Renormalization analysis of continuum models on fractal domains

This thesis builds on the recently begun extension of continuum thermomechanics to fractal media which are specified by a fractional mass scaling law of the resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through renormalization analysis into continuum models, in which the fractal dimension D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D and R, as well as a surface fractal dimension d. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media - the new model is based on a product measure capable of describing local material anisotropy. This measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress. It allows a specification of geometry configuration of the continua by ‘fractal metric’ coefficients, on which the continuum mechanics is subsequently built. Finally, the reciprocity, uniqueness and variational theorems are established for development of approximate numerical solutions.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2012

Author dc:creator

• Li, Jun

Contributors dc:contributor

• Ostoja-Starzewski, Martin

• fractal
• fractional calculus
• product measure
• renormalization
• micropolar continuum
• reciprocal
• uniqueness and variational theorems

Statement dc:rights

• Copyright 2012 Jun Li

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/34287

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/34287