A mathematical study has been undertaken to model various kinds of granular flows including the perfect plasticity flow and the viscous elasto-plasticity flow. The work is mainly based on the double-shearing theory originated by Spencer and developed by many others. The focus of the project is on the formulation of the theory, the construction of mathematical models and the development of robust simulation techniques.Based on a general formulation of the double-shearing theory, the perfect plasticity flow is shown to be governed by a set of highly nonlinear first order hyperbolic partial differential equations with two distinct characteristics. A sophisticated numerical algorithm is then developed based on the method of characteristics to determine the stress discontinuity and the velocity and stress fields. With the method developed, a numerical study is then undertaken to model the flow of granular materials in a hopper in the presence of stress discontinuity and to investigate the influence of various parameters on the distribution of hopper wall pressures.Utilising the double shearing theory, a set of stress-strain constitutive equations in explicit form has been derived, which makes it possible to formulate the double-shearing theory within the framework of the finite element method. Thus, consequently, a sophisticated finite element technique has been developed to solve the general boundary value problem governing the viscous elasto-plasticity flows obeying the double-shearing theory. Numerical implementation of the frictional boundary condition is also presented. The model is then illustrated with a numerical example demonstrating the influence of wall friction on the distribution of pressures on silo walls throughout the dynamic process of material discharge from silos.

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A granular material is a collection of discrete, solid particles of macroscopic size dispersed in an interstitial fluid, in which the fluid has an insignificant effect on the particle dynamics. Because they exhibit fascinating properties because of dissipative interactions, due to their importance in geophysical and industrial processes, flows of granular materials have been the focus of large amount of research involving physicists and engineers. A good understanding of the physics of granular materials is desired in order to design efficient processing and handling systems. Granular materials can be heaped like a solid, and can flow like a fluid. Though the two distinct regimes of granular flows are well described by kinetic theory (rapid flows) and plasticity theories (quasi-static), the intermediate dense flow regime, where collisional and frictional interactions are important, is not yet described successfully. In this thesis, we examine the applicability of kinetic theory for dense granular flows, the structure and dynamics in sheared inelastic hard disks systems and dynamics of sheared non-spherical particles. Two complementary simulation techniques, the discrete element (DE) technique for soft particles and the event driven (ED) simulation technique for hard particles, are used to examine the extent to which the dynamics of an unconfined dense granular flow can be well described by a hard particle model when the particle stiffness becomes large. First, we examine the average co-ordination number for the particles in the flow down an inclined plane using the DE technique using both linear and Hertzian contact models. The simulations show that the average co-ordination number decreases below 1 for values of the spring stiffness corresponding to real materials such as sand and glass, even when the angle of inclination is only 1olarger than the angle of repose. The results of the two simulation techniques for the Bagnold coefficients (ratio of stress and square of the strain rate) and the granular temperature (mean square of the fluctuating velocity) are found to be in quantitative agreement. In addition, we also conduct the comparison of the pre-collisional relative velocities of particles in contact. Since momentum is transported primarily by particle contacts in a dense flow, the relative velocity distribution is a sensitive comparison of the dynamics in the two simulation techniques. It is found that the relative velocity distribution in both simulation techniques are well approximated by an exponential distribution for small coefficients of restitution, indicating that the dynamics of a dense granular flow can be adequately described by a hard particle model. The structure and dynamics of the two-dimensional linear shear flow of inelastic disks at high area fractions are analysed. The event-driven simulation technique is used in the hard-particle limit, where the particles interact through instantaneous collisions. The structure (relative arrangement of particles) is analysed using the bond-orientational order parameter. It is found that the shear flow reduces the order in the system, and the order parameter in a shear flow is lower than that in a collection of elastic hard disks at equilibrium. The distribution of relative velocities between colliding particles is analysed. The relative velocity distribution undergoes a transition from a Gaussian distribution for nearly elastic particles, to an exponential distribution at low coefficients of restitution. However, the single-particle distribution function is close to a Gaussian in the dense limit, indicating that correlations between colliding particles have a strong influence on the relative velocity distribution. This results in a much lower dissipation rate than that predicted using the molecular chaos assumption, where the velocities of colliding particles are considered to be uncorrelated. The orientational ordering and dynamical properties of the shear flow of inelastic dumbbells in two dimensions are studied, as a first step towards examining the effect of shape on the properties of flowing granular materials. The dumbbells are smooth fused disks characterised by the ratio of the distance between centers (L) and the disk diameter (D), and the ratio (L/D)varies between 0 and 1 in our simulations. Area fractions studied are in the range 0.1 to 0.7, while coefficients of normal restitution from 0.99 to 0.6 are considered. The simulations are similar to the event driven simulations for circular disks, but the procedure for predicting collisions is much more complicated due to the non-circular shape of the particles and due to particle rotation. The average orientation is measured using an orientational order parameter S, which varies between 0 (for a perfectly disordered fluid) and 1 (for a fluid with the axis of all dumbbells in the same direction). It is found that there is a gradual increase in ordering as the area fraction is increased, as the aspect ratio is increased or as the coefficient of restitution is decreased, and the order parameter has a maximum value of about 0.5 for the highest area fraction and lowest coefficient of restitution considered here. However, there is no discontinuous nematic transition for all the parameters studied here. The axis of the dumbbells are preferentially oriented along the extensional axis (at an angle of 45ofrom the flow direction) at low area fraction, but the orientation is closer to the flow direction as the area fraction is increased. The orientation distribution is calculated, and it is found that the orientation distribution is well described by a function of the form P(θ) =(1/π)+ (2S/π)cos(2(θ−θp)), where θis the angle from the flow direction and θpis the principal orientation direction. The mean energy of the velocity fluctuations in the flow direction is found to be higher than that in the gradient direction and the rotational energy, though the difference decreases as the area fraction increases, due to the efficient collisional transfer of energy between the three directions. The distributions of the translational and rotational velocity are found to be Gaussian distributions to a very good approximation. The equation of state for the pressure is calculated, and it is found to be remarkably independent of the coefficient of restitution. The pressure and dissipation rate show relatively little variation when scaled by the collision frequency for all the area fractions studied here, indicating that the collision frequency determines the momentum transport and energy dissipation even at the lowest area fractions studied here. The mean angular velocity of the particles is examined in some detail. It is found that the mean angular velocity is equal to half the vorticity at low area fractions, but the magnitude of the mean angular velocity systematically decreases to less than half the vorticity as the area fraction is increased, even though the stress tensor is symmetric.

The purpose of this work is to develop a continuum theory that may be used to predict the effects of anisotropic boundary vibrations on loose granular assemblies. In order to do so, we extend statistical averaging techniques employed in the kinetic theory to derive an anisotropic flow theory for rapid, dense flows of identical, inelastic spheres. The theory is anisotropic in the sense that it treats the full second moment of velocity fluctuations, rather than only its isotropic piece, as a mean field to be determined. In this manner, the theory can, for example, predict granular temperatures that are different in different directions. The flow theory consists of balance equations for mass, momentum, and full second moment of velocity fluctuations, as well as constitutive relations for the pressure tensor, the flux of second moment, and the source of second moment. The averaging procedure employed in deriving the constitutive relations is based on a Maxwellian that is perturbed due to the presence of a deviatoric second and full third moment of velocity fluctuations. Because the theory is anisotropic, it can predict the normal stress differences observed in granular shear flows, as well as the evolution to isotropy in an assembly with granular temperatures that are initially highly anisotropic. In order to complement the theory, we employ similar statistical techniques to derive boundary conditions that ensure that the flux of momentum as well as the flux of second moment are balanced at the vibrating boundary. The bumps are hemispheres arranged in regular arrays, and the fluctuating boundary motion is described by an anisotropic Maxwellian distribution function. The bumpiness of the surface may be adjusted by changing the size of the hemispheres, the spacing between the hemispheres in two separate array-directions, and the angle between the two directions. Statistical averaging consistent with the constitutive theory yields the rates at which momentum and full second moment are transferred to the flow. In order to present results in a form that is easy to interpret physically, the statistical parameters that describe the boundary fluctuations are related in a plausible manner to amplitudes and frequencies of sinusoidal vibrations that may differ in three mutually perpendicular directions, and to phase angles that may be adjusted between the three directions of vibration. The focus of the results presented here is on the steady response of unconfined granular assemblies that are thermalized and driven by horizontal bumpy vibrating boundaries. In a first detailed study of the effects of the boundary geometry and boundary motion on the overall response of the assemblies, the anisotropic theory is reduced to a more familiar isotropic form. The resulting theory predicts the manner in which the profiles of isotropic granular temperature and solid volume fraction as well as the uniform velocity and corresponding flow rate vary with spacings between the bumps, angle of the bump-array, energy of vibration, direction of vibration, and phase angles of the vibration. In a second study, we solve the corresponding, but more elaborate, boundary value problem for anisotropic flows induced by anisotropic boundary vibrations. The main focus in presenting these results is on the differences between granular temperatures in three perpendicular directions normal and tangential to the vibrating surface, and how each is affected by the bumpiness of the boundary and the direction of the vibration. In each case, we calculate the corresponding nonuniform velocity profile, solid volume fraction profile, and mass flow rate.

Les vagues de tsunami sont des ondes longues générées par des événements géophysiques impulsifs de la croûte terrestre, de volcans, d’impacts d’astéroïdes et de glissements de terrain. Si la majorité des tsunamis sont d'origine tectonique, l'effondrement en masse d'un relief côtier peut constituer une source importante de l’aléa tsunami. Après une brève introduction sur les différentes générations de tsunamis dans l'histoire et les enjeux de cette thèse, le chapitre 1 présente les principaux résultats bibliographiques sur la génération et la propagation de tsunamis, ainsi qu’un rappel sur l’établissement des équations s’appliquant à l’étude des vagues extrêmes. Le second chapitre est dédié à la présentation des différents codes numériques utilisés dans ce manuscrit, à savoir, Gerris et SPHysics. Le chapitre 3 s'intéresse à la génération de tsunami par l’impact d’un bloc solide. Les résultats expérimentaux sont comparés aux résultats numériques des deux codes. A partir de là, une étude systématique a été faite, menant à des lois d’échelles sur le temps d’arrivée et l’amplitude de la première vague générée. Dans le chapitre 4, les interactions entre le glissement de terrain et la vague générée sont étudiées expérimentalement à l'aide d'impact granulaire initialement sec dans l'eau. Une étude systématiques des différents paramètres met en lumière l'importance des propriétés du glissement sur la vague générée. Enfin, Le chapitre 5 est dédié à l’étude de l’effondrement du Cap Canaille à Cassis. Cette étude numérique utilise un modèle de génération et de propagation simplifié afin d'estimer le potentiel destructeur d'un éventuel effondrement majeur. / Tsunami waves are long waves generated by impulsive geophysical events of earth's crust, volcanoes, asteroids impacts or landslides. Even if most of the tsunamis are generated by submarine earthquakes, the massive collapse of coastal landscape may constitute an important source of tsunami hazard. After introducing historical tsunami events, chapter 1 presents a state-of-the-art on the generation and propagation of tsunami waves and the main equations dealing with extreme water waves. Chapter 2 presents the numerical codes used in this thesis: Gerris and SPHysics. Chapter 3 focuses on the generation of tsunami by a solid landslide. Experimental results are compared to numerical simulations obtained using both codes. From this results, we derive scaling laws on the arrival time and amplitude of the first generated wave. The chapter 4 deals with the interactions between the slide and the generated wave by taking into account the impact of an initially dry granular media into water. Systematic studies varying the different parameters exhibit the significance of the internal properties of the slide on the generated wave. Finally, chapter 5 is dedicated to the collapse of the Cap Canaille near Cassis. A idealized model for the generation and the propagation are used to estimate the hazard associated to such a massive collapse.

The purpose of this work is to study the effects of boundaries on granular flows down vibrating inclines, on segregation in granular mixtures induced by boundary vibrations, and on flows of granular mixtures through vibrating sieves. In each case, we employ techniques borrowed from the kinetic theory to derive an appropriate set of boundary conditions, and combine them with existing flow theories to calculate the profiles of solid volume fraction, mean velocity, and granular temperature throughout the flows. The boundaries vibrate with full three-dimensional anisotropy in a manner that can be related to their amplitudes, frequencies, and phase angles in three independent directions. At impenetrable surfaces (such as those on the inclines), the conditions derived ensure that momentum and energy are each balanced at the boundary. At penetrable surfaces (such as sieves), the conditions also ensure that mass is balanced at the boundary. In these cases, the momentum and energy balances also are modified to account for particle transport through the boundary. Particular interest in all the applications considered here is in how the details of the boundary geometry and the nature of its vibratory motion affect the resulting flows. In one case, we derive conditions that apply to a monosized granular material that interacts with a bumpy, vibrating, impenetrable boundary, and predict how such boundaries affect steady, fully developed unconfined inclined flows. Results indicate that the flows can be significantly enhanced by increasing the total energy of vibration and are more effectively enhanced by normal vibration than by tangential vibration. Regardless of the direction of vibration, the bumpiness of the boundary has a profound effect on the flows. In a second case, we derive conditions that apply to a binary granular mixture that interacts with a flat, vibrating, penetrable sieve-like boundary, and predict how such boundaries affect the process in which the particles pass through the sieve. In the special case in which the particles are all the same size, the results make clear that energy is more effectively transmitted to the assemblies when either the total vibrational energy or the normal component of the vibrational energy is increased, but that an increase in the energy transferred to the material can sometimes actually decrease the flow rates through the sieve. Consequently, at any instant of time in the sieving process, there is an optimum level of vibrational energy that will maximize the flow rate. For the sieving of binary granular assemblies, the physics associated with the effects of energy transfer on the flow rates still applies. However, in these cases, the flows through the sieve are also profoundly affected by segregation that occurs while the particles reside on sieve before the pass through. For this reason, we also isolate the segregation process from the sieving process by considering the special case in which the holes in the vibrating sieve are too small to allow any particles to pass through. In this case, the results show that under most circumstances the region immediately adjacent to the vibrating surface will be populated almost entirely by the smaller particles or by the more dissipative particles if there is no size disparity, and that the reverse is true in a second region above the first.

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.

Granular materials are composed of solid, discrete particles and exhibit mechanical properties that range from fluid to solid behavior. Some of the complexity exhibited by granular systems arises due to the long-range order that develops due to particle-particle contact. Inter-particle forces in granular materials often form a distributive network of filamentary force-accommodating chains (i.e. force chains), such that a fraction of the total number of particles accommodates the majority of the forces in the system. The force chain network inherent to a system composed of granular materials controls the macroscopic behavior of the granular material. Force transmission by these filamentary chains is focused (or localized) to the grain scale at boundaries such as the granular flow substrate. Recent laboratory experiments have shown that force chains transmit extreme localized forces to the substrates of free surface granular flows. In this work we combine analog and numeric experimental approaches to investigate the forces at the bed of a simplified granular flow. A photoelastic experimental approach is used to resolve discrete forces in the granular flows. We also conduct discrete element method (DEM) simulations, using input parameters derived from measureable physical material properties of experimental and natural materials, which successfully reproduce the analog experimental results. This work suggests that force chain activity may play an unexpected and important role in the bed physics of dense granular flows through substrate modification by erosion and entrainment, and that DEM numerical methods effectively treat force chain processes in simulated granular flows.

Dense flows of grains are commonplace throughout natural and industrial environments, from snow-avalanches down the sides of mountains to flows of cereal down chutes as it is transported from one part of a factory to another. A ubiquitous feature in all of these flows is their ability to separate the different grain types when shaken, stirred, sheared or vibrated. Many flows are sheared through gravity and these flows are particularly efficient at segregating particles based on their size, with small particles percolating to the bottom of the flow and large particles collecting at the top. Within this mechanism, an asymmetry between the large and small particles has been observed, with small particles percolating downwards through many large particles at a faster rate than large particles rise upwards through many small particles. This alternative format thesis presents a revised continuum model for segregation of a bidisperse mixture that can account for this asymmetry. A general class of asymmetric segregation flux functions is introduced that gives rise to asymmetric velocities between the large and small grains. Exact solutions for segregation down an inclined chute, with homogenous and normally graded inflow conditions, show that the asymmetry can significantly enhance the distance for complete segregation. Experiments performed using a classical shear-box with refractive index matched scanning are able to quantify the asymmetry between large and small particles on both bulk and particle scales. The dynamics of a single small particle indicate that it not only falls down faster than a single large particle rises, but that it also exhibits a step-like motion compared to the smooth ascent of the large grain. This points towards an underlying asymmetry between the different sized constituents. The relationship between the segregation-time and the volume fraction of small grains is analysed, and solutions presented for the steady-state balance between segregation and diffusive remixing. These help to show the good agreement between the asymmetric model and experimental data. Segregation at the front of natural avalanches produces a recirculation zone, known as a `breaking size-segregation wave', in which large particles are initially segregated upwards, sheared towards the front of the flow, and overrun before being resegregated again. Solutions for the structure of this recirculation zone are derived using the asymmetric flux model, revealing a novel `lens-tail' structure. Critically, it is seen that a few large particles starting close to the bottom of the flow are swept a long way upstream and take a very long time to recirculate. The breaking size-segregation waves highlight the important interplay between segregation and the bulk velocity field. The properties of flowing monodisperse grains are explored through experiments on a cone that produce a beautiful radial fingering pattern. Equations developed in a conical coordinate system reproduce the measured linear relationship between fingering radius and initial flux, whilst also predicting the slowing and thinning dynamics of the flow. Overall, these results illustrate the complex nature of the granular rheology and provide perspectives for future modelling of segregation in dense granular flows.

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