Continuum Homogenization of Fractal Media
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.
KeywordsBalance laws Fractal Fractional calculus Fractal derivative Homogenization
This work was made possible by the support from NSF (grant CMMI-1462749).
- A.S. Balankin, O. Susarrey, C.A. Mora Santos, J. Patíno, A. Yogues, E.I. García, Stress concentration and size effect in fracture of notched heterogeneous material. Phys. Rev. E 83, 015101(R) (2011)Google Scholar
- J. Li, M. Ostoja-Starzewski, Fractal solids, product measures and fractional wave equations. Proc. R. Soc. A 465, 2521–2536 (2009b); Errata (2010)Google Scholar
- J. Li, M. Ostoja-Starzewski, Edges of Saturn’s rings are fractal. SpringerPlus 4, 158 (2015). arXiv:1207.0155 (2012)Google Scholar
- W. Nowacki, Theory of Asymmetric Elasticity (Pergamon Press/PWN − Polish Sci. Publ., Oxford/Warszawa, 1986)Google Scholar