# Vol 25, No 1 (2021)

**Year:**2021**Articles:**11**URL:**https://journals.eco-vector.com/1991-8615/issue/view/3461**DOI:**https://doi.org/10.14498/vsgtu/v225/i1

## Full Issue

## Differential Equations and Mathematical Physics

### On a problem for the parabolic-hyperbolic type equation of fractional order with non-linear loaded term

#### Abstract

We study the existence and uniqueness of solution of the non-local problem for the parabolic-hyperbolic type equation with non linear loaded term involving Caputo derivative

\[

f(x) = \begin{cases}

{u_{xx}}-_CD_{0y}^\alpha u+a_1(x)u^{p_1}(x,0), & y > 0,

\\

{u_{xx}}-{u_{yy}}+a_2(x)u^{p_2}(x,0), & y<0,

\\

\end{cases}

\]

where

\[

{}_CD_{0y}^{\alpha }f(y) = \frac{1}{{\Gamma (1-\alpha )}}\int_0^y {(y - t)}^{-\alpha}f'(t)\,dt, \quad 0 < \alpha < 1,

\]

\(a_i(x)\) are given functions, \(p_i\), \(\alpha=\mathrm{const}\), besides \(p_i>0\) \((i=1,2)\), \(0 < \alpha < 1\) in the domain \(\Omega\) bounded with segments:

\[

A_1 A_2 = \{ (x,y): x = 1, 0 < y < h\},\quad B_1 B_2 = \{ (x,y): x = 0, 0 < y < h\},

\]

\[

B_2 A_2 = \{ (x,y): y = h, 0 < x < 1\}

\]

at the \(y > 0\), and characteristics:

\[

A_1C: x - y = 1,\quad B_1C: x + y = 0

\]

of the considered equation at \(y < 0\), where \(A_1 (1, 0)\), \(A_2 (1, h)\), \(B_1( 0, 0)\), \(B_2( 0, h)\), and \(C(1/2, -1/2)\).

Uniqueness of solution of the investigated problem was proved by an integral of energy. The existence of solution of the problem was proved by the method of integral equations. The theory of the second kind Fredholm type integral equations and the successive approximations method were widely used. We notice, that boundary value problems for the mixed type equations of fractional order with non linear loaded term have not been investigated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):7-20

### The problem with shift for a degenerate hyperbolic equation of the first kind

#### Abstract

For a degenerate first-order hyperbolic equation of the second order containing a term with a lower derivative, we study two boundary value problems with an offset that generalize the well-known first and second Darboux problems. Theorems on an existence of the unique regular solution of problems are proved under certain conditions on given functions and parameters included in the formulation of the problems under study. The properties of all regular solutions of the equation under consideration are revealed, which are analogues of the mean value theorems for the wave equation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):21-34

### Nonlocal Tricomi boundary value problem for a mixed-type differential-difference equation

#### Abstract

We investigate the Tricomi boundary value problem for a differential-difference leading-lagging equation of mixed type with non-Carleman deviations in all in all arguments of the required function. A reduction is applied to a mixed-type equation without deviations. Symmetric pairwise commutative matrices of the coefficients of the equation are used. The theorems of uniqueness and existence are proved. The problem is unambiguously solvable.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):35-50

### Initial-boundary value problem for the equation of forced vibrations of a cantilever beam

#### Abstract

In this paper, an initial-boundary value problem for the equation of forced vibrations of a cantilever beam is studied. Such a linear differential equation of the fourth order describes bending transverse vibrations of a homogeneous beam under the action of an external force in the absence of rotational motion during bending.

The system of eigenfunctions of the one-dimensional spectral problem, which is orthogonal and complete in the space of square-summable functions, is constructed by the method of separation of variables. The uniqueness of the solution to the initial-boundary value problem is proved in two ways: (i) using the energy integral; (ii) relying on the completeness property of the system of eigenfunctions.

The solution to the problem was first found in the absence of an external force and homogeneous boundary conditions, and then the general case was considered in the presence of an external force and inhomogeneous boundary conditions. In both cases, the solution of the problem is constructed as the sum of the Fourier series.

Estimates of the coefficients of these series and the system of eigenfunctions are obtained. On the basis of the established estimates, sufficient conditions were found for the initial functions, the fulfillment of which ensures the uniform convergence of the constructed series in the class of regular solutions of the beam vibration equation, i.e. existence theorems for the solution of the stated initial-boundary value problem are proved. Based on the solutions obtained, the stability of the solutions of the initial-boundary value problem is established depending on the initial data and the right-hand side of the equation under consideration in the classes of square-summable and continuous functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):51-66

## Mechanics of Solids

### Plane stress state of a uniformly piece-wise homogeneous plane with a periodic system of semi-infinite interphase cracks

#### Abstract

The plane stress state of a uniformly piecewise-homogeneous plane obtained by alternately joining two dissimilar strips is considered, which along the lines of joints of dissimilar strips is weakened by a periodic system of two semi-infinite interfacial cracks and is deformed using normal loads applied to the crack banks. The basic cell of the problem in the form of a two-component strip is considered and, using the generalized Fourier transform, a governing system of equations for the problem is obtained in the form of one singular integral equation of the second kind for a complex combination of contact stresses in the junction zone of the strips.

As a special case, tending the height of the strips to infinity, the governing equation of the problem for a two-component plane of two dissimilar half-planes with two semi-infinite interfacial cracks is obtained and its exact solution is constructed. The governing equation for the stated problem is also obtained in the form of one singular integral equation of the first kind with respect to normal contact stresses in another particular case, when all strips are made of the same material, i.e. in the case of a homogeneous plane, a weakened periodic system of parallel, two semi-infinite cracks.

In the general case, the behavior of the unknown function at the end points of the integration interval is determined and the solution of the problem by the numerical-analytical method of mechanical quadratures is reduced to solving a system of algebraic equations. Simple formulas are obtained to determine the intensity factors, the Cherepanov–Rice integral and crack opening. A numerical calculation has been performed. Regularities of changes in contact stresses and the Cherepanov–Rice integral at the endpoints of cracks are revealed, depending on the elastic characteristics of heterogeneous strips and the geometric parameters of the problem.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):67-82

### Solutions to Lame problems for combined transversally-isotropic spheres with a general center

#### Abstract

The paper deals with obtaining an exact analytical solution of the Lamé problem on the equilibrium state of a combined body consisting of two tightly fitted transversely isotropic spheres with a common center. The body is influenced by uniformly distributed external and internal pressures. The process pressure on the contact surface is determined assuming that it is a consequence of the difference in the geometry of the individual parts of the combined sphere only. We analyzed the laws of the influence of the materials’ anisotropy (the material constants satisfy the relations in the form of inequalities that ensure the positivity of the eigenvalues of the elasticity operator) and the values of the contact process pressure on the stress distribution in the cross sections of pressure vessels. The influence assessment of the materials’ anisotropy shows an opportunity to control the values and nature of the stress distribution in the combined structures that are optimal for the specified operating conditions. The obtained results indicate that a change in the anisotropy index, i.e. an increase in its values in the inner or outer parts of the spheres leads to an increase or decrease in the absolute values of stresses, respectively. This increase or decrease in the anisotropy indices can be realized at the stage of structures’ design due to a change in the reinforcement scheme while maintaining the properties of the individual structural elements. Based on a multicriteria approach, the initial strength of combined pressure vessels was estimated using the mechanisms of tension or compression in the radial and hoop directions. It was found that an increase in the pressure on the contact surface can lead to the material domains that do not resist compression in the hoop direction. These domains are located in the vicinity of the internal surface of the vessel, on which a uniformly distributed pressure acts, which is less in the absolute value than the external pressure. It was found that the points of the combined vessel located on the contact surface become most dangerous from the point of beginning the damage by the compression in the radial direction.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):83-96

### On the conformity of theoretical models of longitudinal rod vibrations with ring defects experimental data

#### Abstract

The paper considers a number of theoretical models for describing longitudinal vibrations of a rod. The most simple and common is based on the wave equation. Next comes the model that takes into account the lateral displacement (Rayleigh correction). Bishop’s model is considered to be more perfect, taking into account both transverse displacement and shear deformation. It would seem that the more perfect the theoretical model, the better it should agree with the experimental data. Nevertheless, when compared with the actually determined experimental spectrum of longitudinal vibrations of the rod on a large base of natural frequencies, it turns out that this is not entirely true. Moreover, the most complex Bishop’s model turns out to be a relative loser. The comparisons were made for a bar with small annular grooves that simulate surface defects, which is considered as a stepped bar. The questions of refinement with the help of experimentally found frequencies of the velocity of longitudinal waves and Poisson's ratio of the rod material are also touched upon.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):97-110

### Unsteady bending function for an unlimited anisotropic plate

#### Abstract

This work is devoted to the study of non-stationary vibrations of a thin anisotropic unbounded Kirchhoff plate under the influence of random non-stationary loads.

The approach to the solution is based on the principle of superposition and the method of influence functions (the so-called Green functions), the essence of which is to link the desired solution to the load using an integral operator of the type of convolution over spatial variables and over time. The convolution core is the Green function for the anisotropic plate, which represents normal displacements in response to the impact of a single concentrated load in coordinates and time, mathematically described by the Dirac delta functions. Direct and inverse integral transformations of Laplace and Fourier are used to construct the Green function. The inverse integral Laplace transform is found analytically. The inverse two-dimensional integral Fourier transform is found numerically by integrating rapidly oscillating functions. The obtained fundamental solution allowed us to present the desired non-stationary deflection in the form of a triple convolution in spatial coordinates and time of the Green function with the non-stationary load function. The rectangle method is used to calculate the convolution integral and construct the desired solution.

The found deflection function makes it possible to study the space-time propagation of non-stationary waves in an unbounded Kirchhoff plate for various versions of the symmetry of the elastic medium: anisotropic, orthotropic, transversally isotropic, and isotropic. Examples of calculations are presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):111-126

## Mathematical Modeling, Numerical Methods and Software Complexes

### Mathematical modeling and noise-proof estimation of shock wave pulse parameters based on the results of an experiment in underwater explosions

#### Abstract

The article deals with the construction of a mathematical model of the underwater shock wave pulse based on the results of the experiment and numerical and analytical scientific research. The results of the development and comparative analysis of various numerical methods for nonlinear estimation of the parameters of this model are presented. A numerical method is proposed for estimating the pulse energy of a shock wave based on the experimental results in the form of an overpressure waveform both over an infinite period of time and at a given pulse duration. The results of testing the developed numerical methods for mathematical modeling of the underwater shock wave pulse when processing the results of the experiment at the explosion of model charge are presented. The reliability and efficiency of the computational algorithms and numerical methods of nonlinear estimation presented in this paper is confirmed by the results of numerical and analytical studies and mathematical models constructed on the basis of experimental data.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):127-162

### A review of methods for developing similarity criteria in mechanics

#### Abstract

Similarity theory is the theoretical basis for modeling and drafting experiments. In addition, it can be used to conduct a comparative analysis of changes in the desired parameters of the problem without solving equations and without conducting experiments. All arguments in similarity theory are based on dimensionless power complexes, which are called similarity criteria (numbers), or invariants. In the literature of different years of release, various methods of obtaining similarity criteria are described, but the author was not able to find a unified classification of these methods and their comparison.

The article provides a review of various methods for obtaining similarity criteria, their classification, which includes five methods from differential equations and seven methods from dimension analysis. All methods are compared on a single problem of mechanics about forced vibrations of the load, which leads to four similarity numbers. This approach helps you compare the labor required to output similarity numbers in different ways. For each method, a list of references is given where it is mentioned, and a brief description of the tasks that are solved there. At the end is a summary table showing which methods are considered in the mentioned works. The table shows the relative popularity of methods.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):163-192

## Short Communications

### Healing of cracks in plates by strong electromagnetic field

#### Abstract

The problem of a pulsed high-energy electromagnetic field action on an edge crack in a thin plate, reproducing the pioneering experiment of Soviet scientists on the destruction of the crack tip by a strong electromagnetic field, is considered. The numerical simulation is based on the proposed electrothermomechanical model of a short-pulse high-energy electromagnetic field (HEMF) action on a material with a crack. The model takes the phase transformations (melting and evaporation) of the material occurring in the vicinity of defects and the corresponding changes in the rheology of the material in the areas of these transformations into account, as well as the possibility of electric current flowing between the free surfaces of the crack (breakdown due to electron emission). All physical and mechanical properties of the material are considered temperature-dependent. The model equations are coupled and solved together on a moving finite element grid using the arbitrary Euler–Lagrangian method. The processes of localization of the current density and temperature fields, phase transformations (melting and evaporation) at the crack tip, autoelectronic and thermoelectronic emissions between free crack surfaces, and the effect of these processes on crack healing are investigated. The simulation results are compared with the available experimental data on the pulse field action on the edge crack in the plate. The average metal heating rate, temperature gradients and time forming of the crater obtained in the vicinity of the crack tip are in good quantitative agreement with the experimental data. Away from the crack, as well as on the crack sides away from the tip, the temperature rises slightly. The process of modeling the electromagnetic field action, similar to the experiment, was accompanied by melting at the crack tip, as well as metal evaporation. Thus, under the considered current action, a crater is formed at the crack tip, which prevents the further spread of the crack, leading to its healing. It was not possible to obtain similar results using the previously proposed models.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2021;25(1):193-202