Date of Award

Summer 2016

Document Type


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Lou Kondic

Second Advisor

Konstantin Michael Mischaikow

Third Advisor

Guillaume Bal

Fourth Advisor

Denis L. Blackmore

Fifth Advisor

Richard O. Moore


Granular materials are collections of objects ranging from sand grains that form sand piles or even sand castles to collections of large objects such as a group of meteors in outer space. The considered range of sizes of granular particles is such that the effect of thermal fluctuations is not relevant. However, the interaction between the particles may be very complex, involving inelasticity and friction, in addition to repulsive and possibly attractive interaction forces. These interactions that may be history dependent, make the systems that consist of a large number of particles complex to analyze and difficult to understand using analytical methods. For this reason, most of the work in the field of granular mater, including the main part of this Thesis, is carried out using discrete element/molecular dynamics type simulations.

At the beginning of this work, the energy propagation is considered in a stochastic granular chain in one spatial dimension (1D). The main finding here is that the properties of the stochastic noise influences strongly the process of energy propagation. As it is shown, the issue of the importance of order and randomness remains significant as other aspects of dense granular systems are considered.

Next, the various aspects of 2D and 3D granular systems are discussed, with the focus on a dense regime, where the particles are in almost continuous contact. One important property of the considered systems is the presence of force networks, that describe how interactions between the particles are organized spatially, and how they evolve in time. These mesoscale structures are known to be related on one hand to the microscopic properties relevant on the particle scale, and on the other hand to the global properties of considered systems as a whole.

Consideration of dense granular systems using the tools of percolation theory illustrates the complex process by which these systems go through percolation and jamming transitions when exposed to compression. One significant finding is that these two transitions may coincide or not, depending on the properties of the granular particles. Furthermore, there is an important influence of the force threshold considered, tracing back to the properties of underlying force networks. These networks are analyzed by considering their scaling properties with respect to the system size. Contrary to the published results, it is found that the properties of these networks are not universal: in particular, the force networks that form in the systems comprising frictionless particles are found to belong to a different universality class.

Another approach to analysis of force networks involves consideration of topological measures. In this direction, a novel study involving direct comparison of computational results is carried out analyzing experimental data using the tools of persistence homology, and in particular Betti numbers, that allow to quantify the properties of the force networks, and make comparisons directly between experiments and simulations. This comparison is important in order to identify additional features that have to be included in simulations to allow for meaningful comparison with experiments.

Finally, the influence of the nature of particle interaction on the properties of the system as a whole is considered in more detail. In one direction, the systems are considered to consist of particles that interact by either purely repulsive, or by both repulsive and attractive interactions. It is found that additional attractive interaction (that may be due to cohesive effects in wet granular systems) play an important role in determining the source of energy loss in sheared systems. In another direction, the computational results are extended to 3D, and the connection of modeling methods to the measures describing the system as a whole is discussed.

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Mathematics Commons