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From fractal media to continuum mechanics

Ostoja-Starzewski, Martin and Li, Jun and Joumaa, Hady and Demmie, Paul N. (2014) From fractal media to continuum mechanics. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 94 (5). pp. 373-401. ISSN 0044-2267. https://resolver.caltech.edu/CaltechAUTHORS:20140620-140845324

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Abstract

This paper presents an overview of modeling fractal media by continuum mechanics using the method of dimensional regularization. The basis of this method is to express the balance laws for fractal media in terms of fractional integrals and, then, convert them to integer-order integrals in conventional (Euclidean) space. Following an account of this method, we develop balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and discuss wave equations in several settings (1d and 3d wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). We then discuss extremum and variational principles, fracture mechanics, and equations of turbulent flow in fractal media. 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. We also point out relations and potential extensions of dimensional regularization to other models of microscopically heterogeneous physical systems.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1002/zamm.201200164DOIArticle
http://onlinelibrary.wiley.com/doi/10.1002/zamm.201200164/abstractPublisherArticle
Additional Information:© 2014 Wiley-VCH Verlag GmbH & Co. Received 30 August 2012, revised and accepted 17 December 2012. Published online 30 January 2013. This work was made possible by the support from Sandia-DTRA (grant HDTRA1-08-10-BRCWMD) and the NSF (grant CMMI-1030940). Also, the support of the first author as Timoshenko Distinguished Visitor in the Division of Mechanics and Computation, Stanford University, is gratefully acknowledged.
Funders:
Funding AgencyGrant Number
Sandia-DTRAHDTRA1-08-10-BRCWMD
NSFCMMI-1030940
Timosheko Distinguished Scholar, Division of Mechanics and Computation, Stanford UniversityUNSPECIFIED
Subject Keywords:Fractals, continuum mechanics, dimensional regularization, balance laws
Issue or Number:5
Record Number:CaltechAUTHORS:20140620-140845324
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20140620-140845324
Official Citation:Ostoja-Starzewski, M., Li, J., Joumaa, H. and Demmie, P.N. (2014), From fractal media to continuum mechanics. Z. angew. Math. Mech., 94: 373–401. doi: 10.1002/zamm.201200164
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:46404
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:20 Jun 2014 22:02
Last Modified:03 Oct 2019 06:44

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