Fractals in mechanics of materials

Fractal concepts have been used in geometric characterizations as well as models of various material microstructures and deformation patterns. The first part of this dissertation studies fractal patterns of plastic regions observed in elastoplastic deformations. As a paradigm, we focus on a random material model with microscale randomness in material properties. When it is subjected to increasing macroscopically uniform loadings, plasticized grains form fractal patterns gradually filling the entire material domain and the sharp kink in the conventional stress-strain curve is replaced by a smooth one. Parametric studies are performed to investigate qualitative influences of material constants or randomness on the elastic-plastic transitions. Following scaling analysis in phase transition theory, we recognize three order parameters in terms of stress-strain, fractal dimension, and plastic volume fraction, which, for the first time, are quantitatively related through proposed scaling functions. A broad range of materials are studied, especially the widely used von Mises models for metals and Mohr-Coulomb models for rocks and soils. Polycrystals and thermo-elasto-plastic materials are also investigated. The fractal character of many porous materials motivates the second part of this dissertation: theoretical modeling of fractally microstructured materials. Using dimensional regularization techniques, a fractional integral is introduced to reflect the mass scaling on fractals. We propose a product measure consistent with generally anisotropic fractals and also simplify previous formulations from decoupling of coordinate variables. Two continuum models are developed – the classical continuum and the micropolar continuum – whereby a consistency of mechanical with variational approaches verifies our formulations. Also, some elastodynamic problems are studied. Finally, we conduct two application case studies: Saturn’s rings and bone microstructures. Their fractal dimensions are measured from public NASA images and our micro-computed tomography (Micro-CT) images, respectively. The values indicate important invariable properties.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2013

Author dc:creator

• Li, Jun

Contributors dc:contributor

• Hubler, Alfred W.
• Ostoja-Starzewski, Martin
• Jasiuk, Iwona M.
• DeVille, Robert E.
• Dahmen, Karin A.

• Fractal
• Fractal dimension
• Elastic-plastic transition
• Random materials
• Scaling function
• Shear band
• Fractional calculus
• Micropolar continuum mechanics

Statement dc:rights

• Copyright 2012 Jun Li

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/42173

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/42173