Key words: box-counting method, mesh size distribution, concrete, fractal dimension, fracture


Advanced variant of optimised box-counting method for estimation of capacitive/ information fractal dimension is presented. Fractal dimensions of representation of fractured concrete surfaces are determined using special mesh size distribution.


1. Introduction

Mechanical/fracture properties of cement based composites (concrete, mortar, hardened cement paste) are used in civil engineering practise. Fracture features of these quasi-brittle materials is studied from many viewpoints - see e.g. [Karihaloo, 1995], [Bažant & Planas, 1998]. Fracture surfaces of these composites and also crack pattern are considered as irregular objects - natural fractals. The beginning of development of fractal characterisation of irregularity of these objects is related to work of e.g. [Mandelbrot, 1982]. Many authors deal with fractal characterisation of fracture surfaces of concrete from the point of view of fractal dimension determination and also correlation to others materials parameters, scaling aspects, etc. - e.g. [Saouma et al., 1990], [Brandt & Prokopski, 1993], [Bažant, 1995], [Carpinteri et al., 1999], [Carpinteri & Invernizzi, 2001], [Mechtcherine & Muller, 2001]. Optimised box-counting method for estimation of capacitive fractal dimension was presented and then fractal dimensions of representation of fractured concrete surfaces of tested specimens was determined in [Frantík et al., 2002]. Advanced variant of fractal dimensions estimation (special mesh size distribution) and some interesting results are presented in this paper.


2. Fractal dimension - optimised box-counting method

We can recall the box-counting method is based on fractal dimension that is called capacity. Capacity dc of any object is defined as the limit:


where N() is the minimal number of n-dimensional cubes of size , which are necessary for covering the measured object nested in n-dimensional space.

Box-counting method is the basic method for numerical estimation of capacity (other generalized dimensions like informational dimension di etc. can be applied as well but not so transparently). Let us cover measured object (see Figure 1) a number of cubic meshes Mi(i), where i = 1, 2, ..., n and 1 > 2 >... > n is size of cubes in mesh Mi (see Figure 2 for illustration). Then we sum the number of cubes Ni(i) for every mesh. Estimation of capacity is first coefficient of linear approximation of points (log(1/i), logNi(i)), i = 1, 2, ..., n. The information dimension is also determined. The standard deviation of the measuring can be established by this method as well.

Figure 1: Measured object


Figure 2: Covering of object by optimised increasing mesh


3. Mesh size

For estimation of fractal dimension we must have a number of measured meshes Mi(i) with different size of cubes i (minimum is two meshes). For linear approximation of measured points (log(1/i), logNi(i)) is usually:


for all i. In simplest = 1, 2, 4, 8, 16,… Let us define kr coefficient of mesh size step:


then for = 1, 2, 4, 8, 16,… is coefficient of mesh size step kr = 2. If we want more measured points, we can decrease coefficient of mesh size step - e.g. coefficient kr = 1.5 gives = 1, 2, 3, 5, 7, 11, 17,…

Now a new parameter for measuring of dimension has been established: coefficient of mesh step. Then dependence of fractal dimension on this coefficient can be investigated. We can write dc = f(kr). The dependence of fractal dimension on coefficient kr for object in Figure 1 is showed in Figure 3. Mean value of measured fractal capacity dimensions dc(kr) and information dimensions di(kr), respectively, can be taken as final fractal dimension.

Figure 3: Fractal dimensions dc, di vs. coefficient of mesh size step kr


4. Determination of fractal dimension of fracture surface

In case of fracture experiment we try to find another invariant surface attribute. We suppose that the capacity of the one-dimensional planar section of the fracture surface has the same invariant properties as the full three-dimensional surface [Mandelbrot, 1982]. We can obtain requested section if we immerse the surface to the coloured solution and take a digital photograph. The visualisation of the boundary between the coloured (immersed) and non-coloured parts can be obtained by the help of standard image processing. An example of representation of fracture surface of specimen and its estimated capacity from linear approximation of measured points is presented in Figure 4 and in Table 4.

Figure 4: Studied sections - representation of fracture surface of specimen (2) - (6)


Table 4: Fractal dimensions (capacitive dc, information dc) standard deviation



Measured boundary has weak fractal properties in case of sections (2) - (6). However experiments showed that this facility is not the consequence of discretization error and we can assume that fracture surface is fractal. To determinate the correlation between the capacity/information of measured boundary and material attributes many more experiments need to be accomplished.

Completely different situation is in case of section (1) - the value of fractal dimension suggests something very interesting. We can see absolute equality of capacitive/information fractal dimensions, which implies very stable, invariant, established, reliable object, without errors! How it was anticipated, value 60 exceeds any reasonable topological dimension a lot!



The authors thank for funding under grant No. 103/03/1350 from the Grant Agency of the Czech Republic (headed by Prof. Zdeněk Bittnar, also his contribution to Figure 4 (1) is gratefully appreciated!) and under the research project reg. No. CEZ: J22/98: 261100009.



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[Bažant & Planas, 1998] Bažant, Z. P. & Planas, J. (1998) Fracture and Size Effect in Concrete and other Quasibrittle Materials, CRC Press, Boca Raton, Florida.

[Brandt & Prokopski, 1993] Brandt, A. M., Prokopski, G. (1993) On the fractal dimension of fracture surfaces of concrete elements, Journal of Material Science, 28, 4762-4766.

[Carpinteri et al., 1999] Carpinteri, A., Chiaia, B., Invernizzi, S. (1999) Three-dimensional fractal analysis of concrete fracture at the meso-level, Theoretical and Applied Fracture Mechanics, 31, 163-172.

[Carpinteri & Invernizzi, 2001] Carpinteri, A., Invernizzi, S. (2001) Uniaxial tensile test and fractal evaluation of softening damage in concrete, Fracture Mechanics of Concrete Structures, de Borst et al. (eds), Swets & Zeitlinger, Lisse, 19- 25.

[Frantík et al., 2002] Frantík, P., Keršner, Z., Macur, J. (2002) Estimation of fractal dimension: Optimised box-counting method, Non-Traditional Cements and Concrete, Bílek & Keršner (eds), Brno, ISBN 80 214 2130 4, 443-446.

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[Mechtcherine & Muller, 2001] Mechtcherine, V., Muller, H. S. (2001) Fractological investigation on the fracture in concrete, Fracture Mechanics of Concrete Structures, de Borst et al. (eds), Swets & Zeitlinger, Lisse, ISBN 90 2651 825 0, 81-88.

[Saouma et al., 1990] Saouma, V. E., Barton, C. C., Gamaleldin, N. A. (1990) Fractal characterization of fracture surfaces in concrete, Engineering Fracture Mechanics, 35, No. 1/2/3, 47-53.