑅))132−𝜈) |𝐹|13        [7]  It should be noted that the Hertz-Mindlin model is a non-linear model which calculates stiffness parameters at each time step based on the particle overlapping conditions. This makes it much more computationally involved than the linear contact model, and therefore it is not always feasible to utilize it for large-scale engineering studies, due to time and cost constraints. For this reason, Hertz-Mindlin model was only used to simulate 9-mm diameter glass beads in this study, as simulations of 1-mm diameter glass beads proved to be too computationally heavy in terms of time.  75  4.2 Description of the Numerical Assembly In this study, a 3D-DEM approach was used to study the effect of particle size on load displacement behavior of materials subjected to one-dimensional compression tests. To exclude the effects of particle shape, only one-dimensional compression tests of spherical glass beads were simulated. All spherical particles were generated using the “particle expansion” method found in the PFC3D software. A closed cylinder with rigid walls was generated, and scaled-down particles were generated randomly within the chamber. The damping coefficient was set to 0.3 and particle rotation was not restricted. After corresponding properties and attributes were assigned to the model, the particle diameters were then multiplied by a scaling factor to obtain the desired particle diameter, and allowed to stabilize under gravity. In order to promote particle movement and packing during the initial stabilization process, considerably smaller particle friction coefficient was assigned to the particles. After the desired packing arrangement was obtained, the correct coefficient of friction was assigned to the particles, and the system allowed to re-stabilize, before beginning the one-dimensional compression test.  In order to match with the laboratory experimental procedure, one-dimensional compression was applied in a displacement-controlled manner and the top plate moved downward at a rate of 0.013 mm/s. Forces generated on the top and bottom plates of the numerical setup, as well as the downward displacement of the top cap were recorded during all simulations. 4.2.1 Numerical configuration for 9-mm diameter glass beads For the simulation of one-dimensional compression test specimen with 9-mm diameter particles, a closed cylinder of inner diameter 76 mm, and inner height of 100 mm was generated to create a full-scale model of the one-dimensional compression chamber. The number of particles in the numerical model were made exactly equal to that of the experimental specimens, with a total of 648 particles. Similar to the experimental tests, a seating load of approximately 5 N was obtained before initializing testing. The numerical assembly for 9-mm glass beads is shown in Figure 76  4-2, with a diameter-particle ratio of 8.4, which is larger than the recommended ratio of 8.   Figure 4-2: Numerical assembly of 9-mm diameter glass beads created in PFC3D  4.2.2 Numerical configuration for 1-mm diameter glass beads Experimental one-dimensional compression tests of 1-mm diameter glass beads were conducted on specimens of 56 mm height and 76 mm diameter. Due to the high computational effort of running a simulation with considerably high numbers of particles, numerical simulations for the 1-mm diameter glass beads were not simulated with a full-scale model. Initially, an attempt was made to re-simulate a full-size specimen equivalent to the experimental specimen tested under the one-77  dimensional laboratory compression tests. However, it quickly became apparent that a numerical assembly with such a large number of particles was too computationally impractical, and would not be feasible for the time-frame of this thesis.  Therefore, it was decided to perform DEM simulations on a specimen of reduced height. Experimental results from one-dimensional compression tests of glass beads in Chapter 3, Section 3.4.1, indicated that observed side wall friction was relatively small (with side wall frictions of approximately 6% of axial stress for 1-mm diameter glass bead tests). With the assumption of negligible side wall friction (represented by the small friction coefficient selected for the smooth, polished steel walls), a numerical specimen with a decreased height (of quarter height) was deemed to be sufficiently representative of the experimental one-dimensional compression tests. The numerical assembly created in PFC3D is shown in Figure 4-3. The diameter of the assembly was kept equal to the experimental setup, at 76 mm inner diameter and the height of the numerical specimen was decreased to a quarter of the experimental specimen height. This resulted in a model geometry consisting of a closed cylinder of 76 mm diameter and 14 mm in height, leading to a diameter-particle ratio of 76 (>8).   Figure 4-3: Numerical assembly of 1-mm diameter glass beads created in PFC3D   The number of particles were determined by creating an assembly with similar bulk density to the experimental specimen. The simulation was deemed acceptable when 78  numerical bulk density was within 50 kg/m3 of the average experimental bulk density. Initial void ratio of the specimen was also checked, to ensure it was within the void ratio range of the experimental tests. The final number of simulated particles in this configuration totaled to some 74,000 particles. 4.3 Input Parameters and Sensitivity Analyses The first set of simulations were performed using the linear contact model, with simulations of both 1-mm and 9-mm diameter glass beads. As mentioned above, similar to many previous studies, it was assumed that normal stiffness equaled shear stiffness (O’Sullivan et al. 2002, O’Sullivan et al. 2004, Coetzee et al. 2009, Coetzee et al. 2010). The Hertz-Mindlin model was also used to simulate 9-mm diameter glass beads, to compare the two different contact models. Surface friction coefficient of the glass beads was assumed to be 0.176, as given in Calvaretta et al. (2010). These authors measured the friction coefficient at the interface between two glass beads using an interparticle testing apparatus, which allowed one particle to be sheared over the top of another stationary particle. As specified in Chapter 3, a specific gravity of 2.5 for the soda lime glass beads was provided by the manufacturer, as well as a Poisson’s ratio of 0.22, which were used in all simulations. In previous studies by Coetzee (2016), the wall stiffness was set to approximately one order of magnitude higher than the particle stiffness, to ensure rigid steel walls. Therefore a relatively high value of wall stiffness (1 x 108 N/m) was used for the steel chamber walls. A low friction coefficient value of 0.05 was used to represent the smoothness of the polished steel surface. All input parameters are summarized in Table 4-1, for both the linear contact model and the Hertz-Mindlin contact model. A particle stiffness sensitivity study was also conducted with the linear contact model, first to identify suitable particle stiffness microparameters in the representation of equivalent experimental material behavior, and second, to understand the effect of particle stiffness on the overall DEM material response. Further insight into the effects of particle stiffness on macroscale response were obtained through observations of stress-strain response and side wall friction response, as detailed in Chapter 5. 79  Sensitivity studies were conducted with 9-mm diameter glass beads only.  Details of the particle stiffness sensitivity study are summarized in Table 4-2.  Table 4-1: Known microparameter inputs for DEM simulations with glass beads, for both the linear and Hertz-Mindlin contact models Parameter Glass beads Steel walls Particle stiffness, k1 Calibrated parameter 1 x 108 N/m Friction coefficient, μ 0.176 0.05 Shear modulus, G Calibrated parameter 77 GPa Poisson’s ratio, v 0.22 0.3 Density, ρ 2500 kg/m3 N/A 1Assuming normal stiffness equal to shear stiffness  Table 4-2: Microparameter inputs for stiffness sensitivity study, conducted with the linear contact model, using 9-mm diameter glass beads Particle stiffness, k Particle friction coefficient, μ Particle density, ρ 7 x 106 N/m 0.176 2500 kg/m3 5 x 106 N/m 0.176 2500 kg/m3 3 x 106 N/m 0.176 2500 kg/m3 1 x 106 N/m 0.176 2500 kg/m3 7 x 105 N/m 0.176 2500 kg/m3 80  5 Numerical Results Numerical results from all one-dimensional compression DEM simulations of glass beads are presented in this chapter. In the following text, calibration of the DEM model refers to the selection of the most suitable DEM input parameters, for successful numerical simulation of laboratory one-dimensional compression tests. Calibration of DEM input parameters (as specified in Chapter 4) is accomplished through comparisons of DEM and experimental results for 9-mm and 1-mm diameter glass beads, respectively. The results presented comprise of evaluations of the variation of the top and bottom axial stresses with applied axial strain for each simulated material type, along with side wall friction values with strain, for the selected range of microparameters as described in Section 4.3.  The stress-strain responses obtained from the experiments, as presented in Chapter 3, were used to identify suitable stiffness microparameters for the simulation of bulk materials in engineering applications. The results are presented in two sections. Section 5.1 comprises of DEM numerical results from one-dimensional compression testing of 9-mm diameter glass beads. Comparisons between the linear and Hertz-Mindlin contact models were also made for the 9-mm glass bead results, to determine the suitability of the two different contact models in capturing bulk material response. Section 5.2 presents the results from DEM numerical analysis of one-dimensional compression testing of 1-mm diameter glass beads using the linear contact model.  In Section 5.3, the final selected particle stiffness parameters for the linear contact models are used to assess the effect of particle size on suitable DEM stiffness input parameters for 1-mm and 9-mm glass beads subjected to one-dimensional compression testing. Comparisons of the 1-mm and 9-mm diameter glass bead DEM models are also made outside of particle stiffness microparameters. Section 5.4 finishes off the chapter with a summary of DEM results obtained in this thesis. As noted in Chapter 4, the normal and shear stiffness were assumed to be equal in all DEM simulations, and referred to as particle stiffness hereafter.  81  5.1  Model Calibration with 9-mm Diameter Glass Beads Two contact models were used in the DEM simulation of 9-mm glass beads under one-dimensional compression. The first contact model considered was the linear contact model. Section 5.1.1 presents a parametric study of the effect of particle stiffness, k, on the bulk material response. The results of the parametric study were utilized in the selection of the most suitable particle stiffness input parameter for representing bulk material behavior under one-dimensional compression. Final calibrated DEM one-dimensional compression results (using the linear contact model) are also presented in Section 5.1.1. The second contact model considered was the Hertz-Mindlin contact model, with calibration of the particle shear modulus, G, to select the most suitable particle shear modulus input parameter for representing bulk material behavior under one-dimensional compression. Final calibrated DEM one-dimensional compression results (using the Hertz-Mindlin contact model) are presented in Section 5.1.2, and compared to experimental testing results. Some researchers have found that the Hertz-Mindlin model can simulate granular material behavior better than the linear contact model, due to the non-linear response of granular assemblies (Dabeet 2014), and therefore comparisons were made between calibrated results obtained using the Hertz-Mindlin and linear contact models in Section 5.1.2. 5.1.1  Linear contact model: microparameter calibration of 9-mm diameter glass beads This section presents a parametric study of the effect of numerical particle stiffness input values on bulk material response. Five different particle stiffness input values were used to investigate the effect of particle stiffness on bulk material response, as summarized in Table 5-1. The axial stress-strain responses obtained with the selected stiffness parameters during the DEM simulation of 9-mm diameter glass beads are presented in Figure 5-1; Figures 5-1(a) and 5-1(b), with the computed stress-strain responses at the top and bottom ends of the numerical specimen, respectively. The computed side-wall friction development is presented in Figure 5-1(c).  82  For clarity of comparison, the five DEM simulations are presented with the counterpart results from one experiment on 9-mm diameter glass beads. This comparison allowed for the selection of the most suitable DEM stiffness input parameter, for the successful simulation of experimental test results (represented by Test D). If the generally linear part of the stress-strain response was considered, “average” constrained moduli values were estimated from the top axial stress-strain response within a range of 0.4% – 1.2% axial strain from the Figure 5-1(a), as presented in Table 5-1.  Table 5-1: Microparameter inputs and “average” constrained modulus estimates for DEM stiffness sensitivity study with the linear contact model – 9 mm diameter glass beads Identification (Legend Label) Particle   stiffness, k Particle friction coefficient, μ “Average” constrained modulus, Mt DEM_k7e6 7 x 106 N/m 0.176 121 MPa DEM_k5e6 5 x 106 N/m 0.176 90 MPa DEM_k3e6 3 x 106 N/m 0.176 53 MPa DEM_k1e6 1 x 106 N/m 0.176 24 MPa DEM_k7e5 7 x 105 N/m 0.176 13 MPa *All simulations used glass bead properties of ρ = 2500 kg/m3   As can be observed from Figure 5-1, particle stiffness parameters have a significant effect on the overall bulk material behavior under one-dimensional compression. It can be clearly noted from Figures 5-1(a) and (b) that increasing particle stiffness results in an increase in the overall stiffness response of the glass-bead specimen under one-dimensional compression. In relation to this, the estimated “average” constrained modulus values of the specimens also follow a similar response (as can be seen in Table 5-1).  83       Figure 5-1: Comparison of one-dimensional compression DEM and experimental results (Test D) for 9-mm diameter glass beads, with the linear contact model for varying particle stiffnesses showing a) top axial stress, b) bottom axial stress and c) side wall friction (refer to Table 5-1 for legend labels)  02004006008001000120014000 0.2 0.4 0.6 0.8 1 1.2Top axial stress (kPa)Axial strain (%)Test DDEM_k7e6DEM_k5e6DEM_k3e6DEM_k1e6DEM_k7e502004006008001000120014000 0.2 0.4 0.6 0.8 1 1.2Bottom axial stress (kPa)Axial strain (%)Test DDEM_k7e6DEM_k5e6DEM_k3e6DEM_k1e6DEM_k7e5051015202530350 0.2 0.4 0.6 0.8 1 1.2Side wall friction (kPa)Axial strain (%)Test DDEM_k7e6DEM_k5e6DEM_k3e6DEM_k1e6DEM_k7e5c) a) b) 84  Figure 5-1(c) shows the notable effect of particle stiffness on the observed side-wall friction. Similar to the effect on observed axial stress responses, an increase in particle stiffness corresponds to increases in observed side-wall friction.  Whilst the DEM results on side-wall friction do not mimic the decidedly non-linear shape of the experimental results in Figure 5-1(c), the computed value of side-wall friction of approximately 2% of axial stress is in general agreement with the range of observed experimental side-wall friction value of about 6% for the 9-mm diameter glass beads.   As seen from Figures 5-1(a) and (b), comparison of the axial stress-strain responses between the numerical simulation and experimental results allow for determination of the most suitable stiffness input parameter; for example, the numerical result with particle stiffness microparameters of k = 7 x 106 N/m seemed to provide an average gradient, or “average” constrained modulus that closely resembles that arising from experimental results (for both top and bottom axial stress strain plots).  It is noted that, as referenced in Section 3.4, the comparison is made at the second portion of stress-strain response as described by Lambe and Whitman (1969), where particle contacts become stable after particle rearrangement, and deformation of the individual glass beads begins to define the stress-strain response.  However, the linear contact model is unable to capture the non-linear load mobilization response seen in the experimental results, which as described by Lambe and Whitman (1969), occurs initially when strain is governed mainly by particle rearrangement. With this shortcoming in mind, the following sections will compare the results obtained through the linear contact model with the Hertz-Mindlin contact model in Section 5.1.3. The ”average” constrained modulus of the DEM result (as estimated from top axial stress) was estimated to be 121 MPa, whilst the “average” constrained modulus from the experimental result was estimated to be 126 MPa (showing agreement within 4% of each other). For clarity of comparison, the final calibrated DEM simulation is presented in Figure 5-2 with the results from five laboratory experimental tests superimposed, to emphasize the suitability of the selected material parameters in the simulation of 9-mm glass beads under one-dimensional compression.  The computed top and bottom load responses are presented in the Figures 5-2(a) and 5-2(b), and with the side-wall friction presented in Figure 5-2(c). All final calibrated DEM input parameters are summarized in Table 5-2. 85     Figure 5-2: Comparison of one-dimensional compression DEM calibration results for 9-mm diameter glass beads with experimental results, for the linear contact model, showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction  020040060080010001200140016000 0.2 0.4 0.6 0.8 1 1.2Top axial stress (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_FDEM_k7e602004006008001000120014000 0.2 0.4 0.6 0.8 1 1.2Bottom axial stress (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_FDEM_k7e6051015202530350 0.2 0.4 0.6 0.8 1 1.2Side wall friction (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_FDEM_k7e6a) b) c) 86  Table 5-2: Final calibrated DEM input parameters for one-dimensional compression testing of 9-mm diameter beads, using the linear contact model Parameter Glass beads Steel walls Stiffness, K1 7 x 106 N/m 1 x 108 N/m Friction coefficient, μ 0.176 0.05 1calibrated parameter  The gradients of the experimental curves are in good agreement with the numerical analysis result, confirming that the use of a particle stiffness input parameter of 7 x 106 N/m of 9-mm diameter glass beads in DEM is reasonable to capture the overall stress-strain response of the 9-mm diameter glass beads under one-dimensional compression. 5.1.2 Hertz-Mindlin contact model: microparameter calibration of 9-mm diameter glass beads This section presents the calibration process for the Hertz-Mindlin contact model, for selection of DEM input parameters for 9-mm diameter glass beads. The Hertz-Mindlin contact model is defined by the shear modulus, Poisson’s ratio and surface friction coefficient of the particles as described in Section 4.1.2. For the glass beads used in the laboratory testing of the present thesis, an accurately measured value of the shear modulus was not available; therefore, two reference values used by previous researchers for DEM studies on glass beads, under one-dimensional compression (Chung and Ooi 2006, Dabeet 2014) were selected for simulations, and a third suitable shear modulus parameter estimated from observed results. Details of the selected shear modulus input parameters are summarized in Table 5-3. The computed axial stress-strain response from the DEM simulation of 9-mm diameter glass beads subjected to one-dimensional compression is presented in Figure 5-3. The computed stress-strain values of the 9-mm diameter glass bead simulations, from load measurements of the top and bottom plates, are presented in the Figures 5-3(a) and 5-3(b), respectively. Side wall friction is presented in Figure 5-3(c). For clarity of comparison, the five DEM simulations are presented with the results from only one experiment (Test D) on 9-mm diameter glass beads. The “average” constrained moduli were based on the predictions for axial stress 87  development at the top cap level (i.e. from Figure 5-3(a)) and presented in Table 5-3. All average constrained moduli reported in this section were estimated considering the stress variations within the range of 0.6% – 1.2% axial strain.   Table 5-3: Microparameter inputs and “average” constrained modulus results for DEM simulations with the Hertz-Mindlin contact model and 9-mm diameter glass beads Identification (Legend Label) Particle shear modulus, G Particle friction coefficient, μ “Average” constrained modulus, Mt DEM_G16.8 16.8 GPa 1 0.176 336 MPa DEM_G6.5 6.5 GPa 0.176 139 MPa DEM_G1.5 1.5 GPa 2 0.176 37 MPa  1 shear modulus used by Chung and Ooi (2006) for 10-mm diameter glass beads 2 shear modulus used by Dabeet (2014) for 2-mm diameter glass beads  Similar to the observations in Section 5.1.1 with the linear contact model, an increase in numerical particle shear modulus corresponds to an increase in the overall stiffness response of the material matrix under one-dimensional compression, as shown in Figures 5-3(a) and (b). Both the specimen top and bottom level axial stress responses show an increase in “average” constrained modulus with increasing particle stiffness, as summarized in Table 5-3. Figure 5-3(c) shows that an increase in particle stiffness also corresponds to increases in observed side-wall friction – similar to those observed with the linear contact model. As seen from Figures 5-3(a) and (b), comparison of the axial stress-strain responses determined the most suitable particle shear modulus input parameter to be 6.5 GPa. The specimen top and bottom level axial stresses show a good agreement between the DEM and experimental results with regard to the estimated “average” constrained modulus values of both tests. The “average” constrained modulus of the DEM results (from the axial stress at the top boundary of the specimen) was estimated to be 139 MPa, whilst the constrained modulus from the experimental result was estimated to be 126 MPa (agreement within 10% of each other).  88     Figure 5-3: Comparison of one-dimensional compression DEM and experimental results (Test D) for 9-mm diameter glass beads, with the Hertz-Mindlin contact model, for varying particle stiffnesses showing a) top axial stress, b) bottom axial stress and c) side wall friction (refer to Table 5-3 for legend labels)  050010001500200025000 0.2 0.4 0.6 0.8 1 1.2Top axial stress (kPa)Axial strain (%)Test DDEM_G16.8DEM_G6.5DEM_G1.5050010001500200025000 0.2 0.4 0.6 0.8 1 1.2Bottom axial stress (kPa)Axial strain (%)Test DDEM_G16.8DEM_G6.5DEM_G1.5051015202530354045500 0.2 0.4 0.6 0.8 1 1.2Side wall friction (kPa)Axial strain (%)Test DDEM_G16.8DEM_G6.5DEM_G1.5a) b) c) 89  The computed results for the side-wall friction presented in Figure 5-3(c) also show good agreement between DEM and experimental results, and markedly so, compared to the linear contact model. Similar to the axial stress-strain response, the Hertz-Mindlin model is able to capture the non-linear side wall friction response of 9-mm diameter glass beads under one-dimensional compression, which the linear contact model was unable to accomplish. For clarity of comparison, the results from the final calibrated DEM simulation for the 9-mm diameter glass beads, using the Hertz-Mindlin contact model, is superimposed with the results from the five laboratory experimental tests with the same size glass beads in Figure 5-4.  The results from computations for the specimen top and bottom levels, are presented in the Figures 5-4(a) and 5-4(b), respectively. The side-wall friction is presented in Figure 5-4(c). All final calibrated DEM input parameters for the Hertz Mindlin contact model are summarized in Table 5-4. These findings emphasize the suitability of the selected material parameters in the simulation of 9-mm glass beads under one-dimensional compression.  A particle shear modulus input parameter of 6.5 GPa was determined to be the most suitable value for simulating one-dimensional compression testing of 9-mm diameter glass beads.   Table 5-4: Final calibrated parameters for one-dimensional compression testing of 9-mm diameter glass beads, using the Hertz-Mindlin contact model Parameter Glass beads Steel walls Friction coefficient, μ 0.176 0.05 Shear modulus, G1 6.5 GPa 77 GPa Poisson’s ratio, v 0.22 0.3 1calibrated parameter  90     Figure 5-4: Comparison of one-dimensional compression DEM calibration results for 9-mm diameter glass beads with experimental results, for the Hertz-Mindlin contact model, showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction  010020030040050060070080090010000 0.2 0.4 0.6 0.8 1 1.2Top axial stress (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_FDEM_G6.5010020030040050060070080090010000 0.2 0.4 0.6 0.8 1 1.2Bottom axial stress (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_FDEM_G6.5051015202530350 0.2 0.4 0.6 0.8 1 1.2Side wall friction (kPa)Axial strain (%)Glass_9mm_AGlass_9mm_CGlass_9mm_DGlass_9mm_EGlass_9mm_FDEM_G6.5a) b) c) 91  As can be noted in Figures 5-4(a), (b) and (c), the shape of the stress-strain responses and side-wall friction obtained from the DEM simulation with Hertz-Mindlin model (using shear modulus of 6.5 GPa as the input parameter) closely agree with the corresponding experimental lab results. The axial stress strain results from Figures 5-4(a) and (b) seem to show very good agreement between DEM and experimental results for both the initial non-linear stress-strain response due to particle rearrangement (i.e., 0 - 0.5% strain range). A good agreement may also be seen with the DEM and experimental side-wall friction results shown in Figure 5-4(c). The above observations are in line with the previous studies by Dabeet (2014), who also noted that the Hertz-Mindlin model better captured the observed laboratory oedometer response (in comparison to that from the linear contact model).  5.2  Model Calibration with 1-mm Diameter Glass Beads The DEM simulation of 1-mm glass beads under one-dimensional compression was accomplished using the linear contact model. As detailed in Chapter 4, a specimen of height equal to H/4 (of the experimental specimen height) was used for all simulations of 1-mm diameter glass beads under one-dimensional compression.  Section 5.2.1 presents findings on the significance of particle rolling resistance, which became an apparent consideration during initial one-dimensional compression simulations of the 1-mm diameter specimens of glass beads. Similar to Section 5.1, DEM results from varying particle stiffness parameters are then presented in Section 5.2.2, to show the selection process of the most suitable particle stiffness input parameter for representing bulk material behavior under one-dimensional compression. Final calibrated DEM one-dimensional compression results are also summarized in Section 5.2.2, and compared with experimental results. 5.2.1  Significance of particle rolling resistance Significant particle spinning/rolling during the particle re-arrangement was noted in the initial simulations of 1-mm glass beads subjected to one-dimensional compression. The energy loss associated with this particle rolling significantly 92  affected the axial load mobilization of the 1-mm diameter glass bead specimen; this was not a consideration for the 9-mm diameter glass bead specimen.  As reviewed in Chapter 2, few DEM studies have been made on the effect of particle size on bulk material behavior, for monodisperse spherical mixtures. The work by Zhou et al. (2001) can be considered as one of the cases, where a combined experimental and DEM study of the angle of repose was done with spherical glass beads. The authors concluded that for coarse spheres, particle size affects the angle of repose mainly through its effect on rolling friction.  Figure 5-5 presents the significant effect of particle size on the average rolling friction torque of glass beads as reported by Zhou et al. (2001). It can be seen that the rolling friction torque decreases considerably with increases in particle size – rolling friction torque of 5-mm diameter particles is approximately 270 times that of 20-mm diameter particles. With reference to Figure 5-5, for the chosen bead diameters of 1 mm and 9 mm in this thesis, average rolling friction torque for 1-mm diameter glass beads may be estimated to be approximately ten times larger than average rolling friction torque for 9-mm diameter glass beads.   Figure 5-5: Effect of particle size on the dimensionless average rolling friction torque, obtained from sandpile tests with glass beads, reproduced from Zhou et al. (2001)  1001000100001000000 5 10 15 20 25Dimensionless average rolling friction torqueParticle size (mm)t = 1 st = 11 s (stable sandpile)93  The outcomes from the present DEM calibration process were in agreement with Zhou et al. (2001). It became evident that the energy loss due to particle spinning and rearrangement was significant, as it was very difficult to obtain a significant seating load before initializing the one-dimensional compression testing phase with 1-mm particles. Lowering the top cap resulted in negligible loads on the top cap, due to excessive particle rearrangement of the material, which were attributed to the higher rolling friction torques acting on the smaller 1-mm particles. This difficulty in obtaining an acceptable seating load culminated in the decision to restrict particle rotation during the compression stages of the 1-mm diameter glass bead simulations - it should be noted that particle rotation was not restricted during the particle generation and stabilization stages. Previous DEM studies have also restricted particle rotation, such as Dabeet (2014), who simulated oedometer and DSS tests with 2-mm diameter glass beads. Therefore, all simulations herein of 1-mm diameter glass beads under one-dimensional compression were conducted with restricted particle rotation. 5.2.2 Linear contact model: microparameter calibration of 1-mm diameter glass beads Five different particle stiffness input values were used to investigate the effect of particle stiffness on bulk material response, as summarized in Table 5-5. The computed stress-strain values of the 1-mm diameter glass bead simulations, from the load measurements at the top and bottom levels of the specimen, are presented in the Figures 5-6(a) and 5-6(b), respectively. Side-wall friction is presented in Figure 5-6(c). It should be noted that though the laboratory experiments were conducted up to 1.6% axial strain, due to computational limitations, the DEM simulations were conducted only up to a range of 0.4% to 1% axial strain. The outcomes from the five DEM simulations are compared with the results from one of the one-dimensional compression tests conducted on 1-mm diameter glass beads. Selection of the most suitable DEM stiffness input parameter was made in relation to the successful simulation of experimental test results (represented by Test D in the following figures). “Average” constrained moduli estimated from computed axial 94  stress-strain response value at the top level of the specimen in Figure 5-6(a) are presented in Table 5-5. Most of the “average” constrained moduli were estimated considering the data within the range of 0.3% – 0.7% axial strain.  Table 5-5: Microparameter inputs and “average” constrained modulus estimates for DEM particle stiffness calibration with the linear contact model – 1 mm diameter glass beads Identification (Legend Label) Particle stiffness, k Particle friction coefficient, μ “Average” constrained modulus, Mt DEM_k2e6 2 x 106 N/m 0.176 233 MPa DEM_k8e5 8 x 105 N/m 0.176 115 MPa DEM_k6e5 6 x 105 N/m 0.176 91 MPa DEM_k5e5 5 x 105 N/m 0.176 81 MPa DEM_k2e5 2 x 105 N/m 0.176 32 MPa *All simulations used glass bead properties of ρ = 2500 kg/m3   Similar to the results obtained in Section 5.1.1 for 9-mm diameter glass beads, it may be observed from Figures 5-6(a) and (b) that increasing particle stiffness results in an increase in the overall stiffness response of the material under one-dimensional compression. Both the responses computed at the top and bottom levels of the specimen show a decrease in “average” constrained modulus with decreasing particle stiffness. Figure 5-6(c) shows the notable effect of particle stiffness on the observed side-wall friction. Similar to the effect on observed axial stress responses, an increase in particle stiffness corresponds to increases in observed side-wall friction.  95     Figure 5-6: Comparison of one-dimensional compression DEM and experimental results (Test D) for 1-mm diameter glass beads, with the linear contact model, for varying particle stiffnesses showing a) top axial stress, b) bottom axial stress and c) side wall friction (refer to Table 5-5 for legend labels)  0200400600800100012000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Top axial stress (kPa)Axial strain (%)Test DDEM_k2e6DEM_k8e5DEM_k6e5DEM_k5e5DEM_k2e50200400600800100012000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Bottom axial stress (kPa)Axial strain (%)Test DDEM_k2e6DEM_k8e5DEM_k6e5DEM_k5e5DEM_k2e5051015202530354045500 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Side wall friction (kPa)Axial strain (%)Test DDEM_k2e6DEM_k8e5DEM_k6e5DEM_k5e5DEM_k2e5a) b) c) 96  As seen from Figures 5-6(a) and (b), comparison of the axial stress-strain responses determined the most suitable particle stiffness input parameter to be 5 x 105 N/m. The responses computed at the top and bottom levels of the specimen show a very good agreement between the DEM and experimental results at this particle stiffness value. The “average” constrained modulus of the DEM result (estimated from top axial stress) was estimated to be 81 MPa, whilst the “average” constrained modulus from the experimental result was estimated to be 80 MPa (agreement within about 1% of each of other).   For clarity of comparison, the final calibrated DEM simulation is presented with the data from five laboratory experimental tests in Figures 5-7(a), (b) and (c), to emphasize the suitability of the selected material parameters in the simulation of 1-mm glass beads under one-dimensional compression.  The stress-strain results computed at the top and bottom levels of the specimen are presented in the Figures 5-7(a) and 5-7(b), respectively. Side wall friction is presented in Figure 5-7(c).  It is clear that the chosen particle stiffness input parameter of 5 x 105 N/m is quite suitable to match with the gradient of the response in simulating one-dimensional compression testing of 1-mm diameter glass beads. All final calibrated DEM input parameters are summarized in Table 5-6.   Table 5-6: Final calibrated DEM input parameters for one-dimensional compression testing of 1-mm diameter beads, using the linear contact model Parameter Glass beads Steel walls Stiffness, k1 5 x 105 N/m 1 x 108 N/m Friction coefficient, μ 0.176 0.05 1calibrated parameter  97     Figure 5-7: Comparison of one-dimensional compression DEM calibration results for 1-mm diameter glass beads with experimental results, for the linear contact model, showing a) top axial stress-strain response, b) bottom axial stress-strain response and c) side wall friction  0200400600800100012000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Top axial stress (kPa)Axial strain (%)DEM_k5e5Glass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_E0200400600800100012000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Bottom axial stress (kPa)Axial strain (%)DEM_k5e5Glass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_E0102030405060700 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Side wall friction (kPa)Axial strain (%)DEM_k5e5Glass_1mm_AGlass_1mm_BGlass_1mm_CGlass_1mm_DGlass_1mm_Ea) b) c) 98  In contrast to the 9-mm diameter glass bead simulation, the 1-mm diameter glass beads simulation with the linear contact model seems to be able to partially capture the non-linear load mobilization response seen in the experimental results, when strain is governed mainly by particle rearrangement. This may be attributed to higher quantities of particle rearrangement in the 1-mm diameter glass bead specimen under one-dimensional compression – it seems reasonable to conclude that a much higher number of simulated particles (with over 100 times more particles than the 9-mm diameter glass beads specimen) results in much more particle rearrangement during the simulation. However, DEM simulation stress-strain responses displayed differences in load mobilization response to the corresponding experimental results, with load mobilization (and therefore particle deformation) observed earlier in the DEM specimen. This is in agreement with results observed in Chung and Ooi (2006), who noted a significantly different loading response at low confining pressures for DEM response of 10-mm glass beads under one-dimensional compression, as compared with experimental results.  5.3  Evaluation of DEM Results The main objective of the DEM study is to provide particle stiffness microparameters for different-sized particles, and thus, investigate the effect of particle size on the behavior of glass beads under one-dimensional compression. This section presents a further evaluation of the effect of particle size on calibrated particle stiffness parameters using the linear contact model for 1-mm and 9-mm diameter particles. Section 5.3.1 investigates the relationship between calibrated DEM stiffness microparameters and the resulting estimated “average” constrained modulus, and Section 5.3.2 comments on further differences noticed between 1-mm and 9-mm diameter glass bead simulations. 99  5.3.1  Effect of particle size on particle stiffness microparameters The final selected DEM particle stiffness input parameters derived for the linear contact models, for 1-mm and 9-mm glass beads subjected to one-dimensional compression testing, are presented in Table 5-7.  The “average” constrained modulus obtained are also presented in the same table. It can be seen that the 9-mm diameter glass beads have a stiffer value than the 1-mm diameter particles (7 x 106 N/m and 5 x 105 N/m, respectively). The results suggest that particle stiffness microparameters increase with particle size, for homogeneous, monodisperse mixtures of spherical particles under one-dimensional compression. For 1-mm and 9-mm diameter glass beads, particle stiffness increased by approximately a factor of 14.  It was deemed prudent to assess the relationship between calibrated particle stiffness microparameters and the corresponding “average” constrained moduli estimated from simulated stress-strain responses (detailed in Sections 5.1 and 5.2). In addition to the two sets of results from the 1-mm and 9-mm diameter glass beads, it could be argued that particles with zero stiffness would have an “average” constrained modulus of zero – in turn, providing a third point for assessing the relationship.   Table 5-7: Comparison of PFC3D results for the linear contact model, showing particle stiffness, k and “average” constrained modulus, Mt results from this thesis DEM Simulation k (N/m) Mt (MPa) 1-mm diameter glass beads 5 x 105 81 9-mm diameter glass beads 7 x 106 121  When these three data points for one-dimensional compression simulations of glass beads were plotted, as shown in Figure 5-8, it became apparent that the use of a hyperbolic function in the form shown in Equation 8 may be suitable to describe the relationship between the “average” constrained modulus and particle stiffness. 100  𝑦 =𝑥|𝑥𝑎|+1𝑏                         [8] Curve-fitting of available particle stiffness and “average” constrained modulus data to Equation 8 results in the coefficients “a” and “b” as follows:  1𝑎 =  0.008 𝑎𝑛𝑑 1𝑏 =  2197.6  Substituting these coefficients back into Equation 8, would lead to Equation 9 to represent the relationship between “average” constrained modulus, Mt and calibrated particle stiffness, k. This curve developed using Equation 9 is superimposed on DEM results in Figure 5-8.  As expected from the curve-fitting, the curve matches well with the data points.  𝑀𝑡 =𝑘|𝑘0.008|+2197.6                 [9] Based on the presentation in Figure 5-8, it appears that the particle stiffness (k) for a given material is less sensitive to particle diameter (d) when particle size is smaller (say d < 1 mm) compared to the when d > 1 mm.  Although this is based on limited data, it is important to note that the results are arising from careful experimentation combined with DEM analysis.  Data from further laboratory testing combined with DEM simulations undertaken on glass beads of different diameters (particularly of diameters less than 9 mm) would add significant value to further confirming this inference.  101   Figure 5-8: Relationship between “average” constrained modulus and particle stiffness for one-dimensional compression simulations of glass beads, with superimposed hyperbolic curve, labelled the “hyperbolic function”.    5.3.2 Further comments on the effect of particle size The effect of particle size on particle rolling resistance was also prominently noted during DEM simulations.  Observations during the DEM calibration process were in agreement with Zhou et al. (2001), where he noted increasing average rolling friction torque with decreasing particle size. Excessive particle rearrangement and material compressibility, attributed to the higher rolling friction torques acting on the smaller 1-mm particles, were encountered during initial DEM simulations. Therefore 1-mm diameter glass beads were simulated with restricted particle rotation, similar to past DEM studies of oedometer testing of glass beads by Dabeet (2014). The effects of the selected contact model on the simulated stress-strain responses of different-sized glass beads under one-dimensional compression was also noted in this study. Using the linear contact model, both 9-mm and 1-mm diameter glass bead DEM specimens successfully simulated the stress-strain response observed in the experiments - and as defined by Lambe and Whitman (1969), where particle contacts 0204060801001201400E+00 1E+06 2E+06 3E+06 4E+06 5E+06 6E+06 7E+06 8E+06"Average" constrained modulus, Mt(MPa)Particle stiffness, k (N/m)PFC calibrated resultsHyperbolic function9-mm diameter glass beads 1-mm diameter glass beads 102  become stable and particle locking and deformation of the individual glass beads begin. Good agreement was achieved between the numerical model and the experiments with estimated “average” constrained modulus values of both tests, as summarized in Section 5.1.1 and Section 5.2.2.  However, some difficulties were encountered in the use of the linear contact model when simulating the non-linear load mobilization response seen in the experimental results – i.e., when the strain seems to be influenced by particle rearrangement. For example, the linear contact model was unable to capture this non-linear stress strain response for the case with the 9-mm diameter glass beads. This suggests that there was very little particle rearrangement obtained in the DEM simulation of 9-mm diameter glass beads. In contrast, some non-linear stress-strain response is seen in the 1-mm diameter glass beads specimen, indicating the occurrence of some particle rearrangement during initial stages of specimen compression. Whilst the load mobilization occurred earlier in the DEM simulation of the specimen of 1-mm diameter glass beads compared to that for the experimental specimen (i.e. faster particle rearrangement and particle locking occurred in the numerical simulation), the DEM results still showed a clear region of non-linear stress-strain response. The above observations suggest that that particle size of a monodisperse granular material can have a significant effect on the bulk material behavior, in turn, on the selection of suitable DEM microparameters for successful simulation of observed bulk responses.   5.4  Summary The DEM models simulating one-dimensional compression response of glass beads were calibrated with respect to results from carefully conducted experiments conducted on 9-mm and 1-mm diameter glass beads using the same loading mode.  The calibrations allowed to identify suitable particle stiffness microparameters for the DEM simulation of granular materials in engineering applications.  The following provides a summary of the key findings from this work. 103  5.4.1  DEM simulations of 9-mm diameter glass beads 1) Increase in particle stiffness (for the linear contact model) resulted in increases in “average” constrained modulus for the particle matrix (from the axial stress-strain response).  This also resulted in increases in side wall friction. 2) A particle stiffness value of k = 7 x 106 N/m was obtained for the 9-mm glass beads from calibrations using the linear contact model; similarly, a particle shear modulus G = 6.5 GPa was obtained when calibrations were performed using the Hertz-Mindlin contact model. 3) The linear contact model was able to satisfactorily capture the relatively linear stress-strain response of 9-mm diameter glass beads under one-dimensional compression, with good agreement between calculated DEM and experimental “average” constrained modulus values. 4) The linear contact model was not able to capture the initial non-linear stress-strain response of 9-mm diameter glass beads under one-dimensional compression, during when strain is mainly governed by particle rearrangement. The load mobilization response initiated almost immediately after the commencement of specimen compression. 5) The Hertz-Mindlin contact model was able to capture both the initial non-linear stress-strain response and the relatively linear stress-strain response of 9-mm diameter glass beads under one-dimensional compression. Therefore, the Hertz-Mindlin contact model results had a better agreement with experimental results, as compared with the linear contact model, which is in agreement with previous observations from Dabeet (2014). 5.4.2   DEM simulations of 1-mm diameter glass beads 1) All simulations of 1-mm diameter glass beads under one-dimensional compression were conducted with restricted particle rotation, due to the increased average rolling friction torque for the smaller glass beads, as discussed by Zhou et al. (2001). 104  2) Increase in particle stiffness (for the linear contact model) resulted in increases in “average” constrained modulus (from the axial stress-strain response) and increases in side wall friction. 3) A calibrated particle stiffness value of k = 5 x 105 N/m was obtained for the 9-mm glass beads with the linear contact model. 4) The linear contact model was able to fittingly capture the latter relatively linear stress-strain response of 1-mm diameter glass beads under one-dimensional compression, with good agreement between DEM and experimental “average” constrained modulus estimations. 5) The linear contact model was able to only partially capture the initial non-linear stress-strain response of 1-mm diameter glass beads under one-dimensional compression, when strain is mainly governed by particle rearrangement. DEM simulation stress-strain responses did differ to those observed in the experimental results during initial stages of compression, with load mobilization (and therefore a relatively linear stress-strain response) observed earlier in the DEM specimen. 5.4.3  Effect of particle size on DEM simulations with the linear contact model 1) Results indicate that particle stiffness microparameters increase with particle size, for homogeneous, monodisperse mixtures of spherical particles under one-dimensional compression. DEM results yields a particle stiffness of 7 x 106 N/m for the 9-mm diameter glass beads – which is about 14 times stiffer than that obtained for the 1-mm diameter particles. 2) Observations during the DEM calibration process were in agreement with Zhou et al. (2001), with increasing average rolling friction torque with decreasing particle size. This was evident through excessive particle rearrangement and difficulties in obtaining seating loads during initial simulations of 1-mm diameter glass beads under one-dimensional compression. 3) A preliminary relationship between “average” constrained modulus and particle stiffness depicted by a hyperbolic curve was suggested based on one-dimensional 105  compression simulations of glass beads of different sizes. Preliminary curve-fitting proved to be successful, however only three data points were available. Additional research needs to be undertaken to investigate this relationship further. 4) In an overall sense, the findings suggest that particle stiffness microparameters should be carefully selected to represent particle size effects in DEM simulations and, in turn, be utilized in quantitative analysis of geotechnical engineering problems.   106  6 Summary and Conclusions A research program was undertaken to study the effect of particle size on micro-parameters required for discrete element modeling (DEM) of granular soils, with particular emphasis on supporting the study of soil-pipe interaction of buried pipelines, and the effect of trench backfill particle size. The results of this thesis will supplement and support the initial studies by Wijewickreme et al. (2014) on the performance of pipelines buried in coarse-grained soils when subjected to lateral ground movements, with particular references to the DEM simulations by Dilrukshi and Wijewickreme (2017) on the significant effects of backfill particle size on the soil restraints on buried pipelines. Laboratory one-dimensional compression tests were conducted on 1-mm and 9-mm diameter glass beads and crushed granite to investigate the effect of particle size on the mechanical response of granular materials. Series of laboratory one-dimensional compression tests were completed for each material type and particle size, and experimental results were assessed in terms of axial-stress strain response, while accounting for the development side wall friction. By calibrating the experimental results obtained from one-dimensional compression tests of glass beads with a 3D DEM simulation of an equivalent numerical specimen, the most suitable DEM particle stiffness microparameters for 1-mm and 9-mm diameter glass beads were identified. Experimental observations and the calibrated DEM microparameters were used to investigate the effect of particle size on the stiffness of the overall granular material matrix, as well as the relationship of particle size to calibrated DEM particle stiffness microparameters. This chapter presents the conclusions arising from the experimental and DEM investigations of this thesis, followed by recommendations for future studies on this subject.  107  6.1 Laboratory One-Dimensional Compression Testing of Glass Beads and Crushed Granite  The results from one-dimensional compression tests indicate that for both glass beads and crushed granite, 9-mm diameter particles have a noticeably stiffer stress-strain response when compared with that for 1-mm diameter particles of the same material type. These results are in accord with previous observations that suggest that an increase in particle size will increase the overall stiffness of the granular material matrix.  Tests conducted with both materials also indicate that 1-mm diameter particles resulted in higher normalized side wall friction compared to the 9-mm diameter particles of the same material type. The increased number of particle-wall contacts in specimens with smaller size particles seem to result in higher friction forces between the material and wall surface.  The dependence of constrained modulus (Mt) on effective confining stress determined experimentally from the tests are in general agreement with the commonly used equation 𝑀𝑡 = 𝑘𝑚𝑃𝑎 (𝜎𝑚′𝑃𝑎)0.5. In particular, for 1-mm diameter glass beads and 1-mm crushed granite, the relationship seems to be suitable up to axial stresses of 500 kPa. For 9-mm diameter glass beads and 9-mm crushed granite, the equation seems to provide a good fit up to slightly higher axial stresses of 600 kPa.   For both materials, experimental constrained modulus values appear to reach a generally constant value after the abovementioned axial stress levels of 500 kPa (for 1-mm sized particles) and 600 kPa (for 9-mm sized particles).  Calculated values of constrained and Young’s modulus coefficients kM and kE (Table 3-4) also agree with the observations of Byrne and Eldridge (1982), where angular particles (i.e. crushed granite) exhibited lower kE values than rounded particles (i.e. glass beads). 108  6.2  DEM Simulation and Calibration of One-Dimensional Compression Testing of Glass Beads  The results reaffirm that DEM simulations of one-dimensional compression tests can be successfully used to calibrate DEM particle stiffness microparameters.  The stress strain responses observed from one-dimensional compression tests on both 9-mm and 1-mm diameter glass bead specimens exhibit a relatively linear relationship - after an initial non-linear response - attributed by Lambe and Whitman (1969) to initial particle rearrangement during initial stages of specimen compression followed by particle contacts becoming stable, as noted based on previous observations of one-dimensional compression stress-strain responses.   The DEM simulation conducted with the linear contact model is able to capture the non-linear stress-strain response observed in the specimens of 1-mm diameter glass beads due to the particle rearrangement that seemingly take place during initial stages of specimen compression. However, for the specimens of 9-mm diameter glass beads, the linear contact model is unable to capture the initial non-linear stress strain response that seem to arise due to particle rearrangement.   In contrast, DEM simulations conducted with the Hertz-Mindlin contact model are able to capture both the initial non-linear stress-strain response and the relatively linear stress-strain response of 9-mm diameter glass beads under one-dimensional compression. Therefore, the Hertz-Mindlin contact model results have a better agreement with experimental results, in comparison to those from the linear contact model; this observation is in agreement with similar previous observations by Dabeet (2014).   DEM results indicate that particle stiffness microparameters increase with increasing particle size, for homogeneous, same-sized spherical particles under one-dimensional compression. DEM results show that 9-mm diameter glass beads have a stiffer k value (7 x 106 N/m) than the 1-mm diameter particles (5 x 105 N/m), i.e., the most suitable particle stiffness microparameter for 9-mm diameter particles is approximately 14 times stiffer than the particle stiffness microparameter for 1-mm diameter particles. 109   Analysis of numerical results indicate that a preliminary relationship between “average” constrained modulus and particle stiffness can be depicted by a hyperbolic relationship.  In agreement with the experimental results, DEM results also indicate that an increase in particle size results in increased stiffness of the overall granular material matrix under one-dimensional compression.  Observations during the DEM calibration process are also in agreement with Zhou et al. (2001), where the average rolling friction torque seems to increase with decreasing particle size. This is evident through excessive particle rearrangement and material compressibility encountered during initial simulations of 1-mm diameter glass beads under one-dimensional compression.  6.3  Recommendations for Future Research In addition to the research findings presented above, this thesis has also led to the identification of a number suggestions for future research as given below:  The current study investigated the effect of particle size for 1-mm and 9-mm diameter particles only (for glass beads and crushed granite, respectively). It would be useful to extend the study to include a wide range of particle diameter sizes representative of coarse backfill materials – say, up to approximately 20-mm in size. During experimentation, the allowable particle sizes were restricted by the size of the one-dimensional compression chamber, and therefore a larger experimental setup would be required to test larger particles.  The present DEM simulation and calibration was restricted to granular packings of monodisperse and spherical particles. Whilst laboratory one-dimensional compression tests were conducted on crushed granite to represent particle size effects on “real-life” backfill materials, further work into the effects of particle shape and particle size distributions are needed for quantitative application of the DEM results to “real-life” pipeline backfill applications. 110   Due to time constraints for this thesis, only two different particle sizes were studied using DEM. Preliminary curve-fitting of calibrated particle stiffness and “average” constrained modulus proved to be successful, however only three data points were available for curve-fitting purposes. Additional research needs to be undertaken to investigate this relationship further. It is recommended that additional one-dimensional compression testing be conducted with different sizes of glass beads for further improvement and validation of this suggested hyperbolic relation.  DEM simulation results indicated the significant effect of particle size on the rolling friction torque experienced by glass beads under one-dimensional compression. Therefore, in this thesis, 1-mm diameter glass beads were simulated with restricted particle rotation. A numerical study of the sensitivity of overall material behavior to selected inputs of particle rolling resistance may be prudent, in order to select the most suitable rolling resistance parameter for a specific particle size, instead of restricting particle rotation altogether. 111  References American Lifeline Alliance (2001) Guidelines for the design of buried steel pipe, <http://www.americanlifelinesalliance.org/pdf/Update061305.pdf>. 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