Scaling of mathematical fractals and box-counting quasi-measure. (English) Zbl 1243.74004
Summary: The same term, ‘fractal’ incorporates two rather different meanings, and it is convenient to split the term into a physical or empirical fractal and a mathematical one. 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. Here, the 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 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 by fractal rough surfaces are discussed.
References:
| [1] | Borodich F.M.: Some fractal models of fracture. J. Mech. Phys. Solids 45, 239–259 (1997) · Zbl 0969.74572 · doi:10.1016/S0022-5096(96)00080-4 |
| [2] | Malcai O., Lidar D.A., Biham O., Avnir D.: Scaling range and cutoffs in empirical fractals. Phys. Rev. E 56, 2817–2828 (1997) · doi:10.1103/PhysRevE.56.2817 |
| [3] | Falconer K.J.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990) · Zbl 0689.28003 |
| [4] | Tricot C.: Curves and Fractal Dimension. Springer, Berlin (1995) · Zbl 0847.28004 |
| [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) · doi:10.1016/j.engfracmech.2004.03.004 |
| [6] | Barenblatt G.I., Monin A.S.: Similarity principles for the biology of pelagic animals. Proc. Natl. Acad. Sci. USA 80, 3540–3542 (1983) · doi:10.1073/pnas.80.11.3540 |
| [7] | Barenblatt G.I.: Scaling. Cambridge University Press, Cambridge (2003) · Zbl 1094.00006 |
| [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) |
| [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) · Zbl 0090.08702 |
| [11] | Turbin A.F., Pratsevityi N.V.: Fractal Sets, Functions, Distributions. Naukova Dumka, Kiev (1992) |
| [12] | Borodich F.M.: Parametric homogeneity and non-classical self-similarity. I. Mathematical background. Acta Mech. 131, 27–45 (1998) · Zbl 0927.74003 · doi:10.1007/BF01178243 |
| [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) · Zbl 0969.74011 · doi:10.1098/rspa.2000.0506 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
