| Granular matter is a discontinuous and complex dissipative form of matter,comprising a large number of particles.Because of its unique behavior,it is considered a fourth state of matter,different from solids,liquids or gas.Granular piles can remain solid at rest,they flow like a fluid when the yield stress on them is exceeded,and they behave like a gas when highly agitated.Consequently,the conventional theories of classical Newtonian mechanics and fluid mechanics are not suitable for the mechanical analysis of granular matter.This makes the currently open problem of rheology of particulate materials is a very challenging.The discrete element method(DEM)has emerged in recent years as a powerful technique to model such systems.This technique makes it possible to construct numerical experiments on relatively large-scale problems of millions of particulates,using the basic small-scale Newtonian dynamics.In this thesis,I use the DEM to simulate and understand the governing mechanisms which desert lizards,modelled as simple selfenergised objects,employ for moving within sand.I investigate the interaction between the granular matter and self-energised intruders as a function of a number of variables,including material parameters,and motion mode,and construct theoretical models to explain the observations.In particular,it is proposed that there is a competition between two mechanisms: climbing over jammed matter,which the intruder pushes,and sinking because of fluidisation of the layer supporting the intruder.A full theoretical model is then constructed,relating the vertical and the horizontal locomotion of the intruder.The thesis is structured as follows.In the first part,I introduce the DEM for granular media,including the theoretical and technical detailed involved.These include the contact detection,contact force calculation,boundary conditions,time step,motion state updating algorithm,etc.The Hertz-Mindlin and soft sphere models,which are at the core of the method,are described the effects of the material parameters of the intruder and bed particles are systematically studied and the optimal parameters for the simulation are determined;In the second part,a range of three-dimensional simulations is carried out in order to understand the interaction of the active intruder with the particulate medium.The intruder oscillates horizontally in sinusoidal mode,which causes it to either rise against gravity or sink,depending on the oscillation amplitude and frequency.A systematic study of the effects of the friction coefficient,size,density,and immersed depth on intruder’s vertical motion is carried out at this stage.The optimisation of the involved parameters of these dynamics is useful for the design and development of desert ‘walking’robots.In the third part,a theoretical model is constructed to describe the physical mechanisms that govern the rising and sinking mechanisms.A competition between two mechanisms drives the vertical motion: below a critical speed,a jammed stagnant zone is generated ahead of the intruder,generating an upward force that causes rising.At speeds above the critical value,the excess kinetic energy fluidises the medium around the intruder,causing the intruder to sink into the bed-layer supporting it.The duration of the rising and sinking phases depend non-trivially on the motion amplitude and frequency,as well as on the intruder’s and bed material parameters,leading to an intricate and rich nonlinear dynamics.An equation of motion is derived from first-principles for the timedependent depth and its solutions are shown to agree well with a wide range of computer simulations,which are performed within the range of parameters allowed by the finiteness of the simulated system.In the fourth part,numerical studies are presented of the forces experienced by an intruder moving horizontally in a granular medium in the presence of gravity.In particular,the drag and lift forces are measured.A discussion is presented of the origin of the lift force and its relation to the stress distribution,which is measured in the simulations.The effects of the drag speed,penetration depth,and shape on the lift and drag forces are studied for both partially-and fully-immersed intruders.Based on these results,a semi-empirical model of drag and lift force is established.In the last part,the self-excitation model is extended from simple horizontal oscillation by including object rotation.The DEM simulations provide clear evidence of the sensitivity of the motion dynamics to rotations.The vertical motion is found to be sensitive to the friction between the intruder and the medium,as well as to the rotational angular velocity.An extrusion and friction mechanisms are proposed,whose combination describes well the vertical locomotion.Different rotational modes have been explored: a continuous rotation,as well as synchronised oscillatory rotation and horizontal motions.The physical mechanisms governing the vertical meandering are analyzed and it is shown that large oscillations are caused by an asynchronous change between the directions of horizontal oscillation and rotation.The intruder’s rising rate is observed to be sensitive to three parameters: the oscillation amplitude,the oscillation frequency,and the rotation angular velocity.Theoretical explanations of these effects are provided.This particular study paves the way to designing efficient bio-inspired robots moving in granular terranes. |
