From fractal media to continuum mechanics

M. Ostoja-Starzewski, J. Li, H. Joumaa, P. N. Demmie

Research output: Contribution to journalArticle

Abstract

This paper presents an overview of modeling fractal media by continuum mechanics using the method of dimensional regularization. The basis of this method is to express the balance laws for fractal media in terms of fractional integrals and, then, convert them to integer-order integrals in conventional (Euclidean) space. Following an account of this method, we develop balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and discuss wave equations in several settings (1d and 3d wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). We then discuss extremum and variational principles, fracture mechanics, and equations of turbulent flow in fractal media. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers. We also point out relations and potential extensions of dimensional regularization to other models of microscopically heterogeneous physical systems.

Original languageEnglish (US)
Pages (from-to)373-401
Number of pages29
JournalZAMM Zeitschrift fur Angewandte Mathematik und Mechanik
Volume94
Issue number5
DOIs
StatePublished - May 2014

Keywords

  • Balance laws
  • Continuum mechanics
  • Dimensional regularization
  • Fractals

ASJC Scopus subject areas

  • Computational Mechanics
  • Applied Mathematics

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