## Vol 23, No 2 (2019)

**Year:**2019**Articles:**11**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1232

Articles

Duality equations on a 4-manifold of conformal torsion-free connection and some of their solutions for the zero signature

#### Abstract

On a 4-manifold of conformal torsion-free connection with zero signature (--++) we found conditions under which the conformal curvature matrix is dual (self-dual or anti-self-dual). These conditions are 5 partial differential equations of the 2nd order on 10 coefficients of the angular metric and 4 partial differential equations of the 1st order, containing also 3 coefficients of external 2-form of charge. (External 2-form of charge is one of the components of the conformal curvature matrix.) Duality equations for a metric of a diagonal type are composed. They form a system of five second-order differential equations on three unknown functions of all four variables. We found several series of solutions for this system. In particular, we obtained all solutions for a logarithmically polynomial diagonal metric, that is, for a metric whose coefficients are exponents of polynomials of four variables.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):207-228

Solvability of a nonlocal problem for a hyperbolic equation with degenerate integral conditions

#### Abstract

In this paper, we consider a nonlocal problem with integral conditions for hyperbolic equation. Close attention focuses on degenerate integral conditions, namely, on the second kind integral conditions which degenerate into the first kind conditions at some points. Such kind of nonlocal conditions inevitably involves some specific difficulties when we try to show solvability of the problem. These difficulties can be overcome by a method suggested in our paper. The essence of this method is the reduction of the problem with degenerate conditions to the problem with dynamical conditions. This technique enables to define effectively a generalized solution to the problem, to obtain a priori estimates and to prove the existence of a unique generalized solution to the problem.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):229-245

Asymmetric tensor representations in micropolar continuum mechanics theories

#### Abstract

In this paper, new representations of three-dimensional asymmetric stress tensor and the corresponding form of the differential equilibrium equations are given. Asymmetric theories of solid mechanics continues to attract attention in connection with the necessity of mathematical modelling of the mechanical behaviour of the advanced materials. The study is restricted to such asymmetric second rank tensors, for which it is still possible to keep the notion of real eigenvalues, but not to accept the mutual orthogonality of the directors of the principal trihedron. The exact algebraic formulation of these asymmetry conditions is discussed. The study extends the dyadic tensor representations of the symmetric stress tensor based on the notion of asymptotic directions. The obtained results are a clear evidence in favor of algebraic hyperbolicity both the symmetric and asymmetric second rank tensors in three-dimensional space.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):246-255

Estimation of mechanical characteristics of salt rock crystals based on the mathematical model of nanoindentation using scanning probe microscope Dimension Icon

#### Abstract

The scanning probe microscope Dimension Icon is used both to assess the relief of the sample surface and to obtain the force response of the sample during the interaction of the cantilever (indenter of special shape, located at the end of the elastic console) with the sample surface. At the same time, unlike other devices, such as NanoTest-600, where as a result of indentation the researcher receives the value of hardness and effective modulus of elasticity with the help of software connected with this device. Dimension Icon gives only the dependence of indenter deviation on the displacement of the cantilever base. For the known flexural stiffness of the elastic console, one can determine the force-displacement curve during loading and unloading. Then here comes the problem of interpretation of this curve: how can we evaluate the mechanical characteristics of the material on its basis? The answer to this question particularly depends on the character of the mechanical behavior of the material. We consider the two-dimensional axisymmetric problem of sample indentation at the stages of loading and unloading for the range of indentation depths significantly exceeding the head spherical part of the cantilever probe, under the assumption of elastic-perfectly-plastic material model. Numerical simulation is carried out in the ANSYS package within the framework of the contact problem, under the assumption of an absolutely rigid cantilever tip. We propose the method for estimating the yield stress and the modulus of elasticity of the sample surface layers and determine the values of mechanical characteristics for several salt rock crystals by processing the results of the computational experiment and the data of previous experiments.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):256-269

Creep and plastic flow of a spherical viscoelastic layer material at its loading and unloading

#### Abstract

The purpose of the research is to investigate the processes of accumulation of irreversible deformations through different mechanisms: creep and plastic flow. Initially irreversible deformations are produced due to the viscous properties of solid material as creep deformations. The mechanism of the production of irreversible deformations changes to plastic when stress state reach the yield surface. On the contrary, this mechanism changes from fast plastic to slow viscous during unloading. The continuity in such increase in irreversible deformations is provided by the corresponding set of creep and plasticity potentials. These processes are considered in the framework of the mathematical theory of small deformations on the example of a one-dimensional problem of the deforming of a viscoelastoplastic hollow sphere under the influence of volumetric pressure changing with time. The processes of creep and plastic flow under increasing pressure, plastic flow at constant pressure, the medium unloading at decreasing pressure and the repeated plastic flow at the unloading were considered. The regularities of the motion of elastic-plastic boundaries in the material of the hollow sphere were established. The parameters of the stress-strain state of the medium were calculated, stress relaxation after the unloading was investigated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):270-283

Determination of elastic constants of rocks

#### Abstract

The A. N. Stavrogin's experimental data are observed during triaxial compression of sandstone samples under proportional loading according to T. Karman's scheme. Sandstones have a sufficiently high porosity in the initial state, so their deformation within elasticity has the following peculiar properties. When the cylindrical sample is uniaxially compressed at small initial stresses (of the order of 0.05÷0.15 of the elastic limit), a nonlinear part is observed on the longitudinal strain diagram, which is associated with the material densification occurring on this section. This circumstance causes a certain difficulty in determining the modulus of elasticity. An elaboration of the method for determination the elastic constants (Young's modulus and Poisson's ratio) are proposed taking into account the initial deformation diagram's special feature, which was mentioned. Earlier A. N. Stavrogin proposed to consider on the longitudinal strain diagram a linear part from the indicated initial stress to the conditional elastic limit. The elastic modulus is determined by this part of the diagram. Linear extrapolation of this segment to zero stress level provides a virtually new point of origin for the longitudinal strain under consideration. In this paper, it is shown that under triaxial compression of a cylindrical specimen, the longitudinal strain (satisfying Hooke's law) can be measured from the same new point of origin, which is established under uniaxial compression. In this case, the lateral strain of the sample is considered in the such range of stress variation, at which the increment of the axial stress causes a negative increment in the lateral strain. Based on the initial experimental values of longitudinal and lateral strain, which were adjusted by this method, the conditional elastic limit was determined.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):284-303

Analysis of the linear viscoelasticity theory capabilities to simulate hydrostatic pressure influence on creep curves and lateral contraction ratio of rheonomous materials

#### Abstract

The Boltzmann-Volterra linear constitutive equation for isotropic non-aging visco-elastic materials is studied analytically in order to examine its capabilities to provide an adequate qualitative description of rheological phenomena related to creep under uni-axial loading combined with constant hydrostatic pressure and of evolution types of the Poisson's ratio (lateral contraction ratio) in creep. The constitutive equation does not involve the third invariants of stress and strain tensors and implies that their hydrostatic and deviatoric parts do not depend on each other. It is governed by two material functions of a positive real argument (that is shear and bulk creep compliances); they are implied to be positive, differentiable, increasing and convex up functions. General properties and characteristic features of the creep curves for volumetric, longitudinal and lateral strain produced by the linear theory (with an arbitrary shear and bulk creep functions) under constant tensile load and constant hydrostatic pressure are investigated. Conditions for creep curves monotonicity and for existence of extrema and sign changes of strains are studied. The Poisson's ratio evolution in time and its dependences on pressure and tensile stress levels and on qualitative characteristics of two creep functions are analyzed. Taking into account compressibility, volumetric creep and pressure influence (governed by the bulk creep function) affects strongly the qualitative behavior of longitudinal creep curves and the Poisson's ratio evolution and its range. In particular, it is proved that the linear theory can simulate non-monotone behavior and sign changes of lateral strain and Poisson’s ratio under constant tensile load (even if the pressure is zero) and the longitudinal strain may start to decrease provided the pressure level is high enough. The expressions for Poisson’s ratio via the strain triaxiality ratio (which is equal to volumetric strain divided by deviatoric strain) and in terms of pressure ratio to axial stress and the creep functions ratio are derived. Assuming creep functions are arbitrary (permissible), general accurate two-sided bounds for the Poisson's ratio range are obtained and the influence of pressure level on the range is studied. Additional restrictions on material functions and loading parameters are derived to provide negative values of Poisson’s ratio. Criteria for the Poisson’s ratio increase or decrease and for its non-dependence on time are found. The analysis revealed the set of immanent features and quantitative characteristics of the theoretic creep curves families and the Poisson's ratio dependence on time and pressure to axial stress ratio which are convenient to check in creep tests (with various levels of pressure and tensile stress) and can be employed as indicators of the linear viscoelasticity theory applicability (or non-applicability) for simulation of a material behavior. The specific properties and restrictions of the model with constant bulk creep compliance which simulates a material exhibiting purely elastic volumetric deformation are considered.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):304-340

Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation

#### Abstract

This article discusses the solvability of an overdetermined system of heat convection equations in the Boussinesq approximation. The Oberbeck-Boussinesq system of equations, supplemented by an incompressibility equation, is overdetermined. The number of equations exceeds the number of unknown functions, since non-uniform layered flows of a viscous incompressible fluid are studied (one of the components of the velocity vector is identically zero). The solvability of the non-linear system of Oberbeck-Boussinesq equations is investigated. The solvability of the overdetermined system of non-linear Oberbeck-Boussinesq equations in partial derivatives is studied by constructing several particular exact solutions. A new class of exact solutions for describing three-dimensional non-linear layered flows of a vertical swirling viscous incompressible fluid is presented. The vertical component of vorticity in a non-rotating fluid is generated by a non-uniform velocity field at the lower boundary of an infinite horizontal fluid layer. Convection in a viscous incompressible fluid is induced by linear heat sources. The main attention is paid to the study of the properties of the flow velocity field. The dependence of the structure of this field on the magnitude of vertical twist is investigated. It is shown that, with nonzero vertical twist, one of the components of the velocity vector allows stratification into five zones through the thickness of the layer under study (four stagnant points). The analysis of the velocity field has shown that the kinetic energy of the fluid can twice take the zero value through the layer thickness.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):341-360

Effective computational procedure of the alternance optimization method

#### Abstract

The article discusses the computational procedure of the alternance optimization method as applied to the problem of semi-infinite programming. These problems are reduced numerous applied problems of optimization of objects with distributed and lumped parameters: robust parametric optimization of dynamic systems, parametric synthesis of control systems, etc. Since the calculation of alternance optimization method is rather difficult, as reduced to the solution, as a rule, the transcendental system of constitutive equations is proposed for efficient computational complexity of an embodiment of a computational procedure. To reduce the complexity of the computational procedure, the properties of the extremum points of the optimality criterion established in the alternance method are used in the region of permissible values of variables. These properties allow creating a topology of this area and thereby minimizing the number of references to it during the search procedure. The proposed computational method is especially effective for non-convex and nonsmooth optimality criteria, to which the technologically sound statements of semiinfinite optimization result. A step-by-step algorithm for preparing data and performing calculations, suitable for implementation in most programming languages, has been developed. The efficiency of the algorithm, which is higher, the larger the number of parameters included in the control vector and the higher the dimension of the optimization domain, is investigated. An estimate of the computational complexity of the computational procedure of the alternance optimization method is proposed, which makes it possible to determine the effectiveness of the application of the proposed algorithm for solving the problem of optimal control of the technological control object.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):361-377

The existence of chaotic regimes of the fractional analogue of the Duffing-type oscillator

#### Abstract

In this paper, we study the chaotic regimes of the fractional Duffing oscillator. To do this, using the Wolf algorithm with Gram-Schmidt orthogonalization, we calculated the spectra of maximum Lyapunov exponents depending on the values of the control parameters, on the basis of which bifurcation diagrams were constructed. Bifurcation diagrams made it possible to determine areas in which a chaotic oscillatory regime exists. Phase trajectories were also constructed, which confirmed the research results.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):378-393

On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functions

#### Abstract

In the theory of ordinary differential equations, the Clairaut equation is well known. This equation is a non-linear differential equation unresolved with respect to the derivative. Finding the general solution of the Clairaut equation is described in detail in the literature and is known to be a family of integral lines. However, along with the general solution, for such equations there exists a singular (special) solution representing the envelope of the given family of integral lines. Note that the singular solution of the Clairaut equation is of particular interest in a number of applied problems.In addition to the ordinary Clairaut differential equation, a differential equation of the first order in partial derivatives of the Clairaut type is known. This equation is a multidimensional generalization of the ordinary differential Clairaut equation, in the case when the sought function depends on many variables. The problem of finding a general solution for partial differential equations of the Clairaut is known to be. It is known that the complete integral of the equation is a family of integral (hyper) planes. In addition to the general solution, there may be partial solutions, and, in some cases, it is possible to find a singular solution. Generally speaking, there is no general algorithm for finding a singular solution, since the problem is reduced to solving a system of nonlinear algebraic equations.The article is devoted to the problem of finding a singular solution of Clairaut type differential equation in partial derivatives for the particular choice of a function from the derivatives in the right-hand side. The work is organized as follows. The introduction provides a brief overview of some of the current results relating to the study of Clairaut-type equations in field theory and classical mechanics. The first part provides general information about differential equations of the Clairaut-type in partial derivatives and the structure of its general solution. In the main part of the paper, we discuss the method for finding singular solutions of the Clairaut-type equations. The main result of the work is to find singular solutions of equations containing power and exponential functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(2):394-401