Numerical study of the arching effect using the discrete element method

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(1)PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE SCHOOL OF ENGINEERING. NUMERICAL STUDY OF THE ARCHING EFFECT USING THE DISCRETE ELEMENT METHOD. ANTONIO FELIPE SALAZAR VÁSQUEZ. Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the degree of Master of Science in Engineering. Advisor: ESTEBAN PATRICIO SÁEZ ROBERT. Santiago de Chile, October 2014 c MMXIV, A NTONIO F ELIPE S ALAZAR V ÁSQUEZ.

(2) PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE SCHOOL OF ENGINEERING. NUMERICAL STUDY OF THE ARCHING EFFECT USING THE DISCRETE ELEMENT METHOD. ANTONIO FELIPE SALAZAR VÁSQUEZ. Members of the Committee: ESTEBAN PATRICIO SÁEZ ROBERT CARLOS OVALLE ORTEGA CLAUDIA MEDINA DÍAZ IGNACIO LIRA CANGUILHEM Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the degree of Master of Science in Engineering. Santiago de Chile, October 2014 c MMXIV, A NTONIO F ELIPE S ALAZAR V ÁSQUEZ.

(3) Gratefully to my parents and siblings.

(4) ACKNOWLEDGEMENTS. To the Fondo Nacional de Desarrollo Cientı́fico y Tecnológico FONDECYT for making this project possible, through grant number 11100157. I would like to express my gratitude to my advisor Esteban Sáez, for the time spent, advise given, and lessons learned, not only referring to my thesis project, but in my whole academic career as a student and assistant. Thank you for helping me strengthen my knowledge in the scientific career that I chose. To my parents, Jorge and Leticia, thank you for supporting and guiding me always. Thank you for all the effort you have put into my education. I hope I have accomplished all that you wished for me. To all my fellow officemates, especially to Alix, Mathias, Toripe, Chubi, Daniel and Tatita. Thank you for creating a dynamic work environment with all of the jokes and stories, but more than that, for the advice and help. Undoubtedly, this process would have been much harder without you. Finally, I would like to thank all the people that were involved, directly or indirectly, during the entire thesis process and that I cannot fit in these lines, especially to my school friends with whom I have shared countless good times. Thank you all.. iv.

(5) TABLE OF CONTENTS. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. iv. LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vii. LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ix. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. x. Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xi. Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 2.. Objectives and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. Chapter 2. Discrete Element Method (DEM) overview . . . . . . . . . . . . . . . . . .. 4. 1.. Normal contact law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.. Tangential contact law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 3.. Time step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 4.. Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 5.. LIGGGHTS code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 6.. Other discontinuous methodologies . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. Chapter 3. Material modeling calibration . . . . . . . . . . . . . . . . . . . . . . . . .. 17. Modeling the direct shear test of a coarse sand using the 3D Discrete Element Method with a rolling friction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. Real sand and DEM model description . . . . . . . . . . . . . . . . . . . . . . . . .. 21. Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27. Calibrated model against experimental data . . . . . . . . . . . . . . . . . . . . . .. 30. Parallel gradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34 v.

(6) Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. Chapter 4. Trapdoor test simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. Numerical study of the arching effect using the Discrete Element Method . . . . . . . .. 42. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42. Tested Trapdoor description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. Discrete Element Method (DEM) Trapdoor model description and methodology . . .. 46. Trapdoor results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56. Chapter 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59. 1.. vi.

(7) LIST OF FIGURES. 1.1. Trapdoor stress distribution scheme . . . . . . . . . . . . . . . . . . . . . . . .. 1. 2.1. DEM computation steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.2. Soft sphere contact example . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.3. Contacts force comparison scheme . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.4. Forces and moments involved in a contact (Johnson & Johnson, 1987) . . . . .. 8. 2.5. Conducted drop test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 2.6. Damping models comparison with a simple drop test scheme . . . . . . . . . .. 12. 2.7. Paralellization process example of four-cores . . . . . . . . . . . . . . . . . . .. 13. 2.8. Benchmark of the direct shear problem . . . . . . . . . . . . . . . . . . . . . .. 15. 3.1. Direct shear test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. 3.2. Grain samples and grain size distribution . . . . . . . . . . . . . . . . . . . . .. 22. 3.3. Real contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 3.4. Mass and Particle number histograms . . . . . . . . . . . . . . . . . . . . . . .. 26. 3.5. Developed DEM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27. 3.6. Sensitivity analysis of the DEM parameters . . . . . . . . . . . . . . . . . . . .. 30. 3.7. Calibrated material response. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32. 3.8. Initial and final chain force . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32. 3.9. Polar distribution of the contact force at the beginning and the end of the shear test on the shear plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33. 3.10. In-plane velocity field (imposed displacement plane) . . . . . . . . . . . . . . .. 34. 3.11. Out-of-plane velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 3.12. Scaled material granulometry . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 3.13. Shear test results with two-times scaled material . . . . . . . . . . . . . . . . .. 37. 3.14. Shear test results with four times bigger material . . . . . . . . . . . . . . . . .. 38. 4.1. Tested Trapdoor scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44 vii.

(8) 4.2. Initial and deformed grid using DIC (Pardo, 2013) . . . . . . . . . . . . . . . .. 45. 4.3. Grain size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. 4.4. Mass and number histograms of the particles . . . . . . . . . . . . . . . . . . .. 47. 4.5. Direct shear results from the calibration process (Salazar, Sáez, & Pardo, 2014) .. 48. 4.6. Tested Trapdoor and DEM simulation . . . . . . . . . . . . . . . . . . . . . . .. 49. 4.7. After settlement stress on the bottom . . . . . . . . . . . . . . . . . . . . . . .. 50. 4.8. Displacement response of the DEM model . . . . . . . . . . . . . . . . . . . .. 51. 4.9. Displacement error of the DEM model . . . . . . . . . . . . . . . . . . . . . .. 52. 4.10. Principal stress rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53. 4.11. Stress field at stage 4 (δ = 2.4 mm) . . . . . . . . . . . . . . . . . . . . . . . .. 53. 4.12. Bottom base stress comparison . . . . . . . . . . . . . . . . . . . . . . . . . .. 54. 5.1. Multi-Sphere simulation model example (CFDEM project, 2014) . . . . . . . .. 58. viii.

(9) LIST OF TABLES. 3.1. Sand properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 3.2. Tested specimens properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 3.3. Calibrated material parameters . . . . . . . . . . . . . . . . . . . . . . . . . .. 31. 3.4. Computational cost comparison . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 4.1. Selected sand properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 4.2. Selected sand properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 4.3. Computational cost comparison . . . . . . . . . . . . . . . . . . . . . . . . . .. 49. 4.4. Error in the bottom stress (∆σch ) . . . . . . . . . . . . . . . . . . . . . . . . .. 55. ix.

(10) ABSTRACT. In the present study, the “arching effect” is reproduced by modeling the widely known Terzaghi’s Trapdoor test using the Discrete Element Method (DEM). The adopted modeling approach describes the soil as an assembly of particles, reproducing the macroscopic soil behavior from micro-mechanics interactions between individual particles. The purpose is to study the ability of this numerical method to reproduce this soil effect, consisting in a load transfer mechanism and shear strain concentrations that naturally occurs when parts of the support system starts yielding or moving, while the rest of the soil mass stays stationary. At previous stages of this research, a Trapdoor instrumented with load cells and tracked with the Digital Correlation (DIC) technique was tested in the laboratory. The bottom base stress and the displacement field obtained with the instrumentations is compared with the model results in order to study the model ability to reproduce the experimental data. The calibration of the DEM’s micro-mechanics parameters was performed from direct shear soil tests using an arrangement of particles with sizes consistent with the real grain size distribution. To understand the effects of the parameters and the initial compactness in the shear response, a sensitivity analysis was performed with two specimens, each having different initial void ratios. Taking advantages of the DEM’s features, such as the capacity to track each particle’s displacement and contact forces, the out-of-plane migration and the rotation of the contact network was also studied. Because of the DEM’s high computational time cost, up-scaled grading curves were tested to speed up the Trapdoor modeling without altering the representativeness of the phenomenon. A good agreement between the numerical and the experimental results was achieved in terms of the stress path, but the displacement path was very difficult to replicate.. Keywords: DEM, direct shear test, micro-parameters, real and scaled granulometry, Trapdoor, soil arching effect. x.

(11) RESUMEN. En el presente trabajo, el denominado “efecto arco” fue reproducido mediante un modelo numérico del ensayo Trapdoor descrito por Terzaghi usando el Método de Elementos Discretos (DEM) tridimensionales. Esta metodologı́a representa el suelo como un conjunto de partı́culas, reproduciendo su respuesta macroscópica mediante las interacciones micro-mecánicas entre partı́culas individuales. Uno de los propósitos de este trabajo es estudiar la capacidad de esta metodologı́a para reproducir el efecto arco, el cual consiste en un mecanismo que ocurre naturalmente en suelos producto de la redistribución de tensiones hacia zonas más rı́gidas cuando existe reacomodamiento de partı́culas debido a desplazamientos impuestos a parte del soporte del sistema. En una etapa previa de este estudio, se ensayó en el laboratorio una prueba Trapdoor instrumentada con celdas de carga y la técnica de la Correlación de Imagenes Digitales (DIC). El campo de desplazamiento y el perfil de cargas en la base obtenido con esta instrumentación, se compara a los resultados de la modelación DEM. La calibración de los parámetros micro-mecánicos fue realizada mediante ensayos de corte directo, usando una distribución de tamaño de partı́culas similar a la granulometrı́a real. Para estudiar el efecto que tienen los parámetros y la densidad inicial, se realizó un análisis de sensibilidad con dos probetas con diferentes indices de vacı́os. Aprovechando las ventajas de esta metodologı́a discontinua, como la capacidad de rastrear el desplazamiento y fuerzas de contacto entre partı́culas, se realizó un análisis de la migración de particulas fuera del plano de corte y de la rotación de las fuerzas intergranulares. Debido al alto costo computacional, se estudió la respuesta de curvas granulométricas escaladas, para disminuir los tiempos de cálculo sin perder la representatividad del fenómeno. En general, se obtuvo una buena correlación entre los resultados experimentales y numéricos en términos de la tensión de corte, pero la respuesta en dilatancia fue difı́cil de replicar.. Palabras Claves: DEM, ensayo de corte directo, parámetros micro-mecánicos, granulometrı́a real y escalada, Trapdoor, efecto arco. xi.

(12) Chapter 1. INTRODUCTION. 1. Motivation The arching effect is the ability of soil to transfer loads from a yielding zone to a stiffer area through shear stresses, in response to a relative displacement between locations (McNulty, 1965). This natural soil effect has been studied by several authors because it is “one of the most universal phenomena encountered in soils both in the field and in the laboratory (Terzaghi, 1943).” The design of short aggregate piers to support highway embankments (White & Suleiman, 2004), reinforcements using geosynthetics in sinking areas (Briancon & Villard, 2008) and stress distribution on discontinuous piling (Sáez & Ledezma, 2012) are typical cases where arching effect plays a predominant role. Terzaghi (1936) introduced a test to reproduce and study the arching effect in the laboratory, known as the Trapdoor test. The Trapdoor consists of soil box with a movable gate in the center of its base. The ascending (passive Trapdoor) or the descending (active Trapdoor) displacement imposed on the gate produces the yielding of the soil above transferring the load to the nearby support through shear bands. Figure 1.1 depicts systematically the active Trapdoor. Right after the displacement is imposed, a stress redistribution takes place due to the arching phenomena.. y. y x. G. x. σyy. ( A ) Initial state. G. σyy. ( B ) After gate displacement. F IGURE 1.1. Trapdoor stress distribution scheme. From the original Terzaghi’s test, several authors have used the Trapdoor results to propose approximated equations that describe the arching effect and the associated redistribution of stresses 1.

(13) for different problems (Terzaghi, 1943; Handy, 1985; Harrop-Williams, 1989; Vermeer, Punlor, & Ruse, 2001; Chen & Martin, 2002). Due to the importance of the arching effect and its wide use, it is important to study new technologies and new methods, allowing a better understanding of the phenomenon and its implications.. 2. Objectives and Methodology This study was divided in two parts. The objectives of the first part are to: • Understand the influence of the Discrete Element Method (DEM) micro-parameters in the macroscopic response of the simulated material. • Reproduce, with a discontinuous numerical model, the actual response of a selected sand under shear conditions with the minimal computational cost. To achieve these initial objectives, a direct shear test was modelled numerically using the Discrete Element Method (DEM) using the 3D Open Source Code LIGGGHTS (Kloss, Goniva, Hager, Amberger, & Pirker, 2012). This commonly used geotechnical laboratory test causes the shear failure condition to a soil specimen by imposing a displacement to the upper half of the specimen while the lower half remains fixed. Two specimens with the same DEM’s properties but with different initial void ratios are implemented. A numerical sensitivity analysis was conducted to highlight the effects of the micro-mechanical parameters and also the initial fabric compactness influence. Afterwards, a third specimen with calibrated parameters to real shear test data was developed to investigate the particle’s out-of-plane migration and the contact forces net. Finally, two models with two and four times up-scaled material are tested to reduce the computational time but keeping the ability to reproduce the sand’s macroscopic behavior. The objective of the second part of this research was to: • Study the ability of the Discrete Element Method to reproduce the soil arching effect. 2.

(14) The arching effect was reproduced by DEM simulation of the active Terzaghi’s Trapdoor test. A laboratory Trapdoor, implemented with displacement transducers, load cells and the Digital Image Correlation technique (DIC) was previously investigated (Pardo, 2013; Pardo & Sáez, 2014). The complete displacement field of the Trapdoor obtained with the DIC technique and the base stress profile from the load cells were compared against the DEM’s model results.. 3.

(15) Chapter 2. DISCRETE ELEMENT METHOD (DEM) OVERVIEW. The DEM was developed by Cundall and Strack (1979) originally to solve rock mechanics problems. From this initial work, thanks to the computation advances and improved mathematical algorithms, the method had been implemented to analyse problems in the physics, chemistry and material sciences. The feature that distinguishes DEM from other Molecular Dynamics methods and makes it applicable to granular problems is the inclusion of the rotational degrees-of-freedom. In this research the soft sphere category of DEM was used, which differs from the hard sphere category in which the overlapping between particles is allowed (Duran & Behringer, 2001; Zhu, Zhou, Yang, & Yu, 2007). The main feature of this method is that it considers the soil as the interaction of individual particles, where each one has its own independent degrees of freedom. The dynamic equilibrium is calculated with Newton’s second law, shown in Eq. 2.1, where m and I are the particle’s mass and inertia, ẍi and Θ̈i are the particle’s linear and angular acceleration, F ji is the force of the particle j over the particle i, Fe is the force produced by external force fields. M ji and M e are the torques produced by F ji and F e respectively.. mẍi =. X. F ji + Fe. (2.1a). M ji + Me. (2.1b). j. I Θ̈i =. X j. The method divides the time in small intervals, where a series of steps are followed to calculate the particles forces and displacements through the simulation. The required steps to complete one time interval are the following (Fig. 2.1): • Initialization: This is a previous stage before the time cycles. The boundary conditions and the particles, with their granulometric and spacial distribution, are created.. 4.

(16) Also in this step, a list of neighbours is created. All particle pairs within a neighbor cut-off distance are found by a binning algorithm and stored into a list. This neighbor list includes all the the potential contacts between particles, and is rebuilt as many times as required. • Contact force calculation: The overlapping is measured for all the neighbour particles and with this information the contact forces are calculated (F ji ). • Resultant force and torque calculation: All the contact forces for each single particle P P are added ( F ji ). Also, if a rolling model is used, the torque is calculated ( M ji ).. • Acceleration integration: The linear and angular acceleration, ẍi and Θ̈i , are calculated using Eq. 2.1 and the velocity is obtained with a chosen integrator scheme.. • Position actualization: The velocity is integrated to obtain each particle displacement. Finally, the whole system is updated to their new position.. Initialization: Geometry & BC Contact model Time step ∆t t=0. t = t1 + ∆t. Contact force calculation. Resultant force and torque calculation. Position actualization. t = t1. t = t1. Acceleration integration. t = t1. F IGURE 2.1. DEM computation steps. In each stage a series of aspects needs to be carefully considered in order to have a stable system and to properly reproduce the problem under study. These aspects are discussed in the following sections. 5.

(17) 1. Normal contact law This is probably the most important aspect in DEM simulations and it is involved in all the processes that control the granular assembly. This relation relates the particles overlapping with the corresponding repulsion or attraction force. The two most used contact laws are: • Linear contact model: This is the simplest contact law for granular materials. Equation 2.2 shows the standard model, where the magnitude of the forces is linearly proportional to the overlapping (δn ), and the user defined contact stiffness (Kn ) is the proportional constant. This model is usually enough to reproduce planar cuasi-static problems (Lätzel, Luding, & Herrmann, 2000).. Fn = Kn δn. (2.2). • Hertz contact model: This more complex normal contact law is shown in Eq. 2.3 and it is based in the relationships proposed by R. Hertz (1882). The equation relates the magnitude of the force to the contact area and the material properties, consequently the force is not linearly proportional to the overlapping. The equivalent Young’s modulus (Ē) and the equivalent radius (R̄), shown in Eq. 2.4 and Eq. 2.5, are calculated from the individual properties (Poisson’s ratio ν, Young’s modulus E and the radius R) of the two particles in contact (i and j), and used in Eq. 2.3 to obtain a single contact force. p 3 4 Fn = (2.3) Ē R̄ δn2 3 (1 − νi2 ) (1 − νj2 ) 1 = + Ei Ej Ē. (2.4). 1 1 1 = + Ri Rj R̄. (2.5). The fundamental difference between these two models is depicted in Fig. 2.2. While linear law considers the overlap (δn ) as single dimension parameter, the Hertz model uses the contact area, which is a two dimensional quantity. This difference leads to Hertz’s force exceeding the 6.

(18) linear relationship for larger overlaps (Fig. 2.3). For high confinements, the linear contact model results larger overlaps than the Hertz’s model, which might change the mechanics of the problem.. Frontal plane view. Contact plane view. Normal contact force. Contact Plane. bc. Contact area. δn bc. F IGURE 2.2. Soft sphere contact example. 2. Tangential contact law This contact model is theoretically impossible to develop between two perfectly spherical particles. Although, a tangential model is almost always included in DEM models to account for particle interlocking produced by asperities and roughness at contact level. Also, the contact between real particles take place in an area (which can be small), not in a single point. Figure 2.4 depicts the contact area, forces and moments involved. According to this figure, the analogy with the sliding block on a flat surfaces seems natural. The tangential model provides a relation to establish the force increment before the sliding occurs. 7.

(19) Contact force. Hertz contact. Linear contact. Overlap δn F IGURE 2.3. Contacts force comparison scheme. F IGURE 2.4. Forces and moments involved in a contact (Johnson & Johnson, 1987). As for the normal contact relation, there are many models to calculate the magnitude of the tangential contact force. The most used is shown in Eq. 2.6, which was proposed by Mindlin (1949) and it also relates the contact area and the deformation material properties (equivalent shear modulus Ḡ in Eq. 2.7) with the normal overlap (δn ). p Ft = 8Ḡ R̄δn δt. (2.6) 8.

(20) 2(2 + νi )(1 − νi ) 2(2 + νj )(1 − νj ) 1 = + Ei Ej Ḡ. (2.7). The major feature of this contact model is that its magnitude is restricted by Coulomb’s frictional model (Eq. 2.8), where µ is the friction coefficient and Fn is the normal contact force (Eq. 2.2 i or Eq. 2.3). Hence, the sliding between particles takes places when the maximum force Fmax is. reached.. t Fmax = µ|Fn |. (2.8). 3. Time step The system convergence and stability can be assured with a small enough time step, but as it decreases the computational time increases. For this reason, there are different methods to find an optimal and efficient time step, known as critical time step. • Time step for linear contact models: Itasca C.G. (2008) proposes the time step shown in Eq. 2.9 as the critical time step to ensure the stability of a system. In Eq. 2.9, m, I, K n and K r are the mass, the inertia, the normal stiffness and the rotational stiffness, of the particle respectively, with the lowest natural period according to equation 2.10.. tcrit.  p m  , translational q Kn = I  rotational Kr 2π T =p m/K. (2.9). (2.10). • Hertz’s time step: Equation 2.11 show the relationship used to calculate the time step for this criteria. This time step is useful when solving a dynamical problem, because it includes the maximum particle’s speed (vmax ). The equivalent radius (R̄) and the equivalent Young’s modulus (Ē) were already defined in Eq. 2.5 and Eq. 2.4, while the equivalent mass is computed according to Eq. 2.12. To determine the critical time step, the collision of each particle with a virtual identical particle is considered, and the smallest time step is used. 9.

(21) r. tcrit = 2.87 0.2. m̄ ¯ R̄E 2 vmax. 1 1 1 + = m̄ mi mj. (2.11). (2.12). • Rayleigh’s time step: For non-linear contacts Li, Xu, and Thornton (2005) proposed that the time step should be small enough to allow a Rayleigh’s wave travel through the medium. To achieve this condition, the time step must comply with Eq. 2.13, which is calculated with each particle’s properties and then the minimum is used.. tcrit. p πR Gρ = 0.16131ν + 0.8766. (2.13). 4. Damping As the formulation of DEM is dynamic some energy dissipation is required to reach static states. Most of this energy dissipation is provided by frictional relative displacement between particles, but for some cases it might not be enough. For this reason, an additional damping force can be added into the Newton’s equilibrium equation (Eq. 2.1) of each particle. The most frequent used models are: • Local damping: The local damping force (Fid ), shown in Eq. 2.14, is added to each particle i. The magnitude of the damping force, in this case, is proportional to its outof-balance force (Fi ) and takes the opposite direction of the velocity (sign(vi )). The out-of-balance force is defined as the resultant force on the particle which produces its acceleration. Fid = −α|Fi |sign(vi ). (2.14). Itasca C.G. (2008) proposed the use of 0.7 for the factor α (constant of proportionality), when solving a quasi-static problem and a lower value when working on a dynamical case. The principal advantage of this kind of damping relationship is that the force is independent of the frequency. 10.

(22) • Viscous damping: This kind of damping has the general structure shown in Eq. 2.15, where m is the particle mass, Kn is the normal contact stiffness and vij is the relative velocity of the particles in contact. The viscous damping force (Fijd ) is added to each contact force (Fij in Eq. 2.1). This kind of damping is appropriate for modelling high speed particles’ collisions and dynamical problems. Fijd = 2β. p mKn |vij |. (2.15). If the Herzt’s and the Mindlin’s contact stiffness are used, the normal and the tangential contact damping force results in Eq. 2.16 and Eq. 2.17. Fijd. Fijd. = −2. r. 5 β 6. q p 2Ē R̄δn m̄|vij |. (2.16). = −2. r. 5 β 6. q p 8Ḡ R̄δn m̄|vij |. (2.17). Figure 2.5 depicts a simple drop test that was conducted to highlight the differences between these two damping models. The test consists in dropping-off a particle under gravity with different damping models and coefficients. Fig. 2.6a shows the restitution coefficient comparison between the models, which was calculated from the maximum ball height after the first rebound. As expected, the restitution coefficient decreases with the damping coefficient. Both models are able to reach a low restitution coefficient with a proper damping. The main difference is that while the viscous damping model keeps the energy of the first impact for all the explored damping coefficients, the local model dissipates energy drastically during the fall (Fig. 2.6b). This effect is not very important for steady state problems, but is critical for dynamic cases. For example, to predict the wear in chute belts, calculating the impact energy is critical.. 5. LIGGGHTS code For this work, we use the Open Source Code LIGGGHTS (Kloss et al., 2012) because it is specially adapted for large size problems thanks to a parallel-oriented implementation. LIGGGHTS works in parallel based on the Message Passing Interface (MPI), proposed as a standard by a 11.

(23) t0. t2 Drop heigh. Rebound. t1. F IGURE 2.5. Conducted drop test. Restitution comparison. Impact energy. Viscous damping Local damping. 1.0. Resitution coeficient. 1.0. 0.8. Normalized energy. 0.8. 0.6. 0.6. 0.4. 0.4. 0.2 0.00.0. 0.2. 0.1. 0.2. 0.3. 0.4. 0.5. Damping. 0.6. 0.7. 0.8. 0.9. ( A ) Restitution coeficient after the first bounce. 0.00.0. Viscous damping Local damping 0.1. 0.2. 0.3. 0.4. 0.5. Damping. 0.6. 0.7. 0.8. 0.9. ( B ) First impact energy. F IGURE 2.6. Damping models comparison with a simple drop test scheme. broadly committee of researchers, vendors, implementors, and users. MPI is not a library, but the specifications of how the library should be. It was designed to work in any hardware architecture: Distributed, Sheared or Cluster Memory System. There are several implementations of MPI, among which can be named OPEN-MPI, MPICH, MPICH2 and MVAPICH. The need of these interfaces is because cores are blind to the others processors data, so the libraries serve to move packages of information between cores. In this way, they can communicate and converge to 12.

(24) one single result. There are other interfaces to program in parallel which are often more easy to implement such as OpenMP, but MPI is the most competent and widely used for complex software programing. The software uses the regional parallelization to speed up the models. This procedure is depicted in Fig. 2.7 and consists in sorting the particles among the cores, according to their spatial position. At the DEM’s initialization phase, the defined simulation space is automatically divided (or as the user defines) into N regions, where N is the number of available cores. This partition is based on the size and shape of the global simulation box in order to minimize the surfaceto-volume ratio of each processor’s sub-domain. Each one of these regions is assigned to one processor. Therefore, when a particle is created into one of the regions (according to their centroid position), all its information is sent to the assigned processor. In every time step, each processor calculates the contact force of their particles according to the global neighbours list. When a contact between particles that belongs to different cores shows up in the list, their information is shared between the involved cores with the MPI’s tools.. 2. 1 1. 1 1 1. 2. 1 1. 1 1. 2 2 2. 2 1. 1 1. 2. 2 2. 2 2. 2. 2 2. F IGURE 2.7. Paralellization process example of four-cores. 13.

(25) The communication between cores is more time-expensive than the internal communication, so the regional parallelization was chosen to minimize the communication between cores. Nevertheless, the particles may be sorted in any way without changing the final result due to of their independent degrees of freedom. In fact, if the centroid of one particle migrates to another region, its information is not immediately passed to the corresponding core until an actualization of the neighbour list is done. Technically one particle can move through regions without changing cores which does not affect the final result, but increases the computational time because of the extra communication required. To complete one time step all the cores must have completed all their contacts calculations and particle actualizations. In order to take take fully advantage of the parallel computing capabilities, it is critically important to divided the regions in order to evenly distribute the amount of particles between cores during the whole simulation. A benchmark was conducted with the direct shear problem using 32, 64 and 96 cores (Fig. 2.8a). The problem has about 70,000 particles and more than 2.5 millions possible contacts in the neighbor list. The speed-up is defined as the ratio of sequential (1 core) and parallel (n cores) execution time. In this case, the speedup is defined with respect to the minimum number of processors Q (Eq. 2.18) because the large amount of data poses a time problem when running it in a single core. The relative efficiency is defined as the ratio between the relative speed-up (SPQ ) and the number of processors (P ) normalized by the minimum number of cores (Q). Equation 2.8a show the Amdahl’s law (1967), which proposed a theoretical maximum speedup, defined by the sequential part of the code (f ). This law proposes that the speedup is not linear with the cores increment (Fig. 2.8a) because of the increase in the communication time between cores and the sequential part of the code (e.g. one time step must be completed to continue with the next). The theoretical efficiency, shown in Eq. 2.19, decreases due to the nonlinearity of the speed-up. Although, the use of 96 cores allows the problem to speed-up 1.2 times with an efficiency of 0.2, which is close to the maximum theoretical speed-up for this problem.. SPQ =. TQ TP. (2.18). 14.

(26) EPQ. 1 f + (1 − f )/P. (2.20). EP ≤. 1 1 + (P − 1)f. (2.21). 1.5. 1.0. Efficiency of the shear problem Measured points Efficiency Law relative to Q 74% sequential portion. Relative Efficiency (EPQ ). Relative Speedup (SPQ ). 0.8. 1.4. 0.6. 1.3. 0.4. 1.2. Measured points Amdahl's Law relative to Q 74% sequential portion. 1.1. 1.0. (2.19). SP ≤. Speedup of the shear problem. 1.6. Spq = P/Q. 102. 103. 104. Number of cores (P) ( A ) Speedup curve of the shear problem. 0.2. 0.0. 102. 103. Number of cores (P). 104. ( B ) Efficiency curve of the shear problem. F IGURE 2.8. Benchmark of the direct shear problem. 6. Other discontinuous methodologies Several other discontinuous methodologies have been developed to solve problems in multiple science areas, in order to improve the results of the simulations. A brief description of some of these methodologies and some advantages and disadvantages are discussed below. • Molecular Dynamics The Molecular Dynamics (MD) method is very similar to DEM. Their main difference is that the molecules are simulated as points, lacking of mass and rotational degrees of freedom. Because of this, MD is not often used for geotechnical applications, but is 15.

(27) use in chemistry and biochemistry to simulate atoms and molecules interactions. As in DEM, the displacement are calculated from interaction forces, but in this case they are not based on the overlap. • Event Driven In Event Driven (ED) method is also called “Hard Sphere” method because the particle overlapping is not allow. In this method only one collision can occurs at a time. Each collision is governed by the momentum exchange equation, while DEM is governed by the Newton’s second law. This method is not often used in geotechnical researches, because it is more suitable for modelling granular flows. • Contact Dynamics The main idea of the Contact Dynamics (CD) method is to determine the interaction force between particles in equilibrium directly through mathematical algorithms, while in DEM the equilibrium is reach as a results of the particles interactions. The CD has been applied to solve granular material problems by several authors (Bratberg, Radjai, & Hansen, 2002; Staron, Vilotte, & Radjai, 2002; Radjai, Roux, & Moreau, 1999). Particularly Azema, Radjai, and Saussine (2009) compared an arrangement of polyhedral particles with perfect spheres, finding that the particles shape influences considerably in the macroscopic and microscopic behave. Although CD seems to manage complex angular shapes better than DEM, this method does not guaranteed a unique solution, but the set of accessible solutions shrinks through time steps (Azema et al., 2009). Renouf, Dubois, and Alart (2004) made an attempt to paralellize CD, which is not easy because of its formulation. Their results show a good speed-up of the problem but fluctuations between the one-core and multiple-cores results can be observed.. 16.

(28) Chapter 3. MATERIAL MODELING CALIBRATION. The direct shear test has been modelled with DEM by several authors to validate the use of this method to represent granular materials and to understand the effect of some DEM’s microparameters. Masson and Martinez (2001) made the first attempt with a 2D DEM direct shear model using 1050 cylinders to simulate the material under shear conditions. The macroscopic results exhibited typical features of the shear response of granular materials. Later, Thornton and Zhang (2003) incremented to 5,000 particles in their 2D DEM model to reproduce the behaviour of cohesionless powders. Their model results shown a qualitative similarity to the experimentally observed behaviour in terms of stress-strain-dilation response. Cui and O’sullivan (2006) tested perfect steel spheres in a direct shear test apparatus. The data was used to compared the results of a 3D DEM model composed of 11,700 particles. Despite the fact that they could not reach a perfect match in the initial void ratio, they obtained a good match between the physical test and the numerical simulations in terms of macro-scale response, validating the use of the method to simulate idealized granular materials. Later, Yan and Ji (2010) were the first to contrast their results with real soil data obtained from direct shear tests with limestone rubbles. To simulate the interlocking between the particles due to the irregular shape, the authors clumped spheres into five different configurations. The 3D model used 2,300 spheres to make 928 of these clumps. They validated the use of clumps to simulate the characteristics of direct shear tests of irregular granular materials. Most recently, Härtl and Ooi (2011) used the Jenike direct shear test to investigate the particle shape and microscopic friction on the bulk friction. They used single sphere particles and a clump of two overlapping spheres to highlight the shape effect. Also, they compared their results with experimental data using the same particles geometry. Their results show that the particle interlocking has a greater effect than the packing density on the bulk friction, this remarks the need of simulating the grain shape in the DEM models. Thornton and Zhang (2003) found differences between 2D and 3D DEM models under nominally two-dimensional shear flow, this due to the discontinuous formulation of the method. They. 17.

(29) used single spheres and discs to model the Ottawa’s standard graded sand without shape considerations. Periodic boundaries where used to simulate the direct ring shear test. The main conclusion of this work is that the 3D DEM framework should be used when a good prediction, from a quantitative point of view, is required. The 2D DEM is acceptable if just a qualitative estimation is required. In order to calibrate the micro-parameters controlling the global behavior of DEM simulations, the first part of this investigation is devoted to reproduce the experimental response of a specific chosen sand on direct shear test. A summary of the work conducted to reproduce the direct shear test using the 3D DEM modeling is presented in the following which has been submitted to Computers & Geotechnics Journal.. 18.

(30) Modeling the direct shear test of a coarse sand using the 3D Discrete Element Method with a rolling friction model Abstract In this research, a direct shear test was modeled using the 3D Discrete Element Method (DEM). The results are compared against experimental data. The real sand was modeled as about 70,000 spheres, with sizes consistent with the real grain size distribution, using a rolling friction model to include the sand’s grain shape. It is showed that, the stress path can be appropriately reproduced, but the dilatancy were very difficult to replicate. Finally, specimens with up-scaled material were tested. Although the unscaled material was the most representative, the response of the particles that were up-scaled twice was slightly less accurate, but with a considerable decrease of computation time. Keywords: DEM, Direct shear test, micro-parameters, rolling friction model, real grain size distribution. Introduction In this paper, a 3D Discrete Element Method (DEM) model of the direct shear test was developed to reproduce the response of coarse sand in order to explore the ability of this modelling approach to reproduce this widely used test considering a realistic grading distribution. Several features of this method were explored, such as the ability to monitor and analyze displacements and contact forces of all particles composing the material. This laboratory test is widely used in geotechnical engineering, because of its simplicity and easy analysis of the results. The test uses a box with two halves, where the lower half is moved horizontally while the upper half remains fixed. A confinement force is applied to the top wall, which is free to move vertically. The registered reaction force depends on the properties of the shear band resulting from the induced relative displacement between both rigid half-boxes (Fig. 3.1). 19.

(31) F IGURE 3.1. Direct shear test. Several studies have shown that it is possible to qualitatively reproduce laboratory results with the discrete element technique using 2D models, despite the planar formulation limitations (Thornton & Zhang, 2003; Masson & Martinez, 2001). Cui et al. (Cui & O’sullivan, 2006) were probably the first to perform a comparison of physical test data and direct shear test DEM models using perfect metallic spheres. Later, Yan et al. (Yan & Ji, 2010) compare their DEM results with real soil direct shear tests performed on irregular limestone rubbles. In their approach, they used clumps to simulate the shape of the real rubbles. In both studies, a good match between numerical results and real test data was achieved. Most recently, Härtl et al (Härtl & Ooi, 2011) used the Jenike direct shear test to investigate the particle shape and microscopic friction on the bulk friction. They used single sphere particles and a clump of two overlapping spheres to investigate the shape effect. Their results show that the particle interlocking has a greater effect than the packing density on the bulk friction. In the present research a 3D approach was chosen, because the 2D models fail to accurately predict the peak and residual friction angles under shear flows (Fleischmann & Drugan, 2013), due to out-of-plane migration of the particles, among other effects. In recent years, the microprocessor industry has made little improvement in the core’s clock speed, giving advantage to multi-core architectures (Geer, 2005; Borkar & Chien, 2011) and the number of processors per machine will continue to increase. This change encourages the use of high-performance parallel software to fully take advantage of this technology. The DEM is especially suitable for this type of code because its elements have independent degrees of freedom, which makes it simple to parallelize. Harnessing this benefit, the coarse sand was initially modeled 20.

(32) without size scaling. Nevertheless, to reduce the computational cost and to allow the use of a large number of particles, simple individual spheres were used with a rolling friction model representing the particle shape effect (Wensrich & Katterfeld, 2012). The effect of the model’s micro-mechanical parameters were studied with a sensitivity analysis. Two specimens with different initial void ratios considered, in order to highlight the effect of the system’s compactness on the response. We present the response of the calibrated model as compared to the laboratory results, which were obtained from previous stages of this research (Pardo & Sáez, 2014; Pardo, 2013). Also, a brief analysis of the chain force and evidence of out-of-plane particles migration is shown. Despite advances in computing technology and numerical algorithms, computational cost is still the major limitation of DEM. Due to this issue, a scaled material was investigated and also compared with the experimental data. Due to laboratory restrictions, several down-scaling methodologies exist to test representative soil samples using small grain sizes. Among them, the parallel graduation method (Lowe, 1964) was found suitable and able to capture the original soil response (Verdugo & Hoz, 2007, 2006; Ramamurthy & Gupta, 1986) with a finer gradation curve. In this paper, we use the parallel graduation method, but in the other direction (up-scaling), to develop specimens with particles twice and four times bigger than the original material. The scaled specimens were tested with the same calibrated parameters as the unscaled specimen.. Real sand and DEM model description Tested material The selected material was an angular non-cohesive coarse sand. Its main properties are listed in Tab. 3.1. A series of direct shear tests were performed on this sand under three different confinement pressures: 160 kP a, 80 kP a and 40 kP a. The laboratory test was instrumented with force and displacement transducers. 21.

(33) TABLE 3.1. Sand properties. Property. Value. Density: γ [kg/m3 ]. 1578. Void ratio: e0. 0.679. Porosity: n0. 0.405. Angle of internal friction: φ′. 35. Cohesion: c′ [kP a]. 8. Figure 3.2a shows a series of photos, taken with a microscope, of the sand’s grains and processed with Balu toolbox for Matlab (Mery, 2011) to extract their apparent boundaries. Under a visual analysis, particles of all sizes have a sphericity of about 0.8 and a roundness of about 0.5, hence a subrounded shape (Mitchell, Soga, et al., 1976). Figure 3.2b shows the sand grain size distribution. According to the Unified Soil Classification System (USCS), it corresponds to a poorly graded sand (SP) and all the material is retained in four sieves. The sand has one percent of fines (diameter less than 0.075 mm) and most of the mass is retained in sieve N ◦ 16, which are the biggest grains (between 2.16 mm and 1.18 mm of diameter).. Cumulative percent passing [%]. 100. 80. 60. 40. 20. 0 10-2. ( A ) Sand grains photos. Grain size distribution Simulated material Real material. 10-1. 100. Particle size [mm]. 101. 102. ( B ) Grain size distribution. of the real and simulated material F IGURE 3.2. Grain samples and grain size distribution. 22.

(34) Brief description of DEM The DEM is a numerical technique used to simulate the behavior of granular aggregates, jointed rock masses and chemical powders, among others. The main feature of this methodology is that the material is simulated as an ensemble of individual particles that interact with each other. This approach allows to capture the relative movements and rotations of the particles without the need of a sophisticated constitutive model. Also, it can achieve large displacements because each particle has its own independent degrees of freedom. For granular applications, the dynamic equilibrium condition uses the Eq. 3.1 (Newton’s second law) and Eq. 3.2, where Mi and Ii are the mass and the moment of inertia of each particle i, Fji and Mji are the contact forces applied by particle j on i and the moment induced by this force; Fe and Me are the resultant of the body-force field over the i’th particle (only gravity in this case) and the torque because of this force.. Mi ẍi =. X. Fji + Fe. (3.1). Mji + Me. (3.2). j. Ii Θ̈i =. X j. The time is divided into intervals, and the displacements are calculated in each step by the double integration of Eq. 3.1 and Eq. 3.2. This time step must be small enough to avoid numerical instabilities. In this research, the maximum time step was selected as the minimum of the Hertz or the Rayleigh’s time step (Li et al., 2005). The DEM software used to solve the problem was LIGGGHTS (Kloss et al., 2012) which is a 3D Open Source Code with several granular features already implemented and ready to run in parallel. Further information can be found in the developer’s website.. Inter granular contact modeling Equation 3.3 shows contact model adopted in this investigation, which is based in Hertz’s work (Hertz, 1882). In this equation, Ē is the equivalent Young modulus (Eq. 3.4), R̄ is the 23.

(35) equivalent particle radius (Eq. 3.5) and δn is the overlap of the contact. νi and Ri in Eq. 3.4 and Eq. 3.5 are the Poisson ratio and the radius of the particle i which is in contact with particle j.. Fn =. . 3 4 p Ē R̄ δn2 3. (3.3). (1 − νi2 ) (1 − νj2 ) 1 = + Ei Ej Ē. (3.4). 1 1 1 = + Ri Rj R̄. (3.5). The magnitude of the tangential force is given by the Mindlin’s theory (Mindlin, 1949) and limited by the Coulomb friction criteria shown in Eq. 3.6. If this limit is exceeded, relative sliding takes place and the tangential force remains constant.. t Fmax ≤ µ|Fn |. (3.6). Since the particles being considered are perfect spheres, the Ai’s (Ai, Chen, Rotter, & Ooi, 2011) Elastic-plastic spring-dashpot rolling friction model was considered to take into account the shape effect between the sand’s grains. This model adds a torque (Mr ) in the dynamical equilibrium of the particles, which includes a spring torque (Mrk ) controlled by the rolling viscous damping coefficient (η) and a viscous torque (Mrd ) controlled by the user by the rolling friction coefficient (µr ). The model implemented is shown in Eq. 3.7 to Eq. 3.10.. Mr = Mrk + Mrd. (3.7). The spring torque (Mrk ) is the product of the rotation of the particle (∆θr ) and the rolling stiffness (kr ). This torque is limited by Eq. 3.9, which is controlled by µr , similar to the Coulomb’s friction model. The rolling spring stiffness is defined in Eq. 3.10, where θrm is the full movilization rolling angle.. 24.

(36) k k Mr,t+∆t = Mr,t − kr ∆θr. (3.8). k |Mr,t+∆t | ≤ µr R̄Fn. (3.9). kr =. µr R̄Fn θrm. (3.10). The viscous torque (Mrd ) is controlled by η (Eq. 3.11) and is only active while the rolling torque is not fully mobilized, assisting particle stabilization more than energy dissipation (Ai et al., 2011).. d Mr,t+∆t. =.  p  −2η Ik ¯ r θ˙r  0. if. k |Mr,t+∆t | < µr R̄Fn. if. k |Mr,t+∆t |. (3.11). = µr R̄Fn. The reason for using this kind of model is because the contact force between two particles does not always pass through the centroid of the particles (an implicit assumption when spheres are used), as depicted in Fig. 3.3.. Fij. ei b. Centroid j b. ej. b. Centroid i. −Fij F IGURE 3.3. Real contact. DEM material The granular material is simulated as an assembly of spheres with a radius according to the grain size distribution of the tested sand. Figure 3.2b shows the comparison between the grain size of the real and simulated material. The model does not include the one-percent of fine sand 25.

(37) because it would increase the computational cost considerably and it does not have any significant influence on the material’s macroscopic behavior. Particle number histogram ASTM sieve Distribution of the simulated material. 25 20 15 10 5 0 2.36. 1.18. 0.6. Sieve size [mm]. 0.3. ( A ) Particle number histogram. Mass histogram. 90. Percentage of the total in mass [%]. Percentage of the total in number [%]. 30. 80. Real material per sieve Cumulated Distribution of the simulated material. 70 60 50 40 30 20 10 0 2.36. 1.18. 0.6. Sieve size [mm]. 0.3. ( B ) Mass histogram. F IGURE 3.4. Mass and Particle number histograms. The percentage of sand mass retained in each of the sieves of the ASTM gradation test was simulated with particles whose diameters were linearly distributed between the opening size of the sieve where they mass was retained and the opening size of the preceding sieve, as can be seen in Fig. 3.4a. The only two exception is that the finest part was not modeled and the diameter of the coarsest particles was limited to 1.6 mm. Figure 3.4b shows that the amount of the modeled mass accumulated per sieve is slightly greater than the real material, because the mass of the finest portion that is not included is distributed among the other sieves. Boundary condition and fabric generation All the walls, except the top one, were modeled as rigid boundaries. This rigid boundary condition does not have inertia and its velocity is imposed to control its position during the simulation. Walls interact with the particles, following the Hertz’s contact model, to calculate the repulsive forces. The top boundary is modeled as a servo-controlled wall, in which its speed is related to the current stress by the algorithm shown in Eq. 3.12. The process is controlled by the constant kp , ki and kd , which multiply the error between the current force in the wall and the target P value (∆et ), the cumulative error ( t ∆et ) and the error derivative, respectively. 26.

(38) Vwall = kp ∆et + ki. X t. ∆et + kd. ∆et − ∆et−∆t ∆t. (3.12). The specimen is composed of approximately 70,000 particles. All the particles are randomly generated inside the box and are divided in three layers of 0.8 cm each. After the first layer is created, the particles fall freely until the static equilibrium is reached due to the gravity; then the process is repeated with the other two layers. After the settlement is completed, the servocontrolled wall is activated to confine the soil. Finally, a finite velocity is imposed on the lower half in order to reach a horizontal target displacement of 8 mm. The model, shown at the beginning of the shear process, is shown in Fig. 3.5. In order to get a loose or a dense fabric (eoi ), different coefficients of friction (µi ) were used in the settlement process and during the shearing modeling. If lower values of µi are used during generation, a dense fabric is obtained. The chosen Poisson’s ratio coefficient is a typical value used to simulate sand, while E was chosen to have a realistic contact stiffness. During shearing, the value of µ, µr & η were modified to highlight their effect on the macroscopic response (Tab. 3.2). No rolling friction is considered during specimen generation.. z y. x. F IGURE 3.5. Developed DEM model. Sensitivity analysis Two specimens were modeled and tested in order to explore the effects of the micromechanical parameters on the macroscopic response. The properties used for the model construction are listed in Tab. 3.2. The difference between these two specimens is the initial coefficient of friction µi and consequently, the obtained void ratio e0i . The unit solid weight γs , used for both 27.

(39) specimens, was obtained from laboratory measurements. The Young Modulus E and the Poisson’s ratio ν are required to calculate the contacts’ stiffness and were selected among values proposed in the literature (Thornton & Zhang, 2003; Masson & Martinez, 2001; Cui & O’sullivan, 2006; Yan & Ji, 2010; Fleischmann & Drugan, 2013), but were small enough to avoid a time step that was too small. TABLE 3.2. Tested specimens properties. Property γs [kg/m3 ] E [kg/m2 ] ν µi ηi µri e0i obtained. Specimen A (dense) Specimen B (loose) 2, 650 5.0 × 108 0.256 0 0 0 0.489. 2, 650 5.0 × 108 0.256 0.3 0 0 0.594. Figure 3.6a, Fig. 3.6b and Fig 3.6c show the results using different combinations of the coefficient of friction µ, the coefficient of rolling viscous damping η and the coefficient of rolling friction µr during the shearing phase. The shear stress, used in these comparisons, was calculated according to Eq. 3.13, where T is the horizontal reaction force of the upper half of the box (Fig. 3.1) and A is the effective shear area which depends directly on the horizontal displacement of the lower half (∆h ).. σs =. T A(∆h ). (3.13). According to Fig. 3.6a, the shear stress comparison shows that the friction coefficient µ has an influence only for small displacements and tends to slightly increase the peak value. In the studied cases, the large-strain stress value is approximately the same for horizontal displacements larger than 4 to 5 mm, indicating that by varying only the Coulomb’s friction coefficient it is not possible to control the response of the model at large strains. Regarding volume evolution, the friction coefficient µ increases the specimen dilatation without considerably affecting the path’s 28.

(40) shape (Fig. 3.6a). Indeed, due to increase of particle friction, a looser configuration could be reached at large strains. Figure 3.6b shows the sensibility on the rolling viscous damping η. In this case, a direct effect of this parameter on the stress path or on the displacement responses is not as clear, but a slight increment in the peak value is detected. The little effect of this parameter indicates that during most of the shear process the viscous torque is fully mobilized (Eq. 3.11) and that the particles are rolling-sliding. Figure 3.6c shows that the most important effect of the coefficient of rolling friction µr is the increase of the shear stress reached at large displacements. This parameter also increases the peak value, but in a less pronounced manner than µ. The displacement comparison in Fig 3.6c shows that µr increases the large-strain volume of the specimens. This influence cannot be noticed at a low-strain, where all the curves seem to follow the same path, highly influenced by the initial material fabric. As µr takes into account the particle’s shape, large values of this parameter can be associated with grains that are less round. In this case, it is reasonable to obtain a supplementary geometric interlocking at large strain, associated with a large apparent macroscopic angle of friction. Due to the shape effect, a looser critical state is also expected. Nevertheless, in general terms, the laboratory displacement path was very difficult to replicate. All three figures show that the main difference between the specimen A and B is that A gives a larger peak at low strain (a typical response of a relatively dense sand), while the response for specimen B correspond to the usual stress-strain path of loose sand. Also, the shear stress comparison shows that specimen A has a higher initial slope (low-strain stiffness), in agreement with the experimental response. This slope can be controlled by the stiffness of the contacts and the initial void ratio. Specimen B has a lower large-strain vertical displacement in comparison to specimen A, which is also a typical experimental result for loose sands. The behavior difference was expected because of the different initial void ratio obtained for both simulated specimens. Compared to the experimental maximum (emax =0.74) and minimum (emin =0.44) void ratios, specimen A has a relative density of 84% classifying it as a very-dense material while specimen B is a medium-dense material with a relative density of 49%.. 29.

(41) Stress comparison. 250. Stress [kPa]. 200 150 100 50 00. 1. 2. 4. 3. 5. 6. Horizontal displacement [mm]. 7. 8. 9. 7. 8. 9. Displacement comparison. Vertical displacement [mm]. 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.20. 1. 2. Laboratory Specimen A. 4. 3. 5. 6. Horizontal displacement [mm] µ=0.4. Specimen B µ=0.3. µ=0.5. µ=0.6. ( A ) µ influence (µr =0.2 & η=0.2). Stress comparison. 250. Stress [kPa]. 200. Stress [kPa]. 200 150. 150. 100. 100. 50. Vertical displacement [mm]. 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.20. 50 1. 2. 3. 4. 5. 6. Horizontal displacement [mm]. 7. 8. 9. 00. 9. 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.20. Displacement comparison. 1. 2. Laboratory Specimen A. 3. 4. 5. 6. Horizontal displacement [mm] Specimen B η=0.3. η=0.4. 7 η=0.5. 8 η=0.6. ( B ) η influence (µ=0.4 & µr =0.5).. Vertical displacement [mm]. 00. Stress comparison. 250. 1. 2. 3. 4. 5. 6. Horizontal displacement [mm]. 7. 8. 9. 7. 8. 9. Displacement comparison. 1. 2. Laboratory Specimen A. 3. 4. 5. 6. Horizontal displacement [mm] Specimen B µr =0.3. µr =0.4. µr =0.5. µr =0.6. ( C ) µr influence (µ=0.4 & η=0.5).. F IGURE 3.6. Sensitivity analysis of the DEM parameters. Calibrated model against experimental data A third specimen was created with the properties shown in Tab. 3.3, a higher contact stiffness was used in order to obtain a higher low-strain stiffness, while using a looser fabric. A greater void ratio was obtained, even when the same µi was used, because higher contact stiffness avoids larger overlaps. The void ratio obtained is slightly lower than the experimental value, which is 30.

(42) expected because of the different shape of the real grains and the simulated particles (Biarez & Hicher, 1997). With a confinement of 160 kP a, the model can appropriately reproduce the shear stress path during the whole test (Stress calibration on Fig. 3.7). For the 80 kP a and 40 kP a confinements, the model does not reproduce the peak value, but the residual stress is accurately reproduced for all the confinements. This might be due to a higher density of the sand in the laboratory tests compared to the DEM models. The dilatancy of the simulated specimen increases as the confinement stress decreases, in agreement with laboratory results (Displacement calibration on Fig. 3.7). Additionally, the shapes of the curves are very similar. Nevertheless, the model fails to accurately predict the final sample’s volume. Indeed, this large strain volume is highly controlled by the particles’ shape, one of the major limitations of the method due to the use of spheres. Thus, the rolling friction model used to approximately include the particles’ shape effect, is not able to fully reproduce the large strain state of the reference soil. In the case of 160 kP a test, the final laboratory void ratio is 0.72; while the simulation gives a value of 0.703. The initial difference was reduced by the increase of the model’s dilatancy, but difference still remains, due to the shape of the sand grains.. TABLE 3.3. Calibrated material parameters. Property 2. E [kg/m ] ν µi ηi µri µ η µr Obtained: γ [kg/m3 ] Obtained: e0i. Specimen C 5.0 × 109 0.256 0.3 0 0 0.3 0.1 0.4 1612 0.643 31.

(43) Stress calibration. Vertical displacement [mm]. Stress [kPa]. 160 140 120 100 80 60 40 20 00. 1. 2. 3. 4. 5. 6. Horizontal displacement [mm]. 7. 8. 9. 7. 8. 9. Displacement calibration. 1.2 1.0 0.8 0.6 0.4 0.2 0.0. −0.20. 1. 2. 3. 4. 5. 6. Horizontal displacement [mm]. Lab 160[kPa] Model Result - 160[kPa]. Lab 80[kPa] Model Result - 80[kPa]. Lab 40[kPa] Model Result - 40[kPa]. F IGURE 3.7. Calibrated material response.. Figure 3.8 shows the initial and final network of contact forces. The forces are evenly distributed at the beginning of the test (Fig. 3.8a), which is expected due to the initial vertical compression. During the test, a diagonal soil strut was formed, as can be seen in Fig. 3.8b. This might be due to the high geometric interlocking of the material used, which prevents the particles of the central part of the sample to relatively slide.. ( A ) Initial force chain. ( B ) Final force chain. F IGURE 3.8. Initial and final chain force. The aforementioned is evidenced when orientations and amplitude of the contact forces are analyzed (Fig. 3.9). Figure 3.9a shows the distribution of the contact force at the beginning of the shear test, after the settlement process, while Fig. 3.9b shows the contact force distribution at the end of the test. At the beginning, the vertical contact forces are almost the double the horizontal 32.

(44) contact forces, because of the vertical confinement imposed to the specimen. At the end of the test, the predominant contact forces are no longer vertical, but almost have the same orientation as the strut (approximately 19.5◦ ). The magnitude of the force in this orientation (between 0◦ and 36◦ ) is about four times larger than the initial vertical forces, while the magnitude of forces in the normal direction to the strut (between 90◦ and 126◦ ) is about one third of the initial vertical magnitude, highlighting the stress-state rotation during the shear test.. Polar force distribution at the beginning of the shear test 108°. 90°. Polar force distribution at the end of the shear test 108°. 72°. 126°. 54°. 72°. 126° 36°. 144°. 90°. 54° 36°. 144°. Force [N]. 162° 1 2. 180°. 3 4. 7 5 6. 8 9 18°. 0°. 198°. 342°. 216°. 324° 234°. 306° 252°. 270°. 288°. Force [N]. 162° 1 2. 180°. 3 4. 7 5 6. 8 9 18°. 0°. 198°. 342°. 216°. 324° 234°. 306° 252°. 270°. 288°. ( A ) Polar contact force distribution at. ( B ) Polar contact force distribution at. the beginning of the shear test. the end of the shear test. F IGURE 3.9. Polar distribution of the contact force at the beginning and the end of the shear test on the shear plane. In Fig. 3.10 and Fig. 3.11 the test was divided into small regions of 3 × 1.2 mm and the particles’ velocities were averaged in each of these regions in order to reduce the number of arrows and to represent the velocity as a continuous field. When the mean radius of the grain size distribution used is 0.5 mm, each of these zones represent the average value of about 360 particles. Figure 3.10 shows a lateral view of the initial and final velocity field of the particles over the imposed displacement plane. At the beginning of the test, all particles have similar velocities, which means that the specimen behaves as a continuous material (Fig. 3.10a). When large strains are reached, 33.

(45) the bottom particles have a higher speed than top ones, hence a sliding surface dividing the specimen in two halves appears (Fig. 3.10b). A shear band of about 2.5 times the mean diameter of the particles is depicted in this figure. The particles do not follow a clear path in the out-of-plane direction (normal to the imposed displacement plane), also the magnitude of the average speed in this plane is about 30 times less than the in-plane speed. Nevertheless, the out-of-plane component of the speed indicates that there was some particle migration across this direction, justifying the choice of a 3D model (Fig. 3.11b). At the beginning of the test, the particles have moved almost completely in the vertical direction, with a very small out-of-plane component (Fig. 3.11a). It is interesting to note that out-of-plane migration particle movement is not symmetric across the observed section, probably due to the random generation of particle size and location among the sample. This asymmetry increases for large strains.. Scaled particle velocity field on plane XZ (∆h = 8 mm). 0.025. 0.020. 0.020. 0.015. 0.015. Z direction [m]. Z direction [m]. Scaled particle velocity field on plane XZ (∆h = 0.72 mm) 0.025. 0.010 0.005 0.000. Shear band. 0.010 0.005 0.000. Scale: 0.6[mm/s]. −0.005 −0.03. −0.02. −0.01. 0.00 0.01 X direction [m]. Scale: 0.6[mm/s]. −0.005 0.02. ( A ) Initial velocity field. 0.03. −0.03. −0.02. −0.01. 0.00 0.01 X direction [m]. 0.02. 0.03. 0.04. ( B ) Final velocity field. F IGURE 3.10. In-plane velocity field (imposed displacement plane). Parallel gradation In order to cut down the running time, the parallel gradation technique has been explored to reduce the number of particles in the simulations (Verdugo & Hoz, 2007). The purpose of using the up-scaled material is to investigate the decrease in the computation time and the accuracy 34.

(46) Scaled particle velocity field on plane YZ (∆h = 0.72 mm). 0.025. 0.020. 0.020. 0.015. 0.015 Z direction [m]. Z direction [m]. 0.025. 0.010 0.005 0.000. 0.010 0.005 0.000. Scale: 0.05[mm/s]. −0.005 −0.010. Scaled particle velocity field on plane YZ (∆h = 8 mm).. −0.03. −0.02. −0.01. Scale: 0.05[mm/s]. −0.005. 0.00 0.01 Y direction [m]. 0.02. 0.03. ( A ) Initial velocity field. −0.010. −0.03. −0.02. −0.01. 0.00 0.01 Y direction [m]. 0.02. 0.03. ( B ) Final velocity field. F IGURE 3.11. Out-of-plane velocity field. degradation with the change in the ratio between the specimen size and the maximum particle diameter (Dspecimen /Dmax ). Two specimens with twice and four times bigger particles were studied. The grain size distribution is shown in Fig. 3.12. The Dspecimen /Dmax ratio of the twice scaled material is 7.5, while the ratio of the four times scaled material is 3.75. Although the Dspecimen/Dmax ratio of the four scaled material is out of the recommended range (Dspecimen /Dmax > 6 (Holtz & Gibbs, 1956)), this scaled factor was tested for completeness. The time-computing comparison shown in Tab. 3.4 was done using only one core, to avoid the performance decrease when using a large number of processors with low data. Also, due to the large amount of time it would take to run the whole test in a single core, and because the purpose of this comparison is to illustrate computing time reduction, each of these simulations consists in a tenth of the required steps to complete the shear and the settlement process. All the simulations were carried out using the same time-step, hence the three specimens include the same number of steps and the same total simulated time. When scaling two-times, the computation time is reduce by 94% (20 times), while the four times scaled is reduce by 99.5% (200 times). This is because the reduction in the number of particles (86.8% and 98.7%) leads to an important decrease in number of contacts (96.4% and 99.8%), which defines the size of the DEM model. It is clear that even 35.

(47) with a small coarse-graining scaling, the computation time decreases dramatically, allowing the development of large models with a reasonable running time. TABLE 3.4. Computational cost comparison. Unscaled. Two-times scaled Four-times scaled. Value. [%]. Value. [%]. Value. [%]. Running time (min). 1903. 100. 107. 5.6. 9.3. 0.5. Number of particles. 64944. 100. 8610. 13.2. 845. 1.3. 3.6. 4033. 0.15. Number of neighbors 2754758 100 100596. Because large fluctuations appear when larger particles are considered, both coarse-grained specimens where tested five times with the same set of parameters shown in Tab. 3.3 and an average was calculated (Fig. 3.13 and Fig. 3.14). The fluctuation in the results of the scaled materials are explained because when using the scaled sample, the number of contacts is significantly reduced, hence any change affects proportionally more in the macroscopic response than the unscaled case where a large number of contacts exist (Fig. 3.14). The paths of the scaled material, which was scaled two times, have a low deviation from the average (Fig. 3.13), while it increases greatly for the specimen that has been scaled four times (Fig. 3.14). This high fluctuation in the stress response is not as clear in the displacement results, because the servo-controlled boundary implementation responds more to a trend than to a single value. As the algorithm relates the stress with the velocities, and not directly to the displacements, the change in the vertical stress must be held for some time steps in order to see changes in the displacements. When averages are compared with laboratory and unscaled material results (Fig. 3.13 and Fig. 3.14), very similar stress paths are obtained for both scales; however, the little increase in the stress path of the material which was scaled four times evidences that the twice scaled is lightly more accurate. The displacement comparison shows that the material which was scaled four times 36.

(48) has a higher displacement than the twice scaled material, which is explained by the increase of the particle size. The material that was scaled four times is an inconvenient choice to reproduce the shear test, because of the high fluctuation of the results, despite the huge decrease of the computational cost. However, the satisfactory enough simulation of the stress paths and the huge decrease in running time make this strategy a suitable choice for problems where the ratio Dspecimen /Dmax is big enough to avoid boundaries effects. Scaled grain size distribution. Cumulative Percent Pasing [%]. 100. Unscaled Scaled two-times Scaled four-times Real sand. 80. 60. 40. 20. 0 10-2. 10-1. 100. Particle size [mm]. 101. 102. F IGURE 3.12. Scaled material granulometry. Stress average. 2. 3. 4. 5. 6. Horizontal displacement [mm]. 7. 8. 9. Displacement average. 1. 2. Lab 160[kPa] Unscaled. 3. 4. 5. 6. Horizontal displacement [mm] Scaled 2 times. 7. Scaled 2 times average. ( A ) Twice scaled average - 160 [kPa]. 8. 9. 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.20. 45 40 35 30 25 20 15 10 5 00. Stress average. Stress [kPa]. Stress [kPa] 1. Vertical displacement [mm]. Vertical displacement [mm]. 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.20. 90 80 70 60 50 40 30 20 10 00. 1. 2. 3. 4. 5. 6. Horizontal displacement [mm]. 7. 8. 9. Displacement average. 1. 2. Lab 80[kPa] Unscaled. 3. 4. 5. 6. Horizontal displacement [mm] Scaled 2 times. 7. Scaled 2 times average. ( B ) Twice scaled average - 80 [kPa]. 8. 9. 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.20. Vertical displacement [mm]. Stress average. Stress [kPa]. 180 160 140 120 100 80 60 40 20 00. 1. 2. 3. 4. 5. 6. 7. 8. 9. 6. 7. 8. 9. Horizontal displacement [mm]. Displacement average. 1. 2. Lab 40[kPa] Unscaled. 3. 4. 5. Horizontal displacement [mm] Scaled 2 times. Scaled 2 times average. ( C ) Twice scaled average - 40 [kPa]. F IGURE 3.13. Shear test results with two-times scaled material. 37.