Mathematical modelling and numerical simulations of various granular dynamical problems

The absence of a general theory that describes the dynamical behaviour of the particulate materials makes the numerical simulations the most current powerful tool that can grasp many mechanical problems relevant to the granular materials. It provides a potential window and useful insight to answer massive questions related to the multi-state dynamical granular systems, including the prediction of internal particles behaviour within the particulate systems. In this thesis, based on a two-dimensional soft particle discrete element method (DEM), a numerical approach was developed to study three different granular dynamical problems, namely, the pulsation stress nature and the stress distribution on hopper wall during filling of granular solids into silos and discharging of granular solids from silos, the penetration of projectiles vertically into granular beds, and the stress distribution underneath granular piles. Moreover, an analytical approach based on the classical continuum mechanical theory was constructed to study the stress distributions at the base of a two-dimensional sand pile in search of the dip vertical stress directly beneath the apex of the sand pile.The granular material is assumed to be an assembly of visco-elastic discs and the motion of the particulate medium is governed by a set of nonlinear first order ordinary differential equations. This system is then solved numerically using the centred finite difference scheme. Based on the presented model, two-dimensional soft particle discrete element simulations have been developed and used to study different features of granular materials during filling and discharging processes in a silo system. The focus is on the stress pulsation mechanism developed on hopper wall and the influence of the wall and internal friction coefficients on the distribution of stresses on hopper wall throughout the static and dynamic processes of material filling and discharging from a silo.The vertical penetrations of projectiles into granular beds are simulated using a two-dimensional soft particle discrete element model to study the influence of various factors such as the impact velocity, lateral and vertical confinements, and the particle size on the projectile penetration distance. Moreover, we study the scaling of the projectile penetration distance with its impact velocity.Finally, to study the normal stress distributions along the base of a two dimensional granular pile, two different mathematical approaches are used for the investigation, including numerical simulations base on a two-dimensional soft particle discrete element model and analytical study based on the classical continuum theory. The influences of particle size and pile height on the normal force distributions beneath granular piles were investigated. Simulations results show the normal force distributions under granular pile that constructed from localised source exhibit a local minimum at the centre of the base under the pile vertex. For the continuum models, the linear elastic theory and plastic theory have been used. These continuum models include an elastic sand pile model and an elasto-plastic sand pile model with an inner plastic region and an outer elastic region. The elasto-plastic model exhibits the “M” shape stress distribution where the location of the local minimum of the vertical stress acting on a horizontal section of the sand pile is at centre