Deformation of Two-Dimensional Amorphous Granular Packings

A microscopic understanding of how amorphous materials deform in response to an imposed disturbance is lacking. In this thesis, the connection between local structure and the observed dynamics is explored experimentally in a disordered granular pillar subjected to a quasi-static deformation. The pillar is composed of a single layer of grains, allowing for easy visualization of all particles throughout the deformation. The addition of a liquid into the system causes capillary bridges form between the grains, making the grains cohesive. The two-dimensionality of the system ensures that the liquid is distributed uniformly throughout the packing, making the cohesive forces between the grains known everywhere. We perform separate experiments to measure these capillarity-induced forces, and we find these measurements to be in excellent agreement with our theoretical model and numerical calculations. In the main experiments presented in this thesis, we explore the quasi-static deformation of granular pillar subjected to uniaxial compression. We find a statistical correlation between the local dynamics, characterized by the deviatoric strain rate, and the local structure, characterized by a new measure, introduced here, akin to a relative free area. This correlation is stronger in the presence of cohesion and indicates that regions that are more (less) well packed relative to their surroundings experience lower (higher) strain rates than their surroundings. The deviatoric strain rate also highlights shear bands within the deforming pillar. These shear bands are transient, moving around as the compression occurs. We have developed a way to identify these extended bands, and we compare the structure within these bands to the structure outside. Preliminary results suggest that these shear bands coincide with paths through the material that tend to have more underpacked regions than other parallel in the vicinity of the shear band

Infinite dimensional moment problem: open questions and applications

Infinite dimensional moment problems have a long history in diverse applied areas dealing with the analysis of complex systems but progress is hindered by the lack of a general understanding of the mathematical structure behind them. Therefore, such problems have recently got great attention in real algebraic geometry also because of their deep connection to the finite dimensional case. In particular, our most recent collaboration with Murray Marshall and Mehdi Ghasemi about the infinite dimensional moment problem on symmetric algebras of locally convex spaces revealed intriguing questions and relations between real algebraic geometry, functional and harmonic analysis. Motivated by this promising interaction, the principal goal of this paper is to identify the main current challenges in the theory of the infinite dimensional moment problem and to highlight their impact in applied areas. The last advances achieved in this emerging field and briefly reviewed throughout this paper led us to several open questions which we outline here.Comment: 14 pages, minor revisions according to referee's comments, updated reference

How to obtain a lattice basis from a discrete projected space

International audienceEuclidean spaces of dimension n are characterized in discrete spaces by the choice of lattices. The goal of this paper is to provide a simple algorithm finding a lattice onto subspaces of lower dimensions onto which these discrete spaces are projected. This first obtained by depicting a tile in a space of dimension n -- 1 when starting from an hypercubic grid in dimension n. Iterating this process across dimensions gives the final result