Statistical Distribution of Forces in Random Packings of Spheres and Honeycomb Structures

This dissertation, "Statistical Distribution of Forces in Random Packings of Spheres and Honeycomb Structures" by Shu-hei, Chan, 陳樹禧, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract Abstract of thesis entitled: Statistical Distribution of Forces in Random Packings of Spheres and Honeycomb Structures Submitted by: Chan Shu Hei for the degree of Master of Philosophy at the University of Hong Kong in September 2004 Many engineering materials used in structural applications have random structures such as open cell foam materials and granular packings. An understanding of how these internal forces scatter is necessary to enable the mechanical failure of these structures to be predicted. The main aim of this study was to perform experiments to investigate the statistical distribution of internal forces in three-dimensional packings of deformable spheres and two-dimensional elastic random honeycomb structures under external loadings. The experimental observations were compared to the predictions from a statistical-mechanics-based theory in which entropy functional was used to represent the structural randomness. A new method for measuring probability distributions of contact forces in packings of deformable spheres was introduced. Polystyrene balls were placed in a large acrylic tank, the inner walls of which were lined with a type of pink, translucent plastic sheets. The plastic sheets possess a special internal reflection property when illuminated by red light. When the sheet is pressed in a normal direction by a polystyrene ball, it exhibits a bright patch with size very sensitive to the pressure. By employing this method, the deformation areas of forces down to 0.3N could be detected. After filling the tank with the balls, a wooden board loaded with deadweights was placed on the top of the balls. A digital camera was used to image the patches on the sides of the tank and the areas of the patches were measured. The effects of three factors (applied pressure, crystallinity of the packing, and grain size) were investigated. The force distribution was found to be independent of the sphere size and the applied load, but was a strong function of the packing structure. When the packing structure changed from random to a hexagonal-close-packed morphology, the force distribution function changed from a monotonically decreasing form to a peaked form. The force distributions in both cases could be described accurately by the family of theoretical curves. In another set of experiments, the probability force distributions of random honeycomb structures were measured. A series of elastic honeycombs were tailor-made by rapid prototyping technology. Mechanical straining experiments were then carried out in a purpose-built apparatus for both uni-axial and 2-D hydrostatic tensile loadings. The displacements of the nodes between struts were measured by assessing the digital photographs with image analyzing software. Axial force and the shear force in each strut could be determined. To verify the experimental measurement methods, finite-element simulations were conducted. The probability distribution of a regular structure under hydrostatic loading was narrowly peaked at the mean force value. In contrast, in the case of uni-axial tension, the probability distribution function of the axial forces was twin-peaked. The force distributions of random structure spread much wider than those of regular structure. The finite-element simulation results of a solid model under uni-axial loading illustrated ve