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Size Effects and Material Length Scales in Nanoindentation for Metals Read chapter Read first chapter

2019 | OriginalPaper | Chapter

25. Continuum Homogenization of Fractal Media

Authors : Martin Ostoja-Starzewski, Jun Li, Paul N. Demmie

Published in: Handbook of Nonlocal Continuum Mechanics for Materials and Structures

Publisher: Springer International Publishing

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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.

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previous chapter Fractional Differential Calculus and Continuum Mechanics
next chapter Modeling High-Speed Impact Failure of Metallic Materials: Nonlocal Approaches
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Metadata
Title
Continuum Homogenization of Fractal Media
Authors
Martin Ostoja-Starzewski
Jun Li
Paul N. Demmie
Copyright Year
2019
Book
Handbook of Nonlocal Continuum Mechanics for Materials and Structures

Print ISBN: 978-3-319-58727-1

Electronic ISBN: 978-3-319-58729-5

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-319-58729-5_18

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