Date of Award
Spring 2023
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mechanical Engineering & Materials Science (ENAS)
First Advisor
O'Hern, Corey
Abstract
Granular materials have fascinating mechanical properties near the onset of jamming. Predicting the bulk mechanical properties of granular materials has been done by applying discrete-element method (DEM) simulations to subsystems with finite system sizes and calculating the ensemble-averaged properties of the subsystems. Many studies using DEM simulations have been done for jammed packings of spherical particles under periodic boundary conditions. However, the most common form of granular materials in the nature and used by industries is non-spherical and in the real world, periodic boundary conditions are not the most representative boundary condition. There have been far less study on jammed packings of non-spherical particles and hardly any with boundary conditions other than periodic boundaries. Therefore, in this thesis, we extend the simple model of soft spherical particles in periodic boundaries by introducing non-spherical particle shapes and physical boundaries and use DEM simulations to study the mechanical properties in such systems. Particularly, we study the pressure-dependent shear modulus of jammed packings of soft grains with macroscopic grain sizes interacting with linear spring potential. The ensemble averaged shear modulus is obtained by sampling the phase space of subsystems usingmolecular dynamics (MD) simulations and averaging the properties of all subsystems. In the first project, we study the pressure-dependent shear modulus of two dimensional (2D) jammed packings of purely repulsive frictionless particles for both disks and circulo-lines. We find that the shape factor plays an important role in the power scaling law that governs the mechanical properties with pressure. For disks, the shear modulus (G) of jammed disk packings decreases linearly with pressure, p, Gf = G0– A p, within geometric families, where the contact network of the packing does not change. In contrast, for circulo-lines, the shear modulus of geometrical families can increase, as well as decrease with pressure: Gf = G0 ± A′p. We derive a formula based on the principle of energy conservation, which relates the sign of A’ to dilation/compression of the system when sheared. We find that while disk packings can only undergo compression when sheared, circulo-lines can undergo either compression or dilation when sheared. Therefore, G only decreases with p for disks while G can increase or decrease with p for circulo-lines. The ensemble-averaged shear modulus does not have the same scaling as the geometric family because of rearrangements. We develop a statistical framework that explains the behavior of ensemble-averaged G from two contributions: 1) geometric families; 2) rearrangements. We show that the ensemble-averaged shear modulus increases as < G > −G0 ∼ pα, where α ∼ 0.5 for disks while α ∼ 0.8 for circulo-lines. We explain the difference in the scaling exponent as a result of both different statistics of rearrangements and the different statistics of increasing and decreasing families. In the second project, we study the mechanical properties of granular metamaterials, which are systems composed of soft grains bounded by physical boundaries. To construct such systems, we construct compartmented granular materials by joining multiple particle-filled compartments together. We focus on compartments that each have four walls and contain a small number of bidisperse disks in two dimensions. We first study the mechanical properties of individual disk-filled compartments with three types of boundaries: periodic boundary conditions, fixed-length physical walls, and flexible physical walls. Hypostatic jammed packings are found for disk-filled compartments with flexible walls, but not in compartments with periodic boundary conditions and fixed-length physical walls, and they are stabilized by quartic modes of the dynamical matrix. The shear modulus of a single compartment depends linearly on p, Gc(θ) = Gc0(θ) + λc(θ)p, where θ is the angle at which the simple shear strain is applied. We find that λc(θ) < 0 for all packings in single compartments with periodic boundary conditions with N ≥ 6. In contrast, single compartments with both fixed-length and flexible walls can possess λc(θ) > 0 for N ≤ 16. We show that we can lock-in the mechanical properties of single compartments in multi-compartment granular materials by constraining the endpoints of the outer walls of the system and enforcing affine response. These studies demonstrate that compartmented granular materials provide a novel platform for the design of soft materials with specified mechanical properties. In particular, we show that we can design compartmented granular materials with a shear modulus that decreases with pressure over more than four orders of magnitude in the large number of compartments limit. In conclusion, we have developed a theoretical framework that predicts the bulk properties from its subsystems for granular materials and granular materials. We direct the future work towards the study of pressure dependent shear modulus of jammed packings of more generous shapes such as circulo-polygons, spherocylinders, and ellipsoids, the study of systems with frictional forces and composite systems where the elastic segment are connected in different ways such as a ring. The ultimate goal is to develop a generalized theory that predicts the bulk property from its subsystems in a wide range of granular materials and granular metamaterials.
Recommended Citation
Zhang, Liheng, "Computational Studies of the Mechanical Properties of Granular Packings and Granular Metamaterials" (2023). Yale Graduate School of Arts and Sciences Dissertations. 982.
https://elischolar.library.yale.edu/gsas_dissertations/982
