Vol 26, No 2 (2022)

February 10, 2022 the famous scientist in mechanics of solids and applied mathematics, teacher, organizer of science and higher education in Russia Yuri N. Radayev is celebrating his 60th anniversary. Yuri N. Radayev is known as a prominent scientist in the field of mechanics and applied mathematics. The principal directions of his academic activity are the Mathematical Theory of Plasticity, Fracture Mechanics, the Theory of Cracks and Microdamages, Coupled Hyperbolic Thermoelasticity and Thermomechanics, Micropolar Elasticity, Mechanics of Granular Solids, Mechanics of Growing Solids. In this biographical background we discuss the scientific and educational work of Prof. Yuri N. Radayev, give an information on his achievements and a list of his main publications.

In this paper, by using the heat kernel and the transport operator on each step of time discretization, approximate solutions for the transport-diffusion equation on the half-plane ${\mathrm{ℝ}}_{+}^2$ are constructed, and their convergence to a function which satisfies the transport-diffusion equation and the initial and boundary conditions is proved. These approximate solutions can be considered as a deterministic version of (the approximation of) the stochastic representation of the solution to parabolic equation, realized by the relationship between the heat kernel and the Brownian motion. But as they are defined only by an integral operator and transport, their properties and their convergence are proved without using probabilistic notions. The result of this paper generalizes that of recent papers about the convergence of analogous approximate solutions on the whole space ${\mathrm{ℝ}}^n$. In case of the half-plane, it is necessary to elaborate (not trivial) estimates of the smoothness of the approximate solutions influenced by boundary condition.

In present paper, an initial-boundary value problem is formulated in a rectangle for a higher even order partial differential equation with the Bessel operator. Applying the method of separation of variables to the considered problem a spectral problem is obtained for an ordinary differential equation of higher even order. The self-adjointness of the last problem is proved, which implies the existence of the system of its eigenfunctions, as well as the orthonormality and completeness of this system. The uniform convergence of some bilinear series and the order of the Fourier coefficients, depending on the found eigenfunctions, is investigated. The solution of the considered problem is found as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. The absolute and uniform convergence of this series, as well as the series obtained by its differentiating, have been proved. The uniqueness of the solution of the problem is proved by the method of spectral analysis. An estimate is obtained for the solution of the problem which implies the continuous dependence of the solution on the given functions.

The paper describes the mechanisms and conditions for the damage-memory effect (Kaiser effect) in rocks subjected to a three-dimensional nonproportional cyclic loading with changes in the rocks' shape and orientation of the Lamé-ellipsoid. The experiments with the cubic samples taken from polymictic sandstone were conducted on Triaxial Independent Loading Testing System with continuous recording of an acoustic emission signals. The results of a nonproportional triaxial compression under the developed protocol, it is 9-cycle loading program, have shown that a dominate mechanism of the damage-memory effect in each ensemble of cracks (vectored differently) is the development of micro-cracks of opening fracture mode oriented subnormally to the minimum main stress. It was found that the Kaiser damage-memory effect is detected not so much to the fact of opening cracks, friendly oriented, as to a discrete growing (increase of length) of already existing and newly emerging micro-cracks. The obtained results can be considered as a trigger for models development oriented to strain and destruction of rocks, taking into account the anisotropic nature of damage accumulation.

The paper studies the results of mathematical modeling of the external flow of Siemens 3D model SWT–3.6–120 (B52 air foil) horizontal axis wind turbine (HAWT), using the Navier–Stokes equations averaged by Reynolds (RANS) closed by k−ε, k−ω Shear Stress Transport (SST) and Eddy Viscosity Transport (EVT) turbulence models. The task of correct determination of the wind speed vector deviation angle over the nacelle of the HAWT is required by operation of the yawing system, which determines in turn the efficiency of the entire turbine. The Struhal number was chosen as a comparison criterion, defined for the transverse flow around the cylinder, describing the frequency of the formation of vortex structure behind the butt part of the blade of the HAWT. The calculated area consists of 3 million tetrahedral volumes with prismatic layer on the surface of the nacelle, using local grinding. The place of flow direction parameters registration is located at a height of 3 m above the nacelle and at a distance of 8 m from the blade shank, which corresponds to the standard location of the weather vane. The analysis of the obtained results showed that the k−ε and EVT turbulence models describe the flow parameters over the HAWT nacelle in almost the same way, but the EVT model represents just one differential equation, thereby it is preferable by the computational cost criterion. Also, one of the advantages of one-parameter turbulence model (EVT model) is a smaller number of closing semi-empirical constants, the analysis of which allows the expanding of the engineering techniques scope for the modeling of turbulent processes in solving the practical problems related to the design of control systems for the wind turbines, increasing their efficiency.

This article is devoted to the study of an inverse source problem for a mixed type equation with a fractional diffusion equation in the parabolic part and a wave equation in the hyperbolic part of a cylindrical domain. The solution is obtained in the form of Fourier–Bessel series expansion using an orthogonal set of Bessel functions. The theorems of uniqueness and existence of a solution are proved.