# Vol 25, No 3 (2021)

**Year:**2021**Articles:**12**URL:**https://journals.eco-vector.com/1991-8615/issue/view/4373**DOI:**https://doi.org/10.14498/vsgtu/v225/i3

## Full Issue

## Differential Equations and Mathematical Physics

### On a nonlocal boundary-value problem for a loaded parabolic-hyperbolic equation with three lines of degeneracy

#### Abstract

The work is devoted to the proof of the uniqueness and existence of a solution of a nonlocal problem for a loaded parabolic-hyperbolic equation with three lines of change of type. Using the representation of the general solution, the uniqueness of the solution is proved, and the existence of the solution is proved by the method of integral equations. Necessary conditions for the parameters and specified functions are established for the unique solvability of Volterra integral equations of the second kind with a shift equivalent to the problem under study.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):407-422

### The second initial-boundary value problem with integral displacement for second-order hyperbolic and parabolic equations

#### Abstract

In this paper, we study the solvability of some non-local analogs of the second initial-boundary value problem for multidimensional hyperbolic and parabolic equations of the second order. We prove the existence and uniqueness theorems of regular solutions (which have all Sobolev generalized derivatives that are summable with a square and are included in the equation). Some generalization and amplification of the obtained results are also given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):423-434

## Mechanics of Solids

### Numerical simulation of the creep process of titanium alloy VT6 under a multi-axis stress state taking into account the influence of an aggressive environment

#### Abstract

The problem of assessing the strength and resource of critical engineering objects is considered. The operating conditions of objects are characterized by high-temperature non-stationary thermomechanical effects, which lead to degradation of the initial strength properties of structural materials by the mechanism of long-term strength.

From the standpoint of the mechanics of a damaged medium, a mathematical model has been developed that describes the kinetics of the stress-strain state and the accumulation of damage during material degradation by the mechanism of long-term strength under conditions of a complex multiaxial stress state.

An experimental-theoretical method for finding the material parameters and scalar functions of the constitutive relations of the mechanics of a damaged medium based on the results of specially set experiments on laboratory samples is proposed.

The results of experimental studies and numerical modeling of the short-term high-temperature creep of VT6 titanium alloy under uniaxial and multiaxial stress states are presented. The numerical results are compared with the data of field experiments. Particular attention is paid to the issues of modeling the process of unsteady creep for complex deformation modes, accompanied by the rotation of the main areas of stress tensors, deformations and creep deformations, taking into account the effect of an aggressive environment, which is simulated by preliminary hydrogenation of laboratory samples to various hydrogen concentrations by mass.

It is shown that the developed version of the constitutive relations of the mechanics of a damaged medium allows, with sufficient accuracy for engineering calculations, to describe unsteady creep and long-term strength of structural alloys under multiaxial stress states, taking into account the effect of an aggressive medium (hydrogen corrosion).

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):435-456

### On the constitutive pseudoscalars of hemitropic micropolar media in inverse coordinate frames

#### Abstract

The paper is devoted to the constitutive pseudoscalars associated with the theory of hemitropic micropolar continuum. The basic concepts of pseudotensor algebra are presented. The pseudotensor form of the hemitropic micropolar elastic potential is given, based on 9 constitutive pseudoscalars (3 are pseudoscalars and 6 are absolute scalars). The weights of the constitutive pseudoscalars are calculated. The fundamental orienting pseudoscalar of weight \(+1\) is used to formulate transformation rules for constitutive pseudoscalars. The governing equations of the hemitropic micropolar elastic continuum are derived. The equations of the dynamics of the hemitropic micropolar continuum are discussed in terms of pseudotensors in right- and left-handed Cartesian coordinate systems. The presence of inverse modes along with normal ones is shown for wave propagation across the hemitropic micropolar continuum.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):457-474

### Rigorous solution of the problem of the state of a linearly elastic isotropic body under the action of polynomial bulk forces

#### Abstract

When solving boundary value problems about the construction of the stress-strain state of an linearly elastic, isotropic body, an important step is finding the internal state generated by the forces, distributed over the area occupied by the body. In the classical version, there is a numerical method for estimating the state at any point of the body based on the singular-integral representation of Cesaro. In the variant of conservative bulk forces, it is possible to construct solutions in an analytical form. With arbitrary regular effects of mechanical and other physical nature the force is not potential and the approaches of Papkovich–Neiber and Arzhanykh–Slobodyansky are powerless. In addition, the solution of nonlinear problems of elastostatics by means of the perturbation method, as well as the use of the Schwarz algorithm in solving problems for the study of multi-cavity solids, lead to the need to solve a sequence of linear problems. At the same time, fictitious bulk forces are necessarily generated, which as a rule have a polynomial nature.

The method proposed by the authors earlier for estimating the stress-strain state of a solid caused by the action of polynomial bulk forces represented in Cartesian coordinates has been improved. The internal state is restored in strict accordance with the forces statically acting on a simply connected bounded linear-elastic body. An effective method for constructing a solution and an algorithm for its computer implementation are proposed and described. Test calculations are demonstrated. The analysis of the state of the ball under the action of a superposition of bulk forces of different nature at different ratios of parameters that emphasize the level of influence of these factors is performed. The results are presented graphically. Conclusions are drawn:

a) the procedure for writing out the stress-strain state on the volume forces represented by polynomials from Cartesian coordinates is justified;

b) the algorithm is implemented in the Mathematica computing system and tested on high-order polynomials;

c) the analysis of the quasi-static state of a linear-elastic isotropic ball exposed to the forces of gravity and inertia at various combinations of parameters corresponding to the variants of slow, fast, compensatory (inertial forces are proportional to the gravitational) rotations is carried out.

The prospects for the development of a new approach to the class of bounded and unbounded bodies containing an arbitrary number of cavities are noted.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):475-490

## Mathematical Modeling, Numerical Methods and Software Complexes

### Exact solutions to the Navier–Stokes equations describing stratified fluid flows

#### Abstract

The paper considers the necessity of constructing exact solutions to the equations of dynamics of a viscous fluid stratified in terms of several physical characteristics, with density and viscosity taken as an example. The application of the families of exact solutions constructed for stratified fluids to modeling various technological processes dealing with moving viscous fluid media is discussed. Based on Lin’s exact solutions, linear in some coordinates, a class of exact solutions to the Navier–Stokes equations is constructed for viscous multilayer media in a mass force field. The class is then extended to the case of the arbitrary relation of kinetic force fields to all three Cartesian coordinates and time. The issues of overdetermination and solvability of the reduced (based on the families under study) Navier–Stokes equation system supplemented by the incompressibility equation are discussed. The case of isobaric shearing flow outside the mass force field is considered in detail as an illustration. Three approaches to obtaining consistency conditions for the overdetermined reduced system of motion equations are discussed, and their interrelation is shown.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):491-507

### Identification of linear dynamic systems of fractional order with errors in variables based on an augmented system of equations

#### Abstract

Equations with derivatives and fractional order differences are widely used to describe various processes and phenomena. Currently, methods of identification of systems described by equations with fractional order differences are actively developing. The paper is devoted to the identification of discrete dynamical systems described by equations with fractional order differences with errors in variables. The problems of identifying systems with errors in variables are often ill-conditioned. The paper proposes an algorithm that uses the representation of a normal biased system as an augmented equivalent system. This representation allows to reduce the number of conditionality of the problem to be solved. Test examples have shown that the proposed algorithm has a higher accuracy than the algorithms based on the decomposition of Cholesky and the minimization of the generalized Rayleigh quotient.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):508-518

### On the place of sonic points in a critical flow

#### Abstract

Stationary irrotational barotropic gas flows are investigated on the basis of the analysis of three-dimensional Euler equations. Critical flows in the article are those in which the Mach number is everywhere less than or equal to one, and at least at one point the Mach number reaches one. In 1954, D. Gilbarg and M. Shiffman showed that if an internal (not lying on the streamlined surface) sonic point exists in a critical flow, then it lies on a flat sonic surface, which at all its points is perpendicular to the gas velocity vector and cannot end inside the flow (theorem about the sonic point). Using this theorem, D. Gilbarg and M. Shiffman obtained a conclusion that is important for the problems of maximizing the critical Mach number. It consists in the fact that in a critical flow for a wide class of bodies in flow, sonic points can be located only on its surface. This conclusion is essentially used in constructing the shapes of streamlined bodies with the maximum value of the critical Mach number (for given isoperimetric conditions).

In this paper, the question of the curvature of streamlines at the internal sonic points of critical flows is considered. It is shown that this curvature is zero. The result is a new necessary condition for the existence of an interior sonic point (and sonic surface). It consists in the fact that at the point of intersection with the sonic surface, the normal curvature of the streamlined surface in the direction normal to the sonic surface should be equal to zero. Examples of streamlined bodies are given for which the theorem by D. Gilbarg and M. Shiffman (on the sonic point) does not answer the question of the location of the sonic points, at the same time a new necessary condition makes it possible to prove that the existence of internal sonic points in a critical flow around these bodies is impossible.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):519-530

### Mathematical modeling and numerical method for estimating the characteristics of non-isothermal creep based on the experimental data

#### Abstract

The desire to reduce the mass of machines and structures while improving their quality, as well as to make the most complete use of the mechanical properties of materials, requires permanent improvement and development of known methods for calculating and analyzing the stress-strain state of materials under creep conditions.

The article proposes a numerical method for estimating the characteristics of the third stage of non-isothermal creep based on a set of creep diagrams constructed when processing test results for various values of nominal stress and temperature.

The method is based on the nonlinear regression model, the root-mean-square estimates of the parameters of which are found by linearization, including on the basis of difference equations describing the experimental results. The proposed numerical method can also be used to estimate the parameters of the third creep stage, when the experimental results are presented in the form of a set of test diagrams for only one temperature.

The results of testing the developed numerical method for processing the experimental results in the form of creep diagrams for the 09G2C alloy at different temperatures are presented. The reliability and efficiency of the calculation algorithms and methods of nonlinear estimation presented in the work are confirmed by the results of numerical and analytical studies and mathematical models of the third stage of non-isothermal creep constructed on the basis of experimental data.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):531-555

### Parameter differentiation method in solution of axisymmetric soft shells stationary dynamics nonlinear problems

#### Abstract

An algorithm of axisymmetric unbranched soft shells nonlinear dynamic behaviour problems solution is suggested in the work. The algorithm does not impose any restrictions on deformations or displacements range, material properties, conditions of fixing or meridian form of the structure. Mathematical statement of the problem is given in vector-matrix form and includes system of partial differential equations, system of additional algebraic equations, structure segments coupling conditions, initial and boundary conditions. Partial differential equations of motion are reduced to nonlinear ordinary differential equations using method of lines. Obtained equation system is differentiated by calendar parameter. As a result problem solution is reduced to solving two interconnected problems: quasilinear multipoint boundary problem and nonlinear Cauchy problem with right-hand side of a special form. Features of represented algorithm using in application to the problems of soft shells dynamics are revealed at its program realization and are described in the work. Three- and four-point finite difference schemes are used for acceleration approximation. Algorithm testing is carried out for the example of hinged hemisphere of neo-hookean material dynamic inflation. Influence of time step and acceleration approximation scheme choice on solution results is investigated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):556-570

### Globalization of the analysis of particle placement models by cells

#### Abstract

The title of the paper means that its goal is a general approach to the pre-asmptotic analysis of schemes with different qualities in all combinations of their distinguishability of their constituent elements (cells and particles). To do this, in each group of such schemes with general restrictions, instead of directly studying them based on the specificity of each scheme, a certain general set of algorithmic procedures for recalculating the results of their pre-asymptotic analysis in the scheme is proposed, starting with the scheme with the greatest differentiation of their outcomes, sequentially for other schemes of the group with differences as one item. The analysis of each scheme is carried out according to the traditional and in a number of new following directions: constructing a random process of formation and numbered non-repeated enumeration of the outcomes of the scheme in the order of their receipt, finding their number, solving the numbering problem for the outcomes of the scheme, which consists in establishing a one-to-one correspondence between their types and numbers, setting their probabilistic distribution and modeling the outcomes of the scheme with this probabilistic distribution.

In particular, the cases of groups of schemes without restrictions on the placement of particles and with a restriction of at most one particle in a cell are studied separately, which lead to some well-known analytical results. Under any restrictions in the considered group of circuits, their analysis is carried out by implementing algorithmic procedures for sequential transformation of the results of the analysis of one circuit of the group for another. Combinations into such pairs of schemes are made on the basis of the difference in the quality of one of their elements.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):571-587

## Short Communications

### Second integral generalization of the Crocco invariant for 3D flows behind detached bow shock wave

#### Abstract

Stationary flows of an ideal gas behind the detached bow shock are investigated in the general 3D case. The well-known integral invariant (V.N. Golubkin, G.B. Sizykh, 2019), generalizing the axisymmetric invariant of (L. Crocco, 1937) to asymmetric flows, is a curvilinear integral over a closed vortex line (such lines lie on isentropic surfaces), in which the integrand is the pressure divided by the vorticity. This integral takes on the same value on all (closed) vortex lines lying on one isentropic surface. It was obtained after the discovery of the fact that the vortex lines are closed in the flow behind the shock in the general 3D case. Recently, another family of closed lines behind the shock was found, lying on isentropic surfaces (G.B. Sizykh, 2020). It is given by vector lines **a** — the vector product of the gas velocity and the gradient of the entropy function. In the general 3D case, these lines and vortex lines do not coincide.

In the presented study, an attempt is made to find the integral invariant associated with closed vector lines **a**. Without using asymptotic, numerical and other approximate methods, the Euler equations are analyzed for the classical model of the flow of an ideal perfect gas with constant heat capacities. The concept of imaginary particles “carrying” the streamlines of a real gas flow, based on the Helmholtz–Zoravsky criterion, is used. A new integral invariant of isentropic surfaces is obtained. It is shown that the curvilinear integral over a closed vector line **a**, in which the integrand is the pressure divided by the projection of the vorticity on the direction **a**, has the same values for all lines a lying on one isentropic surface. This invariant, like another previously known integral invariant (V.N. Golubkin, G.B. Sizykh, 2019), in the particular case of non-swirling axisymmetric flows, coincides with the non-integral invariant of L. Crocco and generalizes it to the general spatial case.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(3):588-595