Design of non-transferred DC electric arc plasma torches (EAPTs) operated with plasma gases containing alkane hydrocarbons, as a promising type of heaters for a number of technologies (thermal spraying, surface hardening, testing of thermal protection systems, etc.), requires taking into account the evaporation rate of surface cathode material (as one of the channels of its ablative degradation). For this procedure, as the first stage, thermodynamic methods can be used to simulate the composition and properties of reactive C–H–O–N–Ar–Me-systems with variable set of such input parameters as the ratio of components of plasma-forming mixture, its pressure and temperature. We theoretically estimated the evaporative degradation of the material for three variants of EAPT cathode with alkane-containing plasmas (“hot” thermochemical zirconium and thermionic tungsten, and “cold” copper) in equilibrium and quasi-equilibrium modes of “plasma gas + surface cathode material”-mixture, with use of generalized thermodynamic properties of the materials. The calculation for conditions with pressure, which is characteristic for EAPT discharge chamber, showed that when varying the initial composition of the plasma-forming mixture (from oxidizers (air or combustion products of alkanes) to reducing gases based on the products of combined partial oxidation and pyrolysis of alkanes), the effect of a difference in the cathode evaporation rate EAI was observed in systems based on (air + alkane)-mixtures near the melting point of surface cathode substances, in a comparison with the case of EAPTs with more conventional gases (commercial N₂, air) and, importantly, for two variants of the analyzed cathodes (with the exception of copper). In addition, the electrode erosion value was compared for simulated zirconium cathode (in terms of erosion evaporative component) when operating on the combustion products of alkanes from “air + CH₄”-mixture, and for some known EAPTs with similar cathodes in other gases. Using the case of earlier tested DC plasma torch with rod Zr-cathode (with microheterogeneous surface) as an example, it was found that our calculation indicates non-monotonic dynamics of EAI value and fractions of Zr-containing vapors as a result of the change of the fuel-air equivalence ratio f of initial reactive mixture. This effect is inconsistent with measured cathode composition, which shows a probability of nonequilibrium character of thermal and diffusion processes in near-electrode plasma and surface layer (~1 mm) of the electrode, at least in the modes with arc current in the torch near 300 A. Besides this, it should be noted that obtained modeling data on the behavior of zirconium compounds (ZrO₂, ZrC) in C–H–O–N–Ar–Zr-system can be used not only for improvement of the torch cathodes, but also for design of new Zr-containing thermal protection systems to predict preliminary their ablation rate in a flow of products of combustion (including incomplete one) of engine-, rocket- and other fuels. Similarly, the results on the copper compounds behavior near the metal evaporation temperature can be useful for optimizing the process of plasma spraying of copper alloy coatings.

Previously, the author obtained an analytical solution for determining the vertical displacements of the homogeneous isotropic quarter-space face, which is affected by a vertical concentrated force. This expression gave an exact solution for determining the vertical displacements of a quarter-space face made of incompressible material and an approximate solution for a Poisson ratio different from 0.5. Later, in the published paper by S. V. Bosakov and P. D. Skachek “Action of Concentrated Force on 1/8 of Homogeneous Isotropic Space”, it was shown that by combining solutions for determining 

the vertical displacements of a quarter-space and a half-space from the action of concentrated forces, one can find vertical displacements for  one-eighth face of a homogeneous isotropic space. The resulting expressions allow solving contact problems for non-classical domains in the form of a quarter of a space and one eighth of a space. Below, the author obtains the first approximation for displacements of a quarter-space face from the action of a vertical concentrated force. In this paper, the author gives the first approximation for determining the vertical displacements of a quarter-space face from the action of a vertical concentrated force, widely using the special approximation method developed in the works of V. M. Alexandrov and allowing to calculate successfully improper integrals. The constructed graphs show good results when determining displacements with a Poisson’s ratio different from 0.5 and equal to 0.5. It should be noted that the approach indicated in the paper by S. V. Bosakov and P. D. Skachek can be successfully used in determining all displacements of the faces of one-eighth face of a homogeneous isotropic sharpened forces tangent to the edge of a quarter-space and this will allow to solve contact problems taking into account friction forces in the contact zone of the beam or plate.

The paper studies the issue of improving the evenness of the prefabricated concrete pavement by changing the design of the slabs from which it is assembled. Irregularities of the prefabricated pavement mainly occur at the junction of the plates and are called “keyboard” effect. The reason for the unevenness is the vertical movement of the ends of the plates relative to each other when the car passes. In order to correct this situation, the authors propose to arrange transverse ribs of rectangular cross section at the ends of the plates. Such ribs will create a reliable support at the edges of the slab, which will reduce the vertical movement of its ends. To determine the effectiveness of the use of transverse ribs, a concrete slab with dimensions of  3´6 m has been examined in the paper. The transverse rib in cross section had the shape of a rectangle with sides 20´10 cm. In order to compare the test results, a concrete slab of the same dimensions, but without transverse ribs, has been studied. Deformations and stresses of the slab under load have been determined by the finite element method using the Lira software package. Calculations have shown the effectiveness of the slabs with transverse ribs mounted on its edges. The vertical displacements of the ends of the slabs with ribs are two times less than those of the slabs without ribs, and the stresses are 2.4 times less. It is obvious that the use of concrete slabs with transverse ribs at the ends will reduce the unevenness of the prefabricated coating at the points of their mating.

An analysis of the probability and conditions for the occurrence of soil liquefaction at the base and in the vicinity of hydraulic structures is presented in the paper. As a rule, hydraulic structures are erected in the valleys of watercourses, the structure of the soil in which contributes to the occurrence of liquefaction processes. These soils are fine-grained, 

non-cohesive, usually consisting of fine- and medium-grained or silty sands, sandy loams, interspersed with layers of loams. Massifs under pressure hydraulic structures are quite water-saturated. Soil liquefaction occurs as a result of the destruction of structural bonds between particles in water-saturated dispersed soils under the action of stresses of various types. An external dynamic or static load applied to a water-saturated massif composed of weak, finely dispersed soils can lead to a complete or partial loss of soil bearing capacity and its transition to a fluid state. The magnitude of soil resistance to shear is determined by the degree of its water saturation; at a moisture content of about 20 %, the angle of repose of sandy soils is significantly reduced. The restoration of the strength properties of soils is prevented by the pore water pressure, the process of compaction (consolidation) of the soil mass occurs after the water is squeezed out of the pores, the flow time of which depends on the filtration properties of the massif. Liquefaction of soils leads to a disruption in the normal functioning of a hydraulic structure, the creation of emergency situations. The paper provides examples of accidents at hydraulic structures in Russia caused by liquefaction phenomena. It is noted that the main directions of protecting the structures of hydraulic structures from dangerous liquefaction are to prevent the possibility of liquefaction and the reduction of its harmful effects.  In this regard, several methods are considered – compaction and strengthening of soils at the base of structures; filtering surcharge device using geotextiles and geogrids; creation of an effective drainage system.

The paper considers a rectangular orthotropic insulated slab on an elastic foundation, modeled by an elastic homogeneous isotropic layer rigidly connected to a non-deformable foundation. Elastic and nonlinear calculations of this plate opening. The calculation of a flexible orthotropic slab on an elastic foundation in a nonlinear formulation is carried out iteratively by the method of B. N. Zhemochkin. To determine the coefficients of the canonical equations and free terms, a mixed method of structural mechanics was used. The deflections of a slab with a pinched normal in the main system of the mixed method due to the action of a concentrated force are determined by the Ritz method when the deflections are represented as a power polynomial in a new original expression, which is proposed by the author for the first time in the studies. This expression satisfies not only the boundary conditions of the pinched slab in terms of displacements, but also the biharmonic equation. In nonlinear calculations, when finding the variable (secant) stiffness for the Zhemochkin section,  at each iteration, the “stiffness – curvature” dependence is used for each of the X and Y directions, approximated by a nonlinear function, the nature of the dependence of which graphically indicates the nonlinear-elastic operation of the orthotropic plate and its deformation taking into account crack formation and crack opening. The algorithm for the above solution is implemented using the Wolfram Mathematica 11.3 computer program.

The description of a plane traveling electromagnetic wave existing in the physical literature by identical solutions of wave equations for the strengths of electric and magnetic fields is physically incorrect, since such solutions contradict the physical meaning of Maxwell’s equations and violate the energy conservation law. The paper gives a physically correct 

description of electromagnetic waves in the framework of Maxwell’s theory. New solutions of Maxwell’s wave equations for traveling electromagnetic wave are proposed, in which the strength of its electric and magnetic components change in time with shifts of a quarter of the period and a quarter of the wavelength along coordinate. The solutions describe a traveling electromagnetic wave, in which the energy of the electrical component is sequentially converted into the energy of the magnetic component and vice versa; the total energy density of the lossless wave remains constant in space at any time; the mutual orientation of the intensity vectors of the electric, magnetic fields and phase velocity changes from a left-handed three to a right-handed three every quarter of the wavelength; the energy flux density of the traveling wave is described by the Umov vector. It is shown that the formation of a standing electromagnetic wave does not require the loss of half a wave of one of the components of the wave reflected at the interface between the media.  In a standing wave, the total energy density remains constant in time, but it is a function of coordinates: there are points in space where the total energy density of the wave at any time is zero – these are nodes, and there are points where it has a maximum value – these are antinodes. Due to the inhomogeneity of the distribution of the total energy density of the wave in space, a standing electromagnetic wave cannot be considered as a harmonic oscillator, but a lossless traveling electromagnetic wave can.

To represent local displacement fields in the problem of the steady-state growth of a crack, which contains a plate of incompressible material, the strain intensity formula is used in the form of a polynomial of the second degree. The case of plane deformation for an elastoplastic material is considered. The solution is obtained by the method of asymptotic expansions. Numerical analysis is carried out for the first term of the expansion. The aim of the work is the process of obtaining analytical solutions to applied problems of the theory of plasticity: finding the components of stress and strain tensors. The paper considers a variant of the method of asymptotic expansions and its application for the problem of the distribution of the stress-strain state in an elastoplastic specimen with a crack. The method of asymptotic expansions has some advantages over the numerical approach in studying the stress-strain state in the vicinity of a crack. It allows to establish exact quantitative relationships between the radial component, the angle, and the components of the stress and strain tensor. Another advantage of this method is the possibility of compiling the mechanical characteristics of an object at the design stage. A system of differential equations has been developed that contains V₀ and its derivatives up to the third order. An example of stress distribution in the vicinity of a crack tip in a steel sample, obtained in a computer system by a numerical method, is given. The deformation diagram has been constructed for the material steel 40. The research results can be used to construct stress and strain fields in the vicinity of a crack, as well as to predict the further direction of crack development.

. Many important questions in the theory of elasticity lead to a variational problem associated with a biharmonic equation and to the corresponding boundary value problems for such an equation. The paper considers the main boundary value problem for the biharmonic equation in the unit circle. This problem leads, for example, to the study of plate deflections in the case of kinematic boundary conditions, when the displacements and their derivatives depend on the circular coordinate. The exact solution of the considered boundary value problem is known. The desired biharmonic function can be represented explicitly in the unit circle by means of the Poisson integral. An approximate solution of this problem is sometimes foundusing difference schemes. To do this, a grid with cells of small diameter is thrown onto the circle, and at each grid node all partial derivatives of the problem are replaced by their finite-difference relations.  As a result, a system of linear algebraic equations arises for unknown approximate values of the biharmonic function, from which they are uniquely found. The disadvantage of this method is that the above system is not always easy to solve. In addition, we get the solution not at any point of the circle, but only at the nodes of the grid. For real calculations and numerical analysis of solutions to applied problems, the authors have constructed its unified analytical approximate representation on the basis of the known exact solution of the boundary value problem while using logarithms. The approximate formula has a simple form and can be easily implemented numerically. Uniform error estimates make it possible to perform calculations with a given accuracy. All coefficients of the quadrature formula for the Poisson integral are non-negative, which greatly simplifies the study of the approximate solution. An analysis of the quadrature sum for stability is carried out. An example of solving a boundary value problem is considered. 

The paper considers an issue of international logistics development in Belarus under world coronavirus COVID-19 pandemic. As a result of analytical reviewing of the up-to-date scientific literature on the problem under study, the expediency is stated for highlighting the logistics of international road freight transportation as an independent area of theoretical research and practical activity. Further development of international road freight transportation logistics in the Republic of Belarus requires a complex economic analysis of the present state and dynamics of the international road freight transportation market, which is the main purpose of the present publication. The paper contains the results of the analysis of the market for international and transit freight road transportation. Separately the paper points out the dynamics of transit goods transportation segment in this market in response to the reorientation of international freight flows. Interconnection is shown between the volume indices of international road freight transit through the territory of Belarus and the dynamics of goods import to the Russian Federation from the EU countries. For the first time a proposal is given to use the methodological tools of systemmatrix diagnostic analysis for investigation of the macroeconomic development process of international road freight transit logistics. The author’s set of initial parameters for the analysis has been formed, and its results are presented on the basis of the data of the State statistical reporting form for 2019–2020. A suggestion is given for joint usage of system-matrix diagnostic tools in combination with author’s methodology for assessing the effectiveness of transit (international) freight transportation while evaluating the competitiveness of economic entities in the industry. The most typical trends of the market development in the short term are stated, taking into account the more complicated market conditions.

Developing and implementing innovative goods, new product groups, services and technologies in all spheres of society in terms of «economic» global digitalization requires advanced training of existing personnel, as well as the emergence of new specialties and professions. In this aspect, it is necessary to forecast the need for personnel in new specialties and specializations, as well as to study the possibilities for advanced training and retraining of existing specialists. The experience of the Comprehensive Forecast of Scientific and Technological Progress (CF STP) of the Republic of Belarus for 2021–2025 and for the period up to 2040 is analyzed in terms of staffing for innovative development. The required human resources potential has been assessed by the method of an expert survey in the process of forming the CF STP. Forecasting the need for personnel in the CF STP had a number of limitations. These limitations could be partially eliminated when developing the next CF STP and other forecasts of the need for personnel in the innovation economy. The paper considers possible directions for improving the forecasting of the provision of the economy with personnel for innovative development. It is advisable to assess the availability of personnel in the economy, taking into account the study of the ratio of supply and demand for them in quantitative and qualitative aspects. Algorithms are proposed for assessing: the needs of the national economy in personnel for innovative development; availability of personnel for innovative development and their competencies in organizations of the real sector of the economy; opportunities of the education system for training personnel for innovative development. The improvement of the methodology for forecasting the availability of personnel will allow to identify the most promising areas of training personnel with the necessary competencies and qualifications capable of developing and implementing technological innovations for the further development of the Republic of Belarus.

Internet marketing in the industrial market has a number of features and differs from that in the consumer market. The first is characterized by specialized and personalized approaches, as well as it must take into account the long cycle of the transaction, and the presence of a large number of decision makers. Marketers at industrial enterprises must have a range of knowledge about the market and industry trends, possess information about the organization of procurement processes and the operation of distribution channels, and be able to build a communication process with a certain type of customer. The paper presents methods for online promotion of industrial products and a study of the features of choosing of internet marketing tools for industrial enterprises. The main idea of the paper is to highlight the expert component for such enterprises as the main tool for communication with customers. To solve this problem, the internet marketing tools of an industrial enterprise has been considered and the expert component in its content strategy has been shown, approaches to the systematization of internet marketing tools have been studied. In the course of the study, general scientific methods have been used: analysis, synthesis, generalization. The result of the research is a model of product promotion in the logic of the industrial consumer's journey, which is based on the connection of the consumer's journey map with internet marketing tools.