Micropolar continuum mechanics of fractal media

Research output: Contribution to journalArticlepeer-review


This paper 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 a technique based on a dimensional regularization, 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. This naturally leads to micropolar continuum mechanics of fractal media. Thereafter, the reciprocity, uniqueness and variational theorems are established.

Original languageEnglish (US)
Pages (from-to)1302-1310
Number of pages9
JournalInternational Journal of Engineering Science
Issue number12
StatePublished - Dec 2011


  • Anisotropy
  • Dimensional regularization
  • Fractal
  • Fractional integrals
  • Micropolar continuum
  • Product measure
  • Reciprocal
  • Uniqueness theorem
  • Variational theorems

ASJC Scopus subject areas

  • General Materials Science
  • General Engineering
  • Mechanics of Materials
  • Mechanical Engineering


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