ContinuumHomogenization of fractal media

Martin Ostoja-Starzewski, Jun Li, Paul N. Demmie

Research output: Chapter in Book/Report/Conference proceedingChapter


This chapter reviews the modeling of fractal materials by homogenized continuum mechanics using calculus in non-integer dimensional spaces. The approach relies on expressing the global balance laws in terms of fractional integrals and, then, converting them to integer-order integrals in conventional (Euclidean) space. Via localization, this allows development of local balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and, in case of elastic responses, formulation of wave equations in several settings (1D and 3D wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). Next, follows an account of extremum and variational principles, and fracture mechanics. 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.

Original languageEnglish (US)
Title of host publicationHandbook of Nonlocal Continuum Mechanics for Materials and Structures
Number of pages31
ISBN (Electronic)9783319587295
ISBN (Print)9783319587271
StatePublished - Jan 10 2019


  • Balance laws
  • Fractal
  • Fractal derivative
  • Fractional calculus
  • Homogenization

ASJC Scopus subject areas

  • General Engineering
  • General Materials Science
  • General Mathematics


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