Vol 24, No 4 (2020)

The paper considers the existence of solutions of the Dirichlet problem for nonlinear elliptic equations of the second order in unbounded domains. Restrictions on the structure of quasilinear equations are formulated in terms of a special class of convex functions (generalized $N$-functions). Namely, nonlinearities are determined by the Musilak–Orlicz functions such that the complementaries functions obeys the condition $ \Delta_2 $. The corresponding Musielak–Orlicz–Sobolev space does not have to be reflexive.This fact is a significant problem, since the theorem for pseudomonotone operators is not applicable here. For the class of equations under consideration, the proof of the existence theorem is based on an abstract theorem for additional systems. An important tool which allowed to generalize available results on the existence of solutions of the considered equations for bounded domains to the case of unbounded domains is an embedding theorem for Musielak–Orlicz–Sobolev spaces.Thus, in this paper, we find conditions on the structure of quasilinear equations in terms of the Musielak–Orlicz functions sufficient for the solvability of the Dirichlet problem in unbounded domains.In addition, we provide examples of equations which demonstrate that the class of nonlinearities considered in the paper is wider than non-power nonlinearities and variable exponent nonlinearities.

The influence of a size of the surface-plastic hardening region on the stress-strain state of a beam with a notch of a semicircular profile is investigated. The problem is reduced to a boundary value problem of fictitious thermoelasticity with the initial (plastic) deformations modeled by temperature anisrotropic deformations in an inhomogeneous temperature field. The solution is based on the finite element method. For model calculations, experimental data on the distribution of residual stresses in a smooth beam made of EP742 alloy after ultrasonic mechanical hardening were used as initial information. A variative numerical analysis of the effect of the notch radius and the size of the hardening zone of the beam face on the distribution of the components of the residual stress tensor in the smallest section from the bottom of the concentrator is carried out. It is shown that when the hardening zone is more than 16–20 % of the entire face area, the stress-strain state in the smallest section is practically stabilized. It was established that if the radius of the semicircular notch is less than the thickness of the hardened layer (the material compression area), an increase (in modulus) of the normal longitudinal component of the residual stress tensor occurs, and if the radius of the notch is greater than the thickness of the hardened layer, then a decrease (in modulus) of this value is observed in comparison with a similar component for a smooth reinforced beam for all values of the hardening zone more than 16–20 % of the entire face area of the beam. An experimental verification of the developed numerical method based on the finite element method for a beam with a fully hardened face is performed.

This article evaluates the accuracy of solutions to differential equations of motion, taking into account relativistic effects obtained on the basis of a new principle of interaction, using the example of studies of the evolution of the orbits of five asteroids.A numerical integration of equations of the asteroids' motion with the initial data referred to different points in time is carried out. Based on a comparison of the results of the study, certain patterns are revealed. At time intervals in the absence of rapprochement of the asteroid with the Earth less than 0.1 au it is possible to apply with equal efficiency the differential equations given in the paper. The loss of accuracy of numerical integration is directly dependent on the magnitude of the rapprochement of the asteroid width the Earth. Due to the fact that in the right sides of the equations of motion we have differences of the coordinates of the asteroid and the planet, with sufficient proximity, the relative accuracy of the coordinates is many times greater than the relative accuracy of the difference. For the studied asteroids, when they approach the Earth, the relative error of the difference in the coordinates of the asteroid and the Earth is approximately 227 to 44900 times higher than the limiting relative error of the coordinates of the asteroid itself. Predicting the motion of Apophis after its close approach to the Earth based on the solution of the equations of motion by modern methods leads to large errors, the reduction of which is possible only by improving the initial data of the elements of the orbits of the asteroid. About the possibility of close approach of Apophis with the Earth on a time interval from April 14, 2029 to January 1, 2100 it can be argued with a certain degree of probability. The results of the research can be generalized to all asteroids of Apollo and Aten groups.

The present paper deals with a pseudotensor formulation of the Neuber theory of micropolar elasticity. The dynamic equations of the micropolar continuum in terms of relative tensors (pseudotensors) are presented and discussed. The constitutive equations for a linear isotropic micropolar solid is given in the pseudotensor form. The final forms of the dynamic equations for the isotropic micropolar continuum in terms of displacements and microrotations are obtained in terms of relative tensors. The refinements of Neuber's dynamic equations are discussed. Those are also considered in the cylindrical coordinate net.

A flat orthotropic geometrically irregular shell of constant torsion, whose thermomechanical parameters are linearly dependent on temperature, is considered. When the temperature reaches a certain value, the change in the shape of the equilibrium occurs abruptly, which causes a change in the initial geometry of the shell. These temperatures are called critical. For practice, the relationships connecting the critical temperatures with the geometrical and thermomechanical parameters of the geometrically irregular shell are of considerable interest. The solution of the problems of static thermal stability of geometrically irregular shells usually begins with an analysis of their initial momentless state. Tangential forces caused by shell heating are defined as solutions of a system of singular differential equations of momentless thermoelasticity. These efforts are contained in the Brian or Reissner forms in the equations of static thermal stability and the further solution of the problem essentially depends on their structure. In this paper, the solution of singular momentless thermoelasticity is found by elementary functions. Using the method of displacement functions, the equations of moment thermoelasticity, written in the components of the displacement field, are reduced to a single singular differential equation in partial derivatives of the eighth order depending on the temperature, which is assumed to be constant. The solution is written as a double trigonometric series. The coefficients of the series, based on the Galerkin procedure, are determined as solutions to a linear homogeneous algebraic system of equations. From the equality to zero of the determinant of this system, an algebraic equation of the fifth degree is obtained for the relative critical temperature. The smallest positive real root of which is the desired temperature. A quantitative analysis of the influence of the geometrical and thermomechanical parameters of the geometrically irregular shell on the value of the critical temperature is carried out.

In the paper, Beltrami flows or helical flows are flows in which the vorticity and velocity vectors are collinear, and the proportionality coefficient between these vectors is nonzero and is the same at all points of the flow. A method is proposed that allows using known helical solutions to obtain new helical solutions of the Euler equations for an incompressible fluid. Some of these new solutions cannot be obtained by the known methods of replicating solutions by shifting and rotating the coordinate system, symmetry, scaling, cyclic permutation of the velocity and coordinate components, vector summation. The new replication method is applied to such parametric families of exact solutions in which the proportionality coefficient between velocity and vorticity remains unchanged for different values of the parameter. The essence of the method is that for such families the derivative of the velocity with respect to the parameter is also the helical velocity. The sequential differentiation of the speed of a new solution with respect to a parameter gives an endless chain of new exact solutions.