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Scaling of mathematical fractals and box-counting quasi-measure

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Abstract

The same term, ‘fractals’ incorporates two rather different meanings and it is convenient to split the term into physical or empirical fractals and mathematical ones. The former term is used when one considers real world or numerically simulated objects exhibiting a particular kind of scaling that is the so-called fractal behaviour, in a bounded range of scales between upper and lower cutoffs. The latter term means sets having non-integer fractal dimensions. Mathematical fractals are often used as models for physical fractal objects. Scaling of mathematical fractals is considered using the Barenblatt–Borodich approach that refers physical quantities to a unit of the fractal measure of the set. To give a rigorous treatment of the fractal measure notion and to develop the approach, the concepts of upper and lower box-counting quasi-measures are presented. Scaling properties of the quasi-measures are studied. As examples of possible applications of the approach, scaling properties of the problems of fractal cracking and adsorption of various substances to fractal rough surfaces are discussed.

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References

  1. Borodich F.M.: Some fractal models of fracture. J. Mech. Phys. Solids 45, 239–259 (1997)

    Article  MATH  Google Scholar 

  2. Malcai O., Lidar D.A., Biham O., Avnir D.: Scaling range and cutoffs in empirical fractals. Phys. Rev. E 56, 2817–2828 (1997)

    Article  Google Scholar 

  3. Falconer K.J.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990)

    MATH  Google Scholar 

  4. Tricot C.: Curves and Fractal Dimension. Springer, Berlin (1995)

    MATH  Google Scholar 

  5. Bazant Z.P., Yavari A.: Is the cause of size effect on structural strength fractal or energetic-statistical?. Eng. Fract. Mech. 72, 1–31 (2005)

    Article  Google Scholar 

  6. Barenblatt G.I., Monin A.S.: Similarity principles for the biology of pelagic animals. Proc. Natl. Acad. Sci. USA 80, 3540–3542 (1983)

    Article  Google Scholar 

  7. Barenblatt G.I.: Scaling. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  8. Borodich F.M.: Fracture energy in a fractal crack propagating in concrete or rock. Trans. (Doklady) Russian Akad. Sci. Earth Sci. Sect. 327, 36–40 (1992)

    Google Scholar 

  9. Borodich, F.M.: Non-classical scaling of microcrack patterns and crack propagation. In: Multiple Scale Analyses and Coupled Physical Systems. Presses de l’école nationale des Ponts et Chaussées, Paris, pp. 493–500 (1997)

  10. Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Measure. The Lebesgue Integral, vol. 2. Hilbert Space. Doven Publications, New York (1999)

  11. Turbin A.F., Pratsevityi N.V.: Fractal Sets, Functions, Distributions. Naukova Dumka, Kiev (1992)

    MATH  Google Scholar 

  12. Borodich F.M.: Parametric homogeneity and non-classical self-similarity. I. Mathematical background. Acta Mech. 131, 27–45 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Borodich F.M., Volovikov A.Y.: Surface integrals for domains with fractal boundaries and some applications to elasticity. Proc. R. Soc. Lond. Ser. A. 456, 1–23 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Engineering, Cardiff University, Cardiff, CF24 3AA, UK

    F. M. Borodich

  2. Faculty of Science, Jiangsu University, 212013, Zhenjiang, Jiangsu, People’s Republic of China

    Z. Feng

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  1. F. M. Borodich
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  2. Z. Feng
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Correspondence to F. M. Borodich.

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Borodich, F.M., Feng, Z. Scaling of mathematical fractals and box-counting quasi-measure. Z. Angew. Math. Phys. 61, 21–31 (2010). https://doi.org/10.1007/s00033-009-0010-6

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  • DOI: https://doi.org/10.1007/s00033-009-0010-6

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