University of Illinois Urbana-Champaign

Renormalization analysis of continuum models on fractal domains

Li, Jun

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Li_Jun.pdf

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

Description

Title
Renormalization analysis of continuum models on fractal domains
Author(s)
Li, Jun
Issue Date
2012-09-18T21:09:39Z
Director of Research (if dissertation) or Advisor (if thesis)
Ostoja-Starzewski, Martin
Committee Member(s)
Ostoja-Starzewski, Martin
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
M.S.
Degree Level
Thesis
Date of Ingest
2012-09-18T21:09:39Z
Keyword(s)
  • fractal
  • fractional calculus
  • product measure
  • renormalization
  • micropolar continuum
  • reciprocal
  • uniqueness and variational theorems
Abstract
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.
Graduation Semester
2012-08
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http://hdl.handle.net/2142/34287
Copyright and License Information
Copyright 2012 Jun Li

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