# 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 27, No 1 (2023)

**Year:**2023**Articles:**11**URL:**https://journals.eco-vector.com/1991-8615/issue/view/7603**DOI:**https://doi.org/10.14498/vsgtu/v227/i1

#### Full Issue

#### Differential Equations and Mathematical Physics

##### Analysis on generalized Clifford algebras

###### Abstract

In this article, we study the analysis related to generalized Clifford algebras ${C}_{n}\left(\overline{)a}\right)$, where $\overline{)a}$ is a non-zero vector. If $\left\{{e}_{1},...,{e}_{n}\right\}$ is an orthonormal basis, the multiplication is defined by relations

${{e}_{j}}^{2}={a}_{j}{e}_{j}-1,$

${e}_{i}{e}_{j}+{e}_{j}{e}_{i}={a}_{i}{e}_{j}+{a}_{j}{e}_{i},$

for ${a}_{j}={e}_{j}\xb7\overline{)a}$. The case $\overline{)a}=\overline{)0}$ corresponds to the classical Clifford algebra. We define the Dirac operator as usual by $D=\sum _{j}{e}_{j}{\partial}_{{x}_{j}}$ and define regular functions as its null solution. We first study the algebraic properties of the algebra. Then we prove the basic formulas for the Dirac operator and study the properties of regular functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):7-22

##### Network of Sobolev spaces and boundary value problems for operators vortex and gradient of divergence

###### Abstract

We will consider the scale of the Sobolev spaces ${\mathbf{\text{H}}}^{m}\left(G\right)$ vector fields in a bounded domain *G* of ${\mathrm{\mathbb{R}}}^{3}$ with a smooth boundary of $\Gamma $. The gradient-ofdivergence and the rotor-of-rotor operators ($\nabla \text{div}$ and ${\text{rot}}^{2}$) and their powers are analogous to the scalar operator ${\Delta}^{m}$ in ${\mathrm{\mathbb{R}}}^{3}$. They generate spaces ${\mathbf{\text{A}}}^{\text{2}\mathrm{k}}\text{(}G\text{)}$ and ${\mathbf{\text{W}}}^{m}\left(G\right)$ potential and vortex fields; where the numbers *k*, *m* > 0 are integers.

It is proven that ${\mathbf{\text{A}}}^{\text{2}\mathrm{k}}\text{(}G\text{)}$ and ${\mathbf{\text{W}}}^{m}\left(G\right)$ are projections of Sobolev spaces ${\mathbf{\text{H}}}^{2k}\left(G\right)$ and ${\mathbf{\text{H}}}^{m}\left(G\right)$ in subspaces $\mathcal{A}$ and $\mathcal{B}$ in ${\mathbf{\text{L}}}_{2}\left(G\right)$. Their direct sums ${\mathbf{\text{A}}}^{2k}\left(G\right)\oplus {\mathbf{\text{W}}}^{m}\left(G\right)$ form a network of spaces. Its elements are classes $\mathbf{\text{C}}\text{(}2k,m\text{)\u2261}{\mathbf{\text{A}}}^{2k}\oplus {\mathbf{\text{W}}}^{m}$.

We consider at the properties of the spaces ${\mathbf{\text{A}}}^{-m}$ and ${\mathbf{\text{W}}}^{-m}$ and proved their compliance with the spaces ${\mathbf{\text{A}}}^{m}$ and ${\mathbf{\text{W}}}^{m}$. We also consider at the direct sums of ${\mathbf{\text{A}}}^{k}\left(G\right)\oplus {\mathbf{\text{W}}}^{m}\left(G\right)$ for any integer numbers *k* and *m>0*. This completes the construction of the ${\left\{\mathbf{\text{C}}(k,m)\right\}}_{k,m}$ network.

In addition, an orthonormal basis has been constructed in the space ${\mathbf{\text{L}}}_{2}\left(G\right)$. It consists of the orthogonal subspace $\mathcal{A}$ and $\mathcal{B}$ bases. Its elements are eigenfields of the operators $\nabla \text{div}$ and $\text{rot}$. The proof of their smoothness is an important stage in the theory developed.

The model boundary value problems for the operators $\text{rot}+\lambda I$, $\nabla \text{div}+\lambda I$, their sum, and also for the Stokes operator have been investigated in the network ${\left\{\mathbf{\text{C}}(k,m)\right\}}_{k,m}$. Solvability conditions are obtained for the model problems considered.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):23-49

##### Lagrange’s representation of the quantum evolution of matter fields

###### Abstract

It is shown that a quantum path integral can be represented as a functional of the unique path that satisfies the principle of least action. The concept of path will be used, which implies the parametric dependence of the coordinates of a point on time x(t), y(t), z(t). On this basis, the material fields, which are identified with a quantum particle, are represented as a continuous set of individual particles, the mechanical motion of which determines the spatial fields of the corresponding physical quantities. The wave function of a stationary state is the complex density of matter field individual particles. The modulus of complex density sets the density of matter normalized in one way or another at a given point in space, and the phase factor determines the result of the superposition of material fields in it. This made it possible to transform the integral equation of quantum evolution to the Lagrange’s representation. By using the description of a quantum harmonic oscillator as an example, this approach is verified. EPRtype experiment is described in detail, and the possibility of the faster-then light communication is proved, as well as the possible rules of thumb of this communication are proposed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):50-63

##### Investigation of the Cauchy problem for one fractional order equation with the Riemann–Liouville operator

###### Abstract

The article is dedicated to solving the Cauchy problem for a differential equation with a Riemann–Liouville fractional derivative. The initial condition is formulated in a natural way and it is proven that the resulting solution is regular. Firstly, a fundamental solution is constructed and its properties are analyzed. Then, based on these properties, the solution to the homogeneous equation in the Cauchy problem is studied. Furthermore, unlike other problems of this type, the solution to the Cauchy problem presented for a nonhomogeneous equation is explicitly obtained in this work using the Duhamel’s principle and the three-parameter Mittag–Leffler function. By applying additional conditions to these problems, it is also demonstrated that this solution is classical.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):64-80

#### Mechanics of Solids

##### Computer simulation of microstructures and processes of granular composites fracture taking into account the influence of grain boundaries

###### Abstract

The paper presents the algorithm for computer simulation of polycrystalline material microstructures with explicitly distinguished grain boundaries. The algorithm is based on the procedure of “growing” structure grains from ellipses, the geometric parameters of which can be set according to different laws of statistical distribution. The grain boundaries of the given thickness are formed from the original granular structure by displacing the boundaries inside the grain. The advantage of the presented algorithm is the possibility of obtaining nonlinear grain boundaries of different thicknesses, the width of which can be specified according to various statistical distribution laws. Polycrystalline material microstructure generation results that contain more than 100 structural elements and have a grain boundary fraction of up to 20 % are presented. New data on computer simulation of the deformation process and fracture of simulated granular materials are presented with different ratios of strength characteristics of grains and grain boundaries. It was revealed that, depending on the strength characteristics ratio value, different fracture mechanisms are realized in the material: intercrystalline, transcrystalline, and mixed forms of fracture.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):81-101

##### The influence of creep deformations on the subsequent plastic flow in a material of rotating cylinde

###### Abstract

The influence of creep deformations on the process of plastic flow in a material is studied by using the example of the rotational motion of a cylinder with an inner cavity (a pipe) that has a rigid coating on its outer boundary to prevent radial expansion. The problem is solved within the frameworks of the theory of infinitesimal deformations. The theory of plastic flow with the associated condition of maximum octahedral stresses of von Mises, generalized to the case of viscoplastic flow, is used to describe the plastic properties of the material. The Norton's power law is used to describe the viscous properties. In the plastic flow region, the irreversible deformation rates are composed of plastic deformation rates and creep deformation rates. The dependencies required to determine the rotational speed at which plastic deformation initiates in the cylinder material are derived from the elastic deformation solution. A system of integro-differential equations is compiled to find the displacements and stresses in the cylinder material for the specified rotational speeds and accumulated irreversible deformations. Numerical calculations show that the presence of creep deformations leads to a later initiation of plastic flow, a reduction in plastic deformation rates, and a decrease in the plastic flow influence area.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):102-118

##### Modeling of non-isothermal elastic-plastic behavior of reinforced shallow shells in the framework of a refined bending theory

###### Abstract

The dynamic problem of non-isothermal and inelastic deformation of flexible shallow multidirectionally reinforced shells is formulated in the frameworks of the refined theory of bending. The temperature is approximated by a 7th order polynomial over the thickness of constructions. The geometric nonlinearity of the problem is modeled by the Karman approximation. The solution of the formulated coupled nonlinear two-dimensional problem is obtained using an explicit numerical scheme. The thermo-elastic-plastic response of fiberglass and metal-composite cylindrical elongated panels with an orthogonal reinforcement structure, loaded frontally with an air blast wave, has been studied. It is shown that, unlike reinforced plates similar in structure and characteristic dimensions, shallow shells under intense short-term loading must be calculated taking into account the occurrence of temperature fields in them. In this case, the refined theory of bending of curved panels should be used instead of the simplified version (the non-classical theory of Ambartsumyan). The temperature increment at separate points of shallow fiberglass shells can reach 14–34 \textcelsius, and in similar metal-composite panels can reach 50–150 \textcelsius. Cylindrical shallow shells are more intensively deformed when they are loaded by an air blast wave from the side of a convex front surface.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):119-141

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

##### On a paradoxical property of solving the problem of stationary flow around a body by a subsonic stratified flow of an ideal gas

###### Abstract

The problem of flow around a smooth convex body moving horizontally at a constant subsonic velocity in a stratified atmosphere at rest consisting of an ideal gas is considered. By the condition of the problem, the (vertical) gradient of the Bernoulli function (taking into account the potential energy of a uniform gravity field) in the atmosphere at rest at all altitudes is nonzero (as is the case in the Earth's standard atmosphere at altitudes up to 51 km), and the flight altitude does not exceed a value equal to the square of the body's flight speed divided by twice the acceleration of gravity. The surface of the earth is considered flat. The coordinate system associated with the body is used. The general spatial case is considered (an asymmetric body or a symmetric body at an angle of attack). We use the generally accepted assumption that in some neighborhood of the stagnation streamline (streamline that ends on the body at the forward stagnation point) there is no second stagnation point, the flow parameters in this neighborhood are twice continuously differentiable, and the stagnation point is spreading point (i.e. in some neighborhood of it, all streamlines on the surface of the body start at this point). Based on a rigorous analysis of the Euler equations, it is shown that the existence of a stationary solution to the problem contradicts this generally accepted (but not strictly proven) idea of the stagnation streamline. This property of the solution of the problem is called paradoxical and casts doubt on the existence of the solution.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):142-158

##### A coupled non-stationary axisymmetric problem of thermoelectroelasticity for a circular piezoceramic hinged plate

###### Abstract

The new closed solution of the coupled non-stationary axisymmetric problem of thermoelectroelasticity for a round axially polarized hinged piezoceramic plate in a three-dimensional formulation is constructed. Its cylindrical surface is hinged. The case of temperature change on the cylindrical surface and front planes of the plate (boundary conditions of the 1st kind) is considered. The front electroded surfaces of the structure are connected to a measuring device with a large input resistance (electric idle).

A plate is investigated, the geometrical dimensions of which and the rate of change of the temperature load do not significantly affect the inertial characteristics of the electroelastic system, making it possible to use the equations of equilibrium, electrostatics and thermal conductivity in the mathematical formulation of the problem. In this case, the initial calculated relations form a non-self-adjoint system of differential equations in partial derivatives.

The problem is solved by sequentially using the Hankel integral transform with respect to the radial coordinate and the generalized method of the biorthogonal finite integral transform (FIT) with respect to the axial variable. The application of the structural FIT algorithm allows one to construct an adjoint operator, without which it is impossible to solve non-self-adjoint linear problems by expanding in terms of eigenvector functions.

The constructed calculation relations make it possible to determine the stress-strain state, temperature and electric fields induced in a piezoceramic element under an arbitrary external temperature action, and also to analyze the effect of the rate of change in body volume and tension on the temperature field.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):159-178

#### Short Communications

##### Creep and long-term strength of hydrogen-containing VT6 titanium alloy with a piecewise constant dependence of tensile stress on time

###### Abstract

We consider the creep of a hydrogenated rod made of VT6 (Ti-6Al-4V) titanium alloy with a piecewise constant dependence of the stress on time up to failure. The results of an experimental and theoretical study on the effect of the concentration of previously introduced hydrogen on the creep and long-term strength of tensile rods made of VT6 titanium alloy at a temperature of 600$\xb0$C and constant nominal tensile stresses in the range from 47 to 217 MPa.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):179-188

##### The effect of bone tissue density on the stress-strain state near dental implants

###### Abstract

The dependence of the stress-strain state of the bone tissue on its density near the dental implant has been studied. The computations were performed by the boundary element method for the plane-deformed state of a model consisting of a cylindrical implant and surrounding bone tissues. Bone tissue is considered as an isotropic and homogeneous elastic material. Simulation the effect of bone density on the stress-strain state when applying a quasi-static load is performed by changing of elasticity modulus of the bone. It has been established that with the increasing in the spongy bone tissue elastic modulus, the maximum equivalent stresses in this bone tissue increase. Stresses in the cortical bone tissue decrease with the increasing in the spongy bone elastic modulus due to the decreasing in the load transferred to this bone part. Stresses in the spongy bone decrease with the increasing in the cortical bone layer elasticity modulus. The level of maximum stress in the cortical layer of the bone increases with the increasing of this bone tissue elastic modulus. The maximum of stresses in the cortical bone tissue are observed near the implant neck.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2023;27(1):189-201