Vol 22, No 1 (2018)

The paper is devoted to Alexander Pavlovich Soldatov, the well known mathematician, and his contribution to the development of scientific theories and applications. Alexander Pavlovich has turned 70 years on the January, 18th, so we give a short biographical background to his anniversary. Alexander Pavlovich Soldatov was a student of Novosibirsk State University, and graduated from with honours. He continued his education, defended a thesis in the famous V. A. Steklov Mathematical Institute of the USSR Academy of Sciences, became a Candidate of Physical and Mathematical Sciences and then Doctor habilitated of Physical and Mathematical Sciences (in Moscow State University). Today he is a professor, the Leading Researcher of Dorodnicyn Computing Centre in Federal Research Center “Computer Science and Control” of RAS, and has a title of Honored Scientist of Russia. The studies of Alexander Pavlovich in partial differential equations are well known to the specialists. He is an author of more then 170 scientific articles and four monographies. Here we give a list (not complete) of A. P. Soldatov publications for the last five years, that should help readers to see the modern interests of scientist. Alexander Pavlovich Soldatov is an excellent teacher, also he works as reviewer for mathematical journals, develops the mathematical competitions for school and university students and organizes mathematical conferences including international. In the paper we introduce the contribution of Alexander Pavlovich to the mathematics and education science, and give a short compilation of his remarkable scientific results.

The subject of this research is a mathematical description of the shape and the stress-strain state of a steel plate subjected to unilateral shot peening, its experimental verification and application of the results for verification of methods for reconstruction of residual stress and strain fields according to experimental data. Such plate is used in manufacturing as a calibrating sample to determine of shot peening duration required for formation of proper compressive tangential stress in the surface layer of the processed product. The method of calibration is convenient and widely applied in different technologies of surface hardening. In that case the source of the residual stresses is plastic strains in surface layer produced by shot peening. For the statement of the problem a plastic strain tensor field is defined up to an arbitrary function. The shape and the stress-strain state of an elastic plate with the surface layer of plastic strains were calculated numerically. The qualitative behavior of numerical solution allowed us to accept the set of hypotheses to find an analytical solution of the spatial problem of elasticity theory and to weaken the boundary conditions. The exact solution has been found analytically. Within the framework of the plane stress state along the thickness and transverse directions, the result exactly corresponds to the Davidenkov-Birger formula connected the tangential residual stress distribution on depth with the function of deflections. An explicit formula for the dependence of the residual (plastic) deformation on the thickness coordinate is obtained. Sources of errors of the received expressions and methods of their correction are analyzed. An experiment has been carried out on the one-sided shot peening of calibration plate made of hardened 65G steel, for which the layer-by-layer etching of the treated surface and the measurement of the flexure of the plate were made (by Davidenkov method). The profiles of residual stresses and strains were reconstructed numerically with reasonable accuracy using the obtained experimental data. The result is applicable to a wide class of problems for elastic bodies with hardened surface layers. It may serve as a base for experimental research of such problems, help to formulate hypotheses and test them by experiment, help to study relation between physical fields in asymptotic case, help to verify applicability of different ways to account residual stresses in numerical solution. The solution found can be used for verification of stress and displacement fields in different cases of preliminarily stressed shell elements in engineering software for calculation of fatigue endurance of different machine parts with hardened surface layer. It also seems to be a reference for the study of surface-hardened bodies with curved free boundary, to which most of the practically important tasks are reduced.

The stability of deformable element of a construction in the form of a plate-strip with its flowing by supersonic flow of ideal gas is investigated. Adopted in paper definitions of stability are consistent with the concept of stability of dynamical systems by Lyapunov. For the description of dynamics of an elastic body the nonlinear mathematical model taking into account transverse and longitudinal deformations of the elastic plate is used. The model describes the associated system of partial differential equations for two unknown functions of deformations. Aerodynamic pressure upon a plate is defined according to Ilyushin’s “piston” theory. On the base of the built functional for the case of hinged motionless fixing the ends of the plate the sufficient conditions of stability of the solution of the system of equations describing the length-cross oscillations of the plate are obtained. The estimation of the amplitude of deformations depending on initial conditions is made. On a specific example of one mechanical system the using of the proved theorems and estimates is shown.

The use of the Taylor polynomial of the second degree when approximating the derivatives by finite differences leads to the second order of approximation of the traditional method of nets in the numerical integration of second-order ordinary differential equations with variable coefficients. In the study of boundary-value problems for the third-order ordinary differential equations with variable coefficients, we offer the previously proposed method of numerical integration, using the means of the matrix calculus, in which approximation of the derivatives by finite differences was not used. According to this method, in the construction of a system of difference equations, an arbitrary power of the Taylor polynomial in the expansion of the desired solution of the problem in a Taylor series can be chosen. The disparity is calculated and an estimate of the order of approximation of the method is given depending on the chosen degree of the Taylor polynomial using the four-point pattern. The regularities between the order of approximation of the matrix method and the degree of the used Taylor polynomial are theoretically revealed. We found out that the order of approximation is proportional to the degree of the used Taylor polynomial and less by two than it. We propose a procedure for constructing a fictitious boundary condition that allows us to construct a closed system of difference equations for the matrix method of numerical integration. The system of difference equations is divided into two subsystems: the first subsystem consists of two equations, the first of which contains the given value of the derivative in the boundary conditions of the problem, the second one contains the value calculated from the fictitious boundary condition; the second subsystem consists of the remaining difference equations of the constructed closed system. The disparity is calculated and an estimate of the order of approximation of the method is given depending on the chosen degree of the Taylor polynomial using the five-point pattern. The regularities between the order of approximation of the matrix method and the degree of the used Taylor polynomial are theoretically revealed. The following is revealed: a) the order of approximation of the first subsystem, the second subsystem with an even value of the degree of the Taylor polynomial and the whole problem is proportional to this degree and less than it by two; b) the order of approximation of the second subsystem with an odd value of the degree of the Taylor polynomial is proportional to this degree and less than it by one.