# Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences

The Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences is the periodical scientific edition published by Samara State Technical University since 1996.

For a long time the journal was an edition where the new scientific results of Russian scientific schools had been published. Now the journal is focused on both Russian and foreign scientists, working in the priority research areas of Samara State Technical University because the main purpose of the journal is an open dissemination of scientific knowledge among Russian and foreign scientists.

Since 2011 the journal is a quarterly printed edition (four issues a year); issue size — 200 p.; language of articles — Russian, English. The journal is published in printed and electronic version.

The editorial board takes and estimates the manuscripts irrespective of race, gender, nationality, heritage, citizenship, occupation, employment, residence, political, philosophic, religious and any other views of the author.

The contributed article should be a completed scientific research. It shouldn't have been published, or be in process of publication in other editions.

The manuscript should contain novel scientific results in the priority research areas of Samara State Technical University, including “Differential Equations and Mathematical Physics”, “Mechanics of Solids”, “Mathematical Modeling, Numerical Methods and Software Systems”.

The journal is published at the expense of publisher. All materials are publishing free of charge, the author's fee is not provided. All materials of the electronic version are freely available.

The target audience of the journal are the scientists working in the following areas:

- “Differential Equations and Mathematical Physics”,
- “Deformable Solid Body Mechanics”,
- “Mathematical Modeling, Numerical Methods and Software Systems”.

The journal is included in the Russian Science Citation Index database on the Web of Science platform. The journal is included in VINITI abstracts databases. The issue details are publishing in ULRICH’S Periodical Directory. The journal articles are indexed in Scholar.Google.com, zbMATH, СyberLeninka.ru, Math-Net.ru. The journal is integrated in CrossRef and FundRef search systems.

## Current Issue

### Vol 26, No 1 (2022)

**Year:**2022**Articles:**10**URL:**https://journals.eco-vector.com/1991-8615/issue/view/5152**DOI:**https://doi.org/10.14498/vsgtu/v226/i1

#### Full Issue

#### Differential Equations and Mathematical Physics

##### Numerical method for solving an initial-boundary value problem for a multidimensional loaded parabolic equation of a general form with conditions of the third kind

###### Abstract

An initial-boundary value problem is studied for a multidimensional loaded parabolic equation of general form with boundary conditions of the third kind. For a numerical solution, a locally one-dimensional difference scheme by A.A. Samarskii with order of approximation $O{h}^{2}+\tau $ is constructed. Using the method of energy inequalities, we obtain a priori estimates in the differential and difference interpretations, which imply uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem in the ${L}_{2}$ norm at a rate equal to the order of approximation of the difference scheme. An algorithm for the computational solution is constructed and numerical calculations of test cases are carried out, illustrating the theoretical calculations obtained in this work.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):7-36

#### Mechanics of Solids

##### On covariant non-constancy of distortion and inversed distortion tensors

###### Abstract

The paper deals with covariant constancy problem for tensors and pseudotensors of an arbitrary rank and weight in an Euclidean space. Requisite preliminaries from pseudotensor algebra and analysis are given. The covariant constancy of pseudotensors are separately considered. Important for multidimensional geometry examples of covariant constant tensors and pseudotensors are demonstrated. In particular, integer powers of the fundamental orienting pseudoscalar satisfied the condition of covariant constancy are introduced and discussed. The paper shows that the distortion and inversed distortion tensors are not actually covariant constant, contrary to the statements of those covariant constancy which can be found in literature on continuum mechanics.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):37-47

##### Modeling the process of equilibrium crack growth in a composite specimen from the standpoints of the postcritical deformation mechanics

###### Abstract

Ensuring the strength and safety of structures requires studying the issues of crack initiation and equilibrium growth. An analogy between the approaches of phenomenological fracture mechanics, which is based on the complete deformation diagrams usage, and crack propagation mechanics is noted. The applying of previously developed postcritical deformation mechanics models, which describes accompanied by softening equilibrium damage accumulation processes, is advisable. On the example of the numerical, with cohesive elements using, simulation of composite specimen interlayer fracture, the realization of the material deformation complete diagram near the crack tip is demonstrated. The calculated loading diagrams are constructed, the points of the postcritical deformation zone initiation and the beginning of crack growth are shown. Relations between softening modulus value and maximum values of load, crack opening and length are revealed. The influence of the loading system rigidity is noted. It is concluded that consideration of the constructions deformation and fracture processes modeling problems using cohesive elements from the postcritical mechanics deformation standpoints is expedient.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):48-61

##### Modelling one-dimensional elastic diffusion processes in an orthotropic solid cylinder under unsteady volumetric perturbations

###### Abstract

A polar-symmetric elastic diffusion problem is considered for an orthotropic multicomponent homogeneous cylinder under uniformly distributed radial unsteady volumetric perturbations. Coupled elastic diffusion equations in a cylindrical coordinate system is used as a mathematical model. The model takes into account a relaxation of diffusion effects implying finite propagation speed of diffusion perturbations.

The solution of the problem is obtained in the integral convolution form of Green’s functions with functions specifying volumetric perturbations. The integral Laplace transform in time and the expansion into the Fourier series by the special Bessel functions are used to find the Green’s functions. The theory of residues and tables of operational calculus are used for inverse Laplace transform.

A calculus example based on a three-component material, in which two components are independent, is considered. The study of the mechanical and diffusion fields interaction in a solid orthotropic cylinder is carried out.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):62-78

##### Numerical simulation of the interaction of a shock wave with a permeable deformable granulated layer

###### Abstract

The article presents a mathematical model that describes, in a one-dimensional approximation, the interconnected processes of unsteady deformation of flat permeable granular layers. The model consists of solid particles and wave processes in pore and surrounding gas. The model is based on nonlinear equations of dynamics of two interpenetrating continua. As interfacial forces, drag forces are taken into account when gas flows around ball particles and friction forces. The numerical solution of the equations is carried out according to the modified scheme of S. K. Godunov, adapted to the problems of the dynamics of interpenetrating media. The contact surfaces of pure gas with the porous granular layer and pore gas are the surface of the fracture of porosity and permeability. The numerical implementation of contact conditions is based on the solution of the problem of disintegration of a gap at a jump in porosity. Solutions are obtained for the effects of plane shock waves on a deformable granular layer. We study the transformation of waves passing through an elastoplastic granular layer with and without taking into account changes in the permeability of the layer. When solving problems, the dependence of the change in the permeability of a layer on its compression is used, which is also obtained numerically when modeling the compression of symmetric fragments of granular layers in a spatial setting. Numerical studies of the processes of nonlinear interaction of shock waves with deformable permeable granular layers have shown that the parameters of transmitted and reflected waves substantially depend on the degree of compression of the granular layers. Assessment of the protective properties of permeable barriers when exposed to strong shock waves should be carried out taking into account changes in their permeability due to deformation layers.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):79-92

##### An application of Mueller's method for determining eigenfrequencies of vibrations of viscoelastic bodies with frequency-dependent characteristics of a material

###### Abstract

A search for optimal damping properties of structures using methods of numerical modelling is as a rule associated with a large number of computations. Alongside this an application of mechanical problem of natural vibrations of structures for this purpose allows estimating damping properties of structures regardless external force and kinematic impacts. This fact leads to sufficient decrease in computational costs. The results of the solution to the problem of natural vibrations of piecewise-homogeneous viscoelastic bodies are complex natural vibration frequencies, the real part of which is a frequency of vibrations and imaginary part is damping index (rate of vibration damping). A mechanical behavior of a viscoelastic material is described by the linear theory of Boltzman–Volterra. Within the frameworks of this theory mechanical properties of a viscoelastic material can be represented as complex dynamic moduli (shear modulus and bulk modulus). As a rule, these properties depend on frequency of external excitation. In current paper an algorithm which allows obtaining solution to the problem on natural vibrations, in case when components of complex dynamic moduli are frequency-dependent, is represented. The algorithm is based on using capabilities of the ANSYS software package and also the Mueller's method which allows solving partial problem of complex eigenvalues. An efficiency and productivity of the algorithm is demonstrated on the example of a two-layered cantilever plate. One layer of the plate is made of an elastic material and the second one is made of a viscoelastic material. Reliability of the obtained results is proved by comparison natural vibration frequencies obtained as a result of solution to the problem of natural vibrations and resonant frequencies at frequency response plots of the displacements obtained as a result of solution to the problem of forced steady-state vibrations using the ANSYS software package.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):93-118

##### Relaxation of residual stresses in a surface-hardened rotating cylinder under creep conditions

###### Abstract

A technique for calculating the relaxation of residual stresses in a cantilevered rotating cylinder after the procedure of surface plastic deformation under creep conditions has been developed, taking into account the effect of a stepwise change in the parameters of temperature-force loading (unloading). The problem simulates the stress-strain state of a surface-hardened cylinder (rod), the end section of which is rigidly fixed on a disk rotating at a constant angular velocity.

The technique includes a method for reconstructing the fields of residual stresses and plastic deformations and a method for calculating the relaxation of residual stresses during creep of a rotating cylindrical rod. Since the tensile stresses caused by rotation along the length of the rod do not change in time, the problem of relaxation of residual stresses for a stretched rod at constant stress is solved in each cross section.

A detailed numerical study of the effect of the number of revolutions on the rate of relaxation of residual stresses was performed for a shot-hardened cylindrical sample with a radius of 3.76 mm made of EI698 alloy at a temperature of 700 ºC.

Analysis of the calculation results allowed to establish a non-trivial effect, which consists in the fact that the relaxation of residual stresses in sections subjected to axial tensile stresses due to rotation occurs less intensively than in the “tail” section, where the axial load from rotation is zero. The results obtained in this work can be useful in evaluating the effectiveness of surface-plastic hardening of parts under high-temperature creep conditions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):119-139

#### Mathematical Modeling, Numerical Methods and Software Complexes

##### The use of pseudoresiduals in the study of convergence of unstable difference boundary value problems for linear nonhomogeneous ordinary second-order differential equations

###### Abstract

The paper considers the previously proposed method of numerical integration using the matrix calculus in the study of boundary value problems for nonhomogeneous linear ordinary differential equations of the second order with variable coefficients. According to the indicated method, when compiling a system of difference equations, an arbitrary degree of the Taylor polynomial in expanding the unknown solution of the problem into a Taylor series can be chosen while neglecting the approximation of the derivatives by finite differences.

Some aspects of the convergence of an unstable second-order difference boundary value problem are investigated. The concept of a pseudo-residual on a certain vector is introduced for an ordinary differential equation. On the basis of the exact solution of the difference boundary value problem, an approximate solution has been built, where the norm of pseudo-residuals is different from the trivial value.

It has been established theoretically that the estimate of the pseudo-residual norm decreases with an increase in the used degree of the Taylor polynomial and with a decrease in the mesh discretization step. The definitions of conditional stability and conditional convergence are given; a theoretical connection between them is established. The perturbed solution has been built on the basis of the found vector of pseudo-residuals, the estimate of the norm of its deviation from the exact solution of the difference boundary value problem has been calculated, which allows one to identify the presence of conditional stability. A theoretical relationship between convergence and conditional convergence is established.

The results of numerical experiments are presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):140-178

##### On a new Lagrangian view on the evolution of vorticity in spatial flows

###### Abstract

The purpose of the study is to extend to the spatial case proposed by G. B. Sizykh approach to a two-dimensional vorticity evolution, which is based on the idea of considering a vorticity evolution in the form of such a motion of vortex lines and tubes that the intensity of these tubes changes over time according to a predefined law.

**Method.** Thorough analysis is determined by describing the flow velocity field of an ideal incompressible fluid and a viscous gas in the general case, using the idea of the movement of imaginary particles.

**Results.** For any given time law of change of velocity circulation (i. e. for an exponential decay) of a real fluid along the contours the method of evaluating the field of velocity of such contours and vortex tubes is proposed (e. g. getting a field of imaginary particles, which transfer them). It is established that for a given time law the velocity of imaginary particles is determined ambiguously, and the method of how to adjust their motion preserving defined law of circulation change is proposed.

**Conclusion.** A new Lagrangian approach to the evolution of vorticity in three-dimensional flows is derived, as well as the expressions for the contours’ velocity, which imply stated changing over the time of the velocity circulation of a real fluid along any contour. This theoretical result can be utilized in spatial modifications of the viscous vortex domain method to limit the number of vector tubes used in calculations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):179-189

##### Poiseuille-type flow in a channel with permeable walls

###### Abstract

In the framework of the Navier–Stokes equations, the flow of a viscous incompressible fluid between immovable parallel permeable walls is considered, on which only the longitudinal velocity component is equal to zero. Solutions are sought in which the velocity component transverse to the plane of the plates is constant. Both stationary and non-stationary solutions are obtained, among which there is a non-trivial solution with a constant pressure and a longitudinal velocity exponentially decaying with time. These solutions show the influence on the profile of the horizontal velocity component of the removal of the boundary layer into the depth of the flow from one plate with simultaneous suction of the boundary layer on the other plate. It is established that for stationary flows the removal of the boundary layer into the depth of the flow from one plate and, with simultaneous suction of the boundary layer on the other plate, leads to an increase in the drag compared to the classical Poiseuille flow. In the case of impermeable walls, an exact non-stationary solution is obtained, the velocity profile of which at fixed times differs from the profile in the classical Poiseuille flow and, in the limit (as time tends to infinity), corresponds to rest.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(1):190-201