Continuum Homogenization of Fractal Media

  • Martin Ostoja-StarzewskiEmail author
  • Jun Li
  • Paul N. Demmie
Reference work entry


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.


Balance laws Fractal Fractional calculus Fractal derivative Homogenization 



This work was made possible by the support from NSF (grant CMMI-1462749).


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Martin Ostoja-Starzewski
    • 1
    Email author
  • Jun Li
    • 2
  • Paul N. Demmie
    • 3
  1. 1.Department of Mechanical Science and Engineering, Institute for Condensed Matter Theory and Beckman InstituteUniversity of Illinois at Urbana–ChampaignUrbanaUSA
  2. 2.Department of Mechanical EngineeringUniversity of MassachusettsDartmouthUSA
  3. 3.Sandia National LaboratoriesAlbuquerqueUSA

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