Mathematical modeling of an effect of refractory metal nanopowders on the strength of concrete

. The quality of concrete and the reliability of its exploitation in structures and structures are largely determined by such characteristic as strength. Additives increasing the strength can significantly improve the quality of concrete. The aim of this work is mathematical modeling of the effect of refractory metal nanopowders on the strength of concrete. Regression equations of the first and second orders are constructed, which adequately find the dependence of the compressive strength of concrete on the input parameters. For presented model the optimal values of the concentrations of refractory metal nanopowders have been calculated.


Introduction
The construction industry is one of the most dynamically developing segments of the Russian economy. Modern trends in the development of materials science in construction are closely related to the development of new resource-saving technologies for production concrete, with improved performance and durability. At the same time, the lack of material resources, developed in a number of regions, urgently requires increasing the profitability of the industry production of building materials due to the wide use as local resources and manmade materials. The downside of this trend is that the use of local substandard raw materials leads to a decrease in the strength of the cement construction composites.
At the same time, the quality of concrete and the reliability of its using in structures and structures are largely determined by such characteristic as strength. Strength is understood as the ability of a material to resist destruction as a result of external loading. This integral characteristic strongly depends on the mechanical and chemical properties of the material (concrete components), on the methods of concrete preparation and the exploitation of the structure. Some theoretical aspects of the dependence of the strength of composite materials on their physical and mechanical properties are considered in [1][2][3][4]. Nevertheless, in modern building materials science there is no complete theory that allows us to calculate its strength based on the given properties of concrete ingredients.
One of the possible solutions to the problem of improving the quality of concrete is the use of a limited number of highly active additives. As well known, concrete resists compression well and is much worse in tension. Due to this, additives, leading to an increase in strength can significantly improve the quality of the concrete. The experimentally demonstrated [5] that micro-and nanomodifiers (nanopowders) are the most promising among the possible additives. A significant limitation of this approach is the high cost of nanopowders, the lack of technologies for their mass production, which does not allow them to be used as additives on an industrial scale.
However, new technologies for the production of nanopowders make it possible to include the use of nanopowders in the cycle of concrete production in the future [6]. Thus, the aim of this work is mathematical modeling of the effect of refractory metal nanopowders on the strength of concrete.

Methods
The purpose of experimental studies is to study the effect of the addition nanopowders of refractory metal on the strength of a concrete sample (size 100 100 100mm  It is usually assumed in experimental analysis of building structures that the response function (concrete compressive strength) is a random variable with a normal distribution law. On this basis, regression analysis methods are used when testing statistical hypotheses about the response function. Tables 1, 2 present the results of the characteristics of the homogeneity of concrete modified with various powders of refractory metals. It should be noted that statistical values are estimated as averages due to the limited sampling. The analysis of the homogeneity of concrete (tables 1, 2) showed, that the intra-batch coefficient of variation is in the range of less than 5%. Thus, the quality of production and testing of samples can be considered satisfactory.

Results of modeling
Let us carry out mathematical modeling of the influence of parameters on the strength of concrete under compression by the first-order planning method, where first-order polynomials  Table 3. The hypothesis of the homogeneity of the dispersion is verified by using the Cochran test, its value is calculated by the formula: x is a fictive variable, which is introduced to obtain complete symmetry; Images of the surface were obtained (Fig. 1), described by objective functions.   are given in Table 4.

Conclusions
The obtained regression equations of the first and second orders allow us to find adequately the dependence of the concrete compressive strength on the ratio of the input parameters x1 and x2. The optimal values of the concentration of nanopowders are calculated as: 2.95% WC , 1.72% 3 WO . According to presented modeling, nanopowder WC is the optimal additive to concrete that increases its operational properties.