Predictive computational tools to study granular materials are important in fields ranging from the geosciences and civil engineering to computer graphics. The simulation of granular materials, however, presents many challenges. The behavior of a granular medium is fundamentally multi-scale, with pair-wise interactions between discrete granules able to influence the continuum-scale evolution of a bulk material. Computational techniques for studying granular materials must therefore contend with this multi-scale nature. This research first addresses both the question of how to accurately model interactions between grains and the question of how to achieve multi-scale simulations of granular materials. We propose a novel rigid body contact model and a time integration technique that, for the first time, are able to simultaneously capture five key features of rigid body impact. We further validate this new model and time integration method by reproducing computationally challenging phenomena from granular physics. We next propose a technique to couple discrete and continuum models of granular materials to one another. This hybrid model reveals a family of possible discretizations suitable for simulation. We derive an explicit integration technique from this framework that is able to capture phenomena previously reserved for discrete treatments, including frictional jamming, while treating bulk regions of the material with a continuum model. To effectively handle the large plastic deformations inherent in the evolution of a granular medium, we further propose a method to dynamically update which regions are treated with a discrete model and which regions are treated with a continuum model. We demonstrate that hybrid simulations of a dynamically evolving granular material are possible and practical, and lay the foundation for further algorithmic development in this space. Finally, as the the tools used in computational science and engineering become progressively more complex, the ability to effectively train students in the field becomes increasingly important. We address the question of how to train students from a computer science background in numerical computation techniques by proposing a new system to automatically vet and identify problems in numerical simulations. This system has been deployed at the undergraduate and graduate level in a course on physical simulation at Columbia University, and has increased both student retention and student satisfaction with the course.

<p>Granular materials are hard to understand due to their discrete and a-thermal nature. The mechanical response of a granular packing under external deformations, although highly relevant in industrial processes, is still poorly understood, partly due to the difficulty to generate a homogeneous granular packing. In this thesis, I present a novel shear apparatus that avoids the formation of inhomogeneities known as shear bands. This apparatus provides quasi-static, quasi-uniform simple shear deformation to a 2D model granular system under fixed packing fraction &phi. The position, orientation and forces for each particle are obtained at each shear step, using the photo-elastic technique. This model granular system exhibits coupling between the shear strain, &gamma, and the pressure, P, which we characterize by the `Reynolds pressure', and a `Reynolds coefficient', R(&phi) = (&partial^2 P/ &partial &gamma^2)/2. Under cyclic shear, this system evolves logarithmically slowly towards limit cycle dynamics, which we characterize in terms of pressure relaxation at cycle n: &Delta P &simeq - &beta ln(n/n_0). &beta depends only on the shear cycle amplitude, suggesting an activated process where &beta plays a temperature-like role. In addition, particles in the sheared system are diffusive. The translational and rotational diffusion, observed under stroboscopic view during cyclic shear, are observed to depend on the packing fraction but not on the stress states of the system. Finally, the structure of the force network, and how that connects to the mechanical behavior, is also briefly discussed.</p> / Dissertation

Understanding animal locomotion requires modeling the interaction of the organism with its environment. Locomotion within granular media like sand, soil, and debris that display both solid and fluid-like behavior in response to stress is less studied than locomotion within fluids or on solid ground. To begin to reveal the secrets of movement in sand, I developed models to explain the subsurface locomotion of the sand-swimming sandfish lizard. I developed a resistive force theory (RFT) with empirical force laws to explain the swimming speed observed in animal experiments. By varying the amplitude of the undulation in the RFT, I found that the range of amplitude used by the animal predicted the optimal swimming speed. I developed a numerical model of the sandfish coupled to a discrete element method simulation of the granular medium to test assumptions in the RFT and to study more detailed mechanics of sand-swimming. Inspired by the shovel-shaped head of the sandfish lizard, I used the simulation to study lift forces in granular media: I found that when a submerged intruder moved at a constant speed within a granular medium it experienced a lift force whose sign and magnitude depended on the intruder shape. The principles learned from the models guided the development of a biologically inspired robot that swam within granular media with similar performance to the lizard.

<p>Flow in a hopper is both a fertile testing ground for understanding fundamental granular flow rheology and industrially highly relevant. Despite increasing research efforts in this area, a comprehensive physical theory is still lacking for both jamming and flow of granular materials in a hopper. In this work, I have designed a two dimensional (2D) hopper experiment using photoelastic particles ( particles' shape: disk or ellipse ), with the goal to build a bridge between macroscopic phenomenon of hopper flow and microscopic particle-scale dynamics. Through synchronized data of particle tracking and stress distributions in particles, I have shown differences between my data of the time-averaged velocity/stress profile of 2D hopper flow with previous theoretical predictions. I have also demonstrated the importance of a mechanical stable arch near the opening on controlling hopper flow rheology and suggested a heuristic phase diagram for the hopper flow/jamming transition. Another part of this thesis work is focused on studying the impact of particle shape of particles on hopper flow. By comparing particle-tracking and photoelastic data for ellipses and disks at the appropriate length scale, I have demonstrated an important role for the rotational freedom of elliptical particles in controlling flow rheology through particle tracking and stress analysis. This work has been supported by International Fine Particle Research Institute (IFPRI) .</p> / Dissertation

Locomotion of animals, whether by running, flying, swimming or crawling, is crucial to their survival. The natural environments they encounter are complex containing fluid, solid or yielding substrates. These environments are often uneven and inclined, which can lead to slipping during footsteps presenting great locomotor challenges. Many animals have specialized appendages for locomotion allowing them to adapt to their environmental conditions. Aquatically adapted animals have fins and flippers to swim through the water, however, some species use their paddle-like appendages to walk on yielding terrestrial substrates like the beach. Beach sand, a granular medium, behaves like a solid or a fluid when stress is applied. Principles of legged locomotion on yielding substrates remain poorly understood, largely due to the lack of fundamental understanding of the complex interactions of body/limbs with these substrates on the level of the Navier-Stokes Equations for fluids. Understanding of the limb-ground interactions of aquatic animals that utilize terrestrial environments can be applied to the ecology and conservation of these species, as well as enhance construction of man-made devices. In this dissertation, we studied the locomotion of hatchling loggerhead sea turtles on granular media integrating biological, robotic, and physics studies to discover principles that govern fin and flipper locomotion on flowing/yielding media. Hatchlings in the field modified their limb use depending on substrate compaction. On soft sand they bent their wrist to utilize the solid features of sand, whereas on hard ground they used a rigid flipper and claw to clasp asperities during forward motion. A sea turtle inspired physical model in the laboratory was used to test detailed kinematics of fin and flipper locomotion on granular media. Coupling of adequate step distance, body lift and thrust generation allowed the robot to move successfully forward avoiding previously disturbed ground. A flat paddle intruder was used to imitate the animal's flipper in physics drag experiments to measure the forces during intrusion and thrust generation.

Numerous research efforts have been devoted to characterizing the behavior of granular materials, which is one of the main concerns of pavement engineers. For better understanding of this behavior, laboratory tests where in-situ stress conditions and traffic loads are adequately simulated are needed. This study makes use of an expanded test protocol called a performance test that includes resilient modulus as well as permanent deformation testing. This test protocol determines three nonlinear resilient modulus parameters (k1, k2, k3) and two permanent deformation parameters (?,??). The resilient modulus test results are required inputs in the Level 1 analysis of the proposed American Association of State Highway and Transportation Officials (AASHTO) Pavement Design Guide. In addition, both resilient modulus and permanent deformation test results provide material property inputs to pavement performance prediction models. This study also evaluated the within laboratory repeatability of the performance test and developed a within laboratory precision statement. Further, a statistical analysis was conducted on the test results to estimate the number of test specimens required for testing for specific reliability levels. Two test specimens are required for a reliability level of 15%. A within laboratory study was also conducted to investigate the influence of specimen size on test results. The specimen height was reduced from 12 in. (304 mm) to 8 in. (203 mm), and there was no difference in test results at a confidence level of 95%. The performance test was further used successfully in subsequent studies to evaluate the behavior of granular materials and the influence of various factors on their behavior. As fines content increased, the resilient modulus values decreased and permanent deformation increased. As the moisture content increased, the resilient modulus value decreased and the resistance to permanent deformation decreased. A simplified laboratory measurement tool that is repeatable, relatively cheap and easy to perform might prompt the use of laboratory measured values of resilient modulus in pavement design and facilitate correlation of these values to field measured values on a large scale. Use of measured data for the base properties rather than estimates would insure improved pavement designs and, in many cases, would save money in construction costs.

Granular materials are composed of solid, discrete particles and exhibit mechanical behavior that differs from those of fluids and solids. The rheology of granular flows is principal to a suite of natural hazards. Laboratory experiments and numerical models have adequately reproduced several features observed in terrestrial gravity driven geophysical flows; however, quantitative comparison to field observations exposes a failure to explain the high mobility and duration of many of these flows. The ability of a granular material to resist deformation is a function of the force chain network inherent to the material. This investigation addresses the evolutionary character of force chains in unconfined, two-dimensional, gravity driven granular flows. Our particular emphasis concerns the effects of stress localization on the substrate by dynamic force chain evolution and the implications for bed erosion in dense granular flows. Experimental systems employing photoelastic techniques provide an avenue for quantitative force analysis via image processing and provide dataset that can be used validate discrete element modeling approaches. We show that force chains cause extreme bed force localization throughout dynamic granular systems in spatial and temporal space; and that these localized forces can propagate extensively into the substrate, even ahead of the flow front.

published_or_final_version / abstract / toc / Mechanical Engineering / Master / Master of Philosophy

Unbound granular materials (UGMs) used in the base and sub-base layers of flexible pavements play a significant role in the overall performance of the structure. Proper understanding and characterization of the deformation behaviour of UGMs in pavement structures are, therefore, vital for the design and maintenance of flexible pavements. In this study, the resilient deformation (RD) and the permanent deformation (PD) behaviour of UGMs were investigated for the better understanding and improved modelling of these deformation characteristics. The study is based on a series of repeated-load triaxial (RLT) tests carried out on several UGMs commonly used in pavement structures. Here, the influences of stress level and moisture content - two of the most significant factors affecting the deformation behaviour of UGMs - were analysed. The effects of the grain size distribution and the degree of compaction were also considered. The study on the RD behaviour indicated that the resilient stiffness (MR)of UGMs increases with the increased bulk stress level, which can be satisfactorily described by the k-θ model. Moisture was found to negatively impact the MR as long as the deformation was mostly resilient with a negligible amount of accumulated PD. Analysis of the influence of moisture on the parameters k1 and k2 of the k-θ model showed that k1 decreases with increased moisture and k2 is relatively insensitive to moisture. Based on these observations, a simple model was developed for the impact of moisture on MR. The performance of this model was comparable to an existing moisture dependent MR model. In contrast, it was further observed that at the later stages of the RLT tests, after a relatively large number of load applications, the MR increased with increased moisture up to the optimum moisture content. This occurred when the RD was accompanied by a significant amount of PD. Further investigation suggested that moisture aided the post-compaction (PC) and possible particle rearrangement that resulted in the increased PD and increased MR. In this case k1 decreased, whereas k2 increased, with increased moisture. The existing MR-moisture model did not work for this behaviour. This suggests that the effect of PC on MRshould be considered in modelling. However, although not explored in this study, it may be possible to simulate this effect of increase in MR with increased moisture due to PC using the proposed model if k2 is expressed as a function of moisture. The PD characteristics of UGMs were investigated based on the multistage (MS) RLT test. In contrast with the single stage (SS) RLT test, the MS RLT test accounts for the effect of stress history and enables a comprehensive study of the material behaviour under cyclic stresses of various magnitudes. Since the existing PD models cannot be directly applied for the MS loading procedure, a general formulation based on the time hardening concept was derived that can be used to extend the models for the MS loading conditions. Based on this formulation, some of the current models were calibrated and their performance in predicting the PD behaviour in MS RLT tests was compared. The investigation regarding the impact of moisture on PD showed that moisture significantly increases the accumulation of PD. Generally, materials with finer grading showed more sensitivity to moisture with regards to both PD and RD. To characterize the impact of moisture, moisture sensitivity of different grain size distributions and the impact of the degree of compaction on PD with reduced effort, a simple model was proposed. Unlike some of the well-performing existing models, this model can be calibrated using a single MS RLT test without requiring any separate static failure triaxial tests. This model was validated using the MS RLT test data with satisfactory results. The sensitivity of the parameters of this model was studied with respect to moisture content, degree of compaction and grain size distribution. Some reasonable trends for the sensitivity of the parameters to these influential factors were obtained, which suggests that these may be further developed to incorporate into the model. / <p>QC 20150325</p>

The research presented in this thesis covers the development, calibration and verification of two thin surfaced unbound granular pavement models: one model to predict the response of a pavement to loading by the monotonic application of a single load event (Response model) and the other model to predict the accumulation of permanent deformation of the pavement when it is subjected to a large number of load applications (Performance model). The response model was developed using the finite element method and used an anisotropic stress dependent stiffness model to represent the granular and subgrade materials. The models were verified with an extensive set of stress, strain and surface deflection measurements collected at the CAPTIF facility. The calibrated models were able to predict the subsurface response of the pavement to a range of dual tyre and FWD load levels (23-72 kN). It was found that the measured stress and strain response of the pavement was different under the two loading mechanisms. It was also found that a particular response at a point in the pavement was linear with respect to load. The performance model was based on similarities observed in the performance of granular materials in both laboratory and full-scale experiments. When the specimen or pavement was showing a steady state response, it was found that the rate of accumulation of permanent deformation was related to the resilient strain. This relationship was then used to predict the deformation of CAPTIF pavements based on the outputs from the response model. The application of laboratory derived models required the use of shift functions to be able to be successfully used in replicating field measurements, this was expected given the differences in boundary conditions and loading mechanisms for the laboratory and field systems.