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Continuum Homogenization of Fractal Media

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

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.

Keywords

Balance laws Fractal Fractional calculus Fractal derivative Homogenization 

Notes

Acknowledgements

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

References

  1. 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
  2. M.F. Barnsley, Fractals Everywhere (Morgan Kaufmann, San Francisco, 1993)zbMATHGoogle Scholar
  3. A. Carpinteri, B. Chiaia, P.A. Cornetti, A disordered microstructure material model based on fractal geometry and fractional calculus. ZAMP 84, 128–135 (2004)MathSciNetzbMATHGoogle Scholar
  4. A. Carpinteri, N. Pugno, Are scaling laws on strength of solids related to mechanics or to geometry? Nat. Mater. 4, 421–23 (2005)CrossRefGoogle Scholar
  5. P.N. Demmie, Ostoja-Starzewski, Waves in fractal media. J. Elast. 104, 187–204 (2011)MathSciNetGoogle Scholar
  6. A.C. Eringen, Microcontinuum Field Theories I (Springer, New York, 1999)CrossRefGoogle Scholar
  7. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, Chichester, 2003).CrossRefGoogle Scholar
  8. E.E. Gdoutos, Fracture Mechanics: An Introduction (Kluwer Academic Publishers, Dordrecht, 1993)CrossRefGoogle Scholar
  9. H.M. Hastings, G. Sugihara, Fractals: A User’s Guide for the Natural Sciences (Oxford Science Publications, Oxford, 1993)zbMATHGoogle Scholar
  10. H. Joumaa, M. Ostoja-Starzewski, On the wave propagation in isotropic fractal media. ZAMP 62, 1117–1129 (2011)MathSciNetzbMATHGoogle Scholar
  11. H. Joumaa, M. Ostoja-Starzewski, Acoustic-elastodynamic interaction in isotropic fractal media. Eur. Phys. J. Spec. Top. 222, 1949–1958 (2013)CrossRefGoogle Scholar
  12. H. Joumaa, M. Ostoja-Starzewski, P.N. Demmie, Elastodynamics in micropolar fractal solids. Math. Mech. Solids 19(2), 117–134 (2014)MathSciNetCrossRefGoogle Scholar
  13. H. Joumaa, M. Ostoja-Starzewski, On the dilatational wave motion in anisotropic fractal solids. Math. Comput. Simul. 127, 114–130 (2016)MathSciNetCrossRefGoogle Scholar
  14. G. Jumarie, On the representation of fractional Brownian motion as an integral with respect to (dt)a. Appl. Math. Lett. 18, 739–748 (2005)MathSciNetCrossRefGoogle Scholar
  15. G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Appl. Math. Lett. 22(3), 378–385 (2009)MathSciNetCrossRefGoogle Scholar
  16. A. Le Méhauté, Fractal Geometry: Theory and Applications (CRC Press, Boca Raton, 1991)zbMATHGoogle Scholar
  17. J. Li, M. Ostoja-Starzewski, Fractal materials, beams and fracture mechanics. ZAMP 60, 1–12 (2009a)MathSciNetzbMATHGoogle Scholar
  18. 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
  19. J. Li, M. Ostoja-Starzewski, Fractal solids, product measures and continuum mechanics, chapter 33, in Mechanics of Generalized Continua: One Hundred Years After the Cosserats, ed. by G.A. Maugin, A.V. Metrikine (Springer, New York, 2010), pp. 315–323CrossRefGoogle Scholar
  20. J. Li, M. Ostoja-Starzewski, Micropolar continuum mechanics of fractal media. Int. J. Eng. Sci. (A.C. Eringen Spec. Issue) 49, 1302–1310 (2011)MathSciNetCrossRefGoogle Scholar
  21. J. Li, M. Ostoja-Starzewski, Edges of Saturn’s rings are fractal. SpringerPlus 4, 158 (2015). arXiv:1207.0155 (2012)Google Scholar
  22. B.B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman & Co, NewYork, 1982)zbMATHGoogle Scholar
  23. G.A. Maugin, The Thermomechanics of Nonlinear Irreversible Behaviours (World Scientific Pub. Co., Singapore, 1999)CrossRefGoogle Scholar
  24. G.A. Maugin, Non-classical Continuum Mechanics: A Dictionary (Springer, Singapore, 2016)zbMATHGoogle Scholar
  25. W. Nowacki, Theory of Asymmetric Elasticity (Pergamon Press/PWN − Polish Sci. Publ., Oxford/Warszawa, 1986)Google Scholar
  26. K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, San Diego, 1974)zbMATHGoogle Scholar
  27. M. Ostoja-Starzewski, Fracture of brittle micro-beams. ASME J. Appl. Mech. 71, 424–427 (2004)CrossRefGoogle Scholar
  28. M. Ostoja-Starzewski, Towards thermomechanics of fractal media. ZAMP 58(6), 1085–1096 (2007)MathSciNetzbMATHGoogle Scholar
  29. M. Ostoja-Starzewski, On turbulence in fractal porous media. ZAMP 59(6), 1111–1117 (2008a)MathSciNetzbMATHGoogle Scholar
  30. M. Ostoja-Starzewski, Microstructural Randomness and Scaling in Mechanics of Materials (CRC Press, Boca Raton, 2008b)zbMATHGoogle Scholar
  31. M. Ostoja-Starzewski, Extremum and variational principles for elastic and inelastic media with fractal geometries. Acta Mech. 205, 161–170 (2009)CrossRefGoogle Scholar
  32. M. Ostoja-Starzewski, Electromagnetism on anisotropic fractal media. ZAMP 64(2), 381–390 (2013)MathSciNetzbMATHGoogle Scholar
  33. M. Ostoja-Starzewski, J. Li, H. Joumaa, P.N. Demmie, From fractal media to continuum mechanics. ZAMM 94(5), 373–401 (2014)MathSciNetCrossRefGoogle Scholar
  34. M. Ostoja-Starzewski, S. Kale, P. Karimi, A. Malyarenko, B. Raghavan, S.I. Ranganathan, J. Zhang, Scaling to RVE in random media. Adv. Appl. Mech. 49, 111–211 (2016)CrossRefGoogle Scholar
  35. D. Stoyan, H. Stoyan, Fractals, Random Shapes and Point Fields (Wiley, Chichester, 1994)zbMATHGoogle Scholar
  36. V.E. Tarasov, Fractional hydrodynamic equations for fractal media. Ann. Phys. 318(2), 286–307 (2005a)MathSciNetCrossRefGoogle Scholar
  37. V.E. Tarasov, Wave equation for fractal solid string. Mod. Phys. Lett. B 19(15), 721–728 (2005b)CrossRefGoogle Scholar
  38. V.E. Tarasov, Continuous medium model for fractal media. Phys. Lett. A 336, 167–174 (2005c)CrossRefGoogle Scholar
  39. V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, Berlin, 2010)CrossRefGoogle Scholar
  40. V.E. Tarasov, Anisotropic fractal media by vector calculus in non-integer dimensional space. J. Math. Phys. 55, 083510-1-20 (2014)MathSciNetCrossRefGoogle Scholar
  41. V.E. Tarasov, Electromagnetic waves in non-integer dimensional spaces and fractals. Chaos, Solitons Fractals 81, 38–42 (2015a)MathSciNetCrossRefGoogle Scholar
  42. V.E. Tarasov, Vector calculus in non-integer dimensional space and its applications to fractal media. Commun. Nonlinear Sci. Numer. Simul. 20, 360–374 (2015b)MathSciNetCrossRefGoogle Scholar
  43. H. Ziegler, An Introduction to Thermomechanics (North-Holland, Amsterdam, 1983)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Martin Ostoja-Starzewski
    • 1
  • 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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