Fractal express methods evaluation of a breaking stress of concrete

We have offered the methods of the operative evaluation of a breaking stress in testing strength on compression of heavy concrete of grade 400 with the use of fractal formalism. The methods are based on the setting of the relations of the values of a breaking stress and fractal measurement of concrete: 1. Areas with crushed stone prevailing (R2 = 0.7224); 2. Areas with sand prevailing (R2 = 0.6102); 3. Pin holes (R2 = 0.6874). During fractal experimenting the indexes of breaking were increased from 391.63 to 515.13 kH in reducing fractal dimension of areas 1 from 1.866 to 1.588; areas 3 from 1.826 to 1.684 and internal quality (boundary of its elements) from 1.617 to 1.353. Increase of fractal dimension was fixed only for areas containing sand (areas 2) from 1.755 to 1.944. These results make possible to apply fractal dimension as the indicator of structural changes of concrete in the prediction of its properties. In fractal modelling the accuracy of the results depends on the option of the task way for space metric that is proved by the obtained linear model (R2 = 0.9254), which describes the connection between the elements of macrostructure and strength criterion of concrete. Such methods provide a satisfactory in practical purposes operative prediction for the values of breaking stress of concrete of grade 400 with significant reduction of time and money expenditures on full-scale testing and the application of microscopy.


Introduction
A great number of parameters influence the quality criteria of concretes including the conditions of its obtaining, mineralogical and phase compositions [1,2]. Real structure of concrete on various scale levels is characterised by a great number of elements that reflect its properties in a varying degree [3]. Approximation of the elements of concrete structure by Euclid's integral figures (appraisal of points, square etc.) not always allows to use these results in the models of the prediction of its quality. One of the reasons of such a phenomenon is insufficiency of formal axiomatics in the identification of a real structure of material with complex geometric configuration of its elements.
For partial compensation of the existing insufficiency of formal axiomatics to identify materials structure, you can use the methods of mathematical modelling [4,5], system analysis [6,7], fractal theory [8,9]. One of the advantages of using the fractal approach in order to describe qualitative transformations of materials is an arbitrary way of metrics definition. Fractal dimension as a quantative feature of fractal set gives more differential evaluation for similar objects at first view. That is why fractal formalism is used in quality ranking of multiparameter technology [10,11]. Basing on the results of the work for the application of fractal theory to model the structure and qualitative features of concretes, for example [12][13][14], the work dealt with the evaluation of breaking stress on the compression of heavy concrete on the base of fractal characteristics of the elements of its macrostructure. The application of such an approach allows to model strength criterion according to the photos of structure without any additional expenditures on full-scale testing and microscopy.

Materials and methods
Concrete of grade 400 produced from Portland Cement was investigated. Five samples from one pilot run were produced (Fig. 1). There are three main constitunts on the surface of concrete: 1. Coarse aggregate (crushed stone) of 10… 20 mm fraction. Percentage 54…67 %. On the expanding fragment of Sample 5 Fig.1, these are the darker areas 1 of the structure of concrete containing crushed stone coming to the surface. 2. Fine aggregate (sand) with fractions not less 1.5 mm. Lighter areas 2 of the structure of concrete in Fig. 1  To increase the reliability of the obtained values of fractal dimension we used patented methods described in detail in [15]. The main point of the methods lies in the determination of fractal dimension of tested object on the base of precision of cell and point-like dimensions.
Cell dimension D was calculated by F.Hausdorff equation (1) [16]. At the core of this method, there is an idea of covering an object with cells by the number N with linear size l:   Boundaries in a multicomponent system i.e. concrete, contributes in its strength properties [1,2]. In order to study the influence of boundaries between the elements of the structure of concrete on a breaking stress, the values of their fractal dimensions between areas of the structure were calculated separately.

Results and discussion
The values of the calculated values of fractal dimensions of the areas of the structure and their boundaries are shown in Table 2.  On account of the analysis of the obtained ratio, fractal dimension of a coarse aggregate (crushed stone R² = 0,7224 in Fig. 3 а) and boundary dimension (R² = 0.7851 in Fig. 3 d) have the significant influence on the strength of concrete. Sensitivity of indicators of a breaking stress to fractal characteristics of crushed stone forming matrix of concrete and to interphase boundaries as failures reducing strength indicators [1,2], is proved by the processes of their physical and chemical influence on the properties of concrete. The obtained results correspond to the data from [13] for ceramsite concrete where in all cases there is the reduction of strength indicators on compression fck.cube in increasing the values of fractal dimensions of the elements of macrostructure (expanded clay gravel, cement-sand matrix, large feldspar grains in the sand, large pores, large fractions of quartz). Particularly, in [13] indicators fck.cube are reduced from 31.3 to 42.6 MPa in the reduction of fractal dimension of a coarse aggregate (gravel) from 1.998 to 1.910. Such an effect shows the influence of space metrics in evaluation of concrete structure. Lower sensitivity is determined between indicators of a breaking stress and fractal dimensions of areas with sand prevailing (light areas of the structure) in Fig. 3 b, and fractal dimensions of pores (Fig. 3 с). There are many publications devoted to research of fractal dimensions, e.g. works [17,18] confirm their fractal nature. In [18] fractal dimension of pores of aerated concrete is calculated at the range of 1.775…1.805, it is increased linearly with porosity, area, size and an average diameter of pores. Results from [18] gave the opportunity to use fractal dimension of pores as an integral indicator of pore structure of aerated concrete. In our case the values of a breaking stress of concrete correlate with all the elements of the structure of concrete.
As all the elements under research of macrostructure influence on physical and chemical properties of concrete, in particular, its strength, there was obtained a multiparameter fractal model (3) of their complex influence.

Summary
The application of fractal approach in the prediction of concrete quality criteria is due to its multicomponent structure, in which real elements of the structure are described with the help of Euclid's integral geometry, however, they have a complex geometric configuration of shape. Thus, the use of traditional results of the evaluation of the structure in present models of prediction structure-properties is limited.
There was considered the approach for the evaluation of a breaking stress of heavy concrete on compression on the base of fractal analysis for determined elements of macrostructure (areas with crushed stone, sand, pores, their boundaries). The advantage of the given methods over traditional ones is the use of fractal dimension as an indicator of structural changes that is proved by a number of publications. To calculate fractal dimension of the elements of the structure of concrete, photos of 256-coloured format were used. Obtained results show the efficiency of the chosen methods for prediction of strength indicator on the base of obtained model with correlation coefficient R² = 0.9254.