# Vol 22, No 1 (2018)

**Year:**2018**Articles:**9**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1227

### To the 75th Anniversary of Professor Evgeniy Vladimirovich Radkevich

#### Abstract

One of the most influential scientists in mathematical science, Evgeniy Vladimirovich Radkevich, has celebrated his 75th anniversary on the January, 26th. Here we give the biographical background and discuss his well known researches in different areas of science and technology. Evgeniy Vladimirovich Radkevich began his mathematical education in Moscow State University, Faculty of Mechanics and Mathematics. He has studied under Olga Arsenievna Oleinik, a famous mathematician, one of the best specialists in the theory of partial differential equations, and has become a Candidate of Physical and Mathematical Sciences. E. V. Radkevich worked in the leading mathematical institutes-Institute for Problems in Mechanics of the USSR Academy of Sciences, Moscow Institute of Electronics and Mathematics, Moscow Institute of Radio-Engineering, Electronics and Automation, Moscow State University. Now he is Professor, Doctor habilitated of Physical and Mathematical Sciences, his interests are in many areas of mathematics (differential equations and boundary value problems, asymptotical methods, phase transition problems and so on) and also he is an experienced teacher and lecturer. In the paper we give a list of Evgeniy Vladimirovich publications for the last five years to show the level of his scientific researches. Also here we discuss his greatest scientific and technological contributions and concern the other important area of interest of E. V. Radkevich-the theatre arts, and his work as theatre director.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(1):7-14

7-14

### To the 70th Anniversary of Professor Alexander Pavlovich Soldatov

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(1):15-22

15-22

### The use of piecewise linear plastic potentials in the nonstationary theory of temperature stresses

#### Abstract

Under the conditions of a piecewise linear plastic potential defining in the space of principal stresses the plasticity condition for the maximum reduced tangential stresses, a solution is obtained for a one-dimensional quasistatic problem of the theory of temperature stresses about the local heating of a circular plate made of an ideal elastoplastic material. The yield point is assumed to be temperature dependent. A comparison is made in the distribution of the current and residual stresses during heating and cooling of the plate metal obtained both with the dependence of the elastic models on the temperature, and without taking this dependence into account. It is shown that, depending on the rate and temperature of heating, the regime of plastic flow can change under the transition of stressed states from one face of the loading surface to another. In this case, the possibility of an inclined prism of fluidity on the edge is excluded, the surface of which in the space of principal stresses is the loading surface.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(1):23-39

23-39

### On the reconstruction of residual stresses and strains of a plate after shot peening

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(1):40-64

40-64

### Analysis of properties of creep curves generated by the linear viscoelasticity theory under arbitrary loading programs at initial stage

#### Abstract

The general equation of creep curves family generated by the linear integral constitutive relation of viscoelasticity (with an arbitrary creep compliance function) under arbitrary non-decreasing stress histories at initial stage of loading up to a given stress level is derived and analyzed. Basic qualitative properties of the theoretic creep curves and their dependence on a rise time magnitude, on a loading program shape at initial stage and on creep function characteristics are studied analytically in the uni-axial case assuming creep compliance is an increasing convex-up function of time. Monotonicity and convexity intervals of creep curves, their asymptotic behavior at infinity and conditions for convergence to zero of the deviation from the creep curve under instantaneous (step) loading to a constant stress with time tending to infinity are examined. Two-sided bounds have been obtained for such creep curves and for deviation from the creep curve under step loading and for differences of creep curves with different initial programs of loading up to a given stress level. The uniform convergence of the theoretic creep curves family (with fixed loading law at initial stage) to the creep curve under step loading with the rise time tending to zero has been proved. The analysis revealed the importance of convexity restriction imposed on a creep compliance and the key role of its derivative limit value at infinity. It is proved that the derivative limit value equality to zero is the criterion for memory fading. General properties and peculiarities of the theoretic creep curves and their dependence on loading program shape at initial stage are illustrated by the examination of the classical rheological models (consisting of two or three spring and dashpot elements), fractional models and hybrid models (with piecewise creep function). The main classes of linear models are considered and specific features of their theoretic creep curves are marked. The results of the analysis are helpful to examine the linear viscoelasticity theory abilities to provide an adequate description of basic rheological phenomena related to creep and to indicate the field of applicability or non-applicability of the linear theory considering creep test data for a given material. The results constitutes the analytical foundation for obtaining precise two-sided bounds and correction formulas for creep compliance via theoretic or experimental creep curves with initial stage of loading (ramp loading, in particular) and for development of identification, fitting and verification techniques.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(1):65-95

65-95

### Dynamic stability of deformable elements of designs at supersonic mode of flow

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(1):96-115

96-115

### Development of mathematical models and research of strongly nonequilibrium phenomena taking into account space-time nonlocality

#### Abstract

Based on the principles of locally-nonequilibrium thermodynamics, the mathematical models of heat, mass, momentum transfer processes were developed taking into account spatial and temporal nonlocality. The derivation of the differential equations is based on the taking into account accelerate in time as the specific fluxes (heat, mass, momentum) and the gradients of the corresponding variables in the Fourier’s, Fick’s, Newton’s, Hooke’s, Ohm’s, etc. diffusion laws. Studying of exact analytical solutions of the obtained models allowed us to discover new regularities of the changes of the desired parameters at low and ultra low values of temporal and spatial variables, and for all fast processes, time change which is comparable with the relaxation time. And, in particular, from the analysis of the exact analytical decision the fact of a time lag of acceptance of a boundary condition of the first kind demonstrating that in view of resistance of the body shown to warmth penetration process, its instantaneous warming up on boundary is impossible under no circumstances heat exchange with the environment is found. Therefore, the heat emission coefficient on a wall depends not only on heat exchange conditions (environment speed, viscosity and so forth), but also on physical properties of a body and it, in the first, is variable value in time and, in the second, it can not exceed some value, limit for each case.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(1):116-152

116-152

### Numerical integration by the matrix method of boundary value problems for linear inhomogeneous ordinary differential equations of the third order with variable coefficients

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(1):153-183

153-183

### The equiconvergence theorem for an integral operator with piecewise constant kernel

#### Abstract

The paper is devoted to the equiconvergence of the trigonometric Fourier series and the expansions in the eigen and associated functions of the integral operator A, the kernel of which has jumps on the sides of the square inscribed in the unit square. An equivalent integral operator in the space of 4-dimension vector-functions is introduced. This operator is remarkable for the fact that the components of its kernel have discontinuities only on the line diagonal. Necessary and sufficient conditions of the invertibility of the operator A are obtained in the form that a certain 4th order determinant is not zero. The Fredholm resolvent of the operator A is studied and its formula is found. The constructing of this formula is reduced to the solving of the boundary value problem for the first order differential system in the 4-dimension vector-functions space. To overcome the difficulties of this solving the transformation of the boundary value problem is carried out. Conditions analogous to Birkhoff regularity conditions are also obtained. These conditions mean that some 4th order determinants are not zero and can be easily verified. Under these conditions the determinant, which zeros are the eigenvalue of the boundary value problem, can be estimated. The equiconvergence theorem for the operator A is formulated. The basic method used in the proof of this theorem is Cauchy-Poincare method of integrating the resolvent of the operator A over expanding contours in the complex plane of the spectral parameter. An example is also given of the integral operator with piecewise constant kernel, which satisfies all the requirements obtained in the paper.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(1):184-197

184-197