TY - THES
AU - DeGiuli, Eric
PY - 2012
TI - Continuum limits of granular systems
KW - Thesis/Dissertation
LA - eng
M3 - Text
AB - Despite a century of study, the macroscopic behaviour of quasistatic granular materials remains poorly understood. In particular, we lack a fundamental system of continuum equations, comparable to the Navier-Stokes equations for a Newtonian fluid. In this thesis, we derive continuum models for two-dimensional granular materials directly from the grain scale, using tools of discrete calculus, which we develop.
To make this objective precise, we pose the canonical isostatic problem: a marginally stable granular material in the plane has 4 components of the stress tensor σ, but only 3 continuum equations in Newton’s laws ∇ ‧σ = 0 and σ = σT. At isostaticity, there is a missing stress-geometry equation, arising from Newton’s laws at the grain scale, which is not present in their conventional continuum form.
We first show that a discrete potential ψ can be defined such that the stress tensor is written as σ = ∇ × ∇ × ψ, where the derivatives are given an exact meaning at the grain scale, and converge to their continuum counterpart in an appropriate limit. The introduction of ψ allows us to understand how force and torque balance couple neighbouring grains, and thus to understand where the stress-geometry equation is hidden.
Using this formulation, we derive the missing stress-geometry equation ∆(F^ : ∇∇ψ) = 0, introducing a fabric tensor F^ which characterizes the geometry. We show that the equation imposes granularity in a literal sense, and that on a homo- geneous fabric, the equation reduces to a particular form of anisotropic elasticity.
We then discuss the deformation of rigid granular materials, and derive the mean-field phase diagram for quasistatic flow. We find that isostatic states are fluid states, existing between solid and gaseous phases. The appearance of iso- staticity is linked to the saturation of steric exclusion and Coulomb inequalities.
Finally, we present a model for the fluctuations of contact forces using tools of statistical mechanics. We find that force chains, the filamentary networks of con- tact forces ubiquitously observed in experiments, arise from an entropic instability which favours localization of contact forces.
N2 - Despite a century of study, the macroscopic behaviour of quasistatic granular materials remains poorly understood. In particular, we lack a fundamental system of continuum equations, comparable to the Navier-Stokes equations for a Newtonian fluid. In this thesis, we derive continuum models for two-dimensional granular materials directly from the grain scale, using tools of discrete calculus, which we develop.
To make this objective precise, we pose the canonical isostatic problem: a marginally stable granular material in the plane has 4 components of the stress tensor σ, but only 3 continuum equations in Newton’s laws ∇ ‧σ = 0 and σ = σT. At isostaticity, there is a missing stress-geometry equation, arising from Newton’s laws at the grain scale, which is not present in their conventional continuum form.
We first show that a discrete potential ψ can be defined such that the stress tensor is written as σ = ∇ × ∇ × ψ, where the derivatives are given an exact meaning at the grain scale, and converge to their continuum counterpart in an appropriate limit. The introduction of ψ allows us to understand how force and torque balance couple neighbouring grains, and thus to understand where the stress-geometry equation is hidden.
Using this formulation, we derive the missing stress-geometry equation ∆(F^ : ∇∇ψ) = 0, introducing a fabric tensor F^ which characterizes the geometry. We show that the equation imposes granularity in a literal sense, and that on a homo- geneous fabric, the equation reduces to a particular form of anisotropic elasticity.
We then discuss the deformation of rigid granular materials, and derive the mean-field phase diagram for quasistatic flow. We find that isostatic states are fluid states, existing between solid and gaseous phases. The appearance of iso- staticity is linked to the saturation of steric exclusion and Coulomb inequalities.
Finally, we present a model for the fluctuations of contact forces using tools of statistical mechanics. We find that force chains, the filamentary networks of con- tact forces ubiquitously observed in experiments, arise from an entropic instability which favours localization of contact forces.
UR - https://open.library.ubc.ca/collections/24/items/1.0071936
ER - End of Reference