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Continuum Homogenization of Fractal Media

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Handbook of Nonlocal Continuum Mechanics for Materials and Structures

Abstract

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.

Sandia National Laboratories is a multimission laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mechanical Science and Engineering, Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana–Champaign, Urbana, IL, USA

    Martin Ostoja-Starzewski

  2. Department of Mechanical Engineering, University of Massachusetts, Dartmouth, MA, USA

    Jun Li

  3. Sandia National Laboratories, Albuquerque, NM, USA

    Paul N. Demmie

Authors
  1. Martin Ostoja-Starzewski
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  2. Jun Li
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  3. Paul N. Demmie
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Corresponding author

Correspondence to Martin Ostoja-Starzewski .

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Editors and Affiliations

  1. Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA, USA

    George Z. Voyiadjis

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Ostoja-Starzewski, M., Li, J., Demmie, P.N. (2019). Continuum Homogenization of Fractal Media. In: Voyiadjis, G. (eds) Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-58729-5_18

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