This article advances continuum-type mechanics of porous media having a generally anisotropic, product-like fractal geometry. Relying on a fractal derivative, the approach leads to global balance laws in terms of fractal integrals based on product measures and, then, converting them to integer-order integrals in conventional (Euclidean) space. Proposed is a new line transformation coefficient that is frame invariant, has no bias with respect to the coordinate origin and captures the differences between two fractal media having the same fractal dimension but different density distributions. A continuum localization procedure then allows the development of local balance laws of fractal media: conservation of mass, microinertia, linear momentum, angular momentum and energy, as well as the second law of thermodynamics. The product measure formulation, together with the angular momentum balance, directly leads to a generally asymmetric Cauchy stress and, hence, to a micropolar (rather than classical) mechanics of fractal media. The resulting micropolar model allowing for conservative and/or dissipative effects is applied to diffusion in fractal thermoelastic media. First, a mechanical formulation of Fick's Law in fractal media is given. Then, a complete system of equations governing displacement, microrotation, temperature and concentration fields is developed. As a special case, an isothermal model is worked out. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Original languageEnglish (US)
Number of pages1
JournalPhilosophical transactions. Series A, Mathematical, physical, and engineering sciences
Issue number2172
StatePublished - May 29 2020


  • diffusion
  • fractal derivative
  • homogenization
  • thermomechanics

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

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