# Vol 22, No 3 (2018)

**Year:**2018**Articles:**12**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1229

### In memory of Prof. Oleg A. Repin

#### Abstract

Oleg A. Repin, age 71, our colleague, editorial board member of the Journal, Professor, Doctor habilitated of Physical and Mathematical Sciences, passed away on August 4th, 2018, after a long battle with serious disease. Oleg A. Repin was a leading scientist from the field of the theory of partial differential equations. He performed the ambitious scientific activities, kept in touch with the top scientific centers in Russia and was well known abroad. In this biographical background we discuss the scientific and educational work of Prof. Oleg A. Repin, give an information on his achievements and awards and a list of his main publications for the last five years.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):401-406

401-406

### Modeling of phase transformations and superelastic hardening of unstable materials

#### Abstract

The article presents models of superelastic hardening of materials with unstable phase structure at a constant temperature. The kinetic equation of the process of formation and growth of spherical nuclei of a new phase is formulated depending on the level of development of inelastic structural deformations, according to which the new phase first represents separate inclusions from embryos, developing it forms the structures of the matrix mixture in the form of interpenetrating skeletons, and finally the new phase is transformed in a matrix with separate inclusions from the material of the remains of the old phase. The influence of structural deformations on the features of phase transformations and nonlinear hardening of inhomogeneous unstable materials with different degree of connectivity of the constituent phases is studied. Various variants of the microstructure material formed in the conditions of the phase transition in the form of separate inclusions and in the form of interpenetrating components are considered. New macroscopic determining relationships for unstable microinhomogeneous materials are established and their effective elastic moduli are calculated. Macroscopic conditions of direct and inverse phase transitions are obtained, their effective limits and hardening coefficients are calculated. It is shown that the values of the macroscopic elasticity moduli of the obtained models lie inside the fork of the lower and upper Hashin-Shtrikman boundaries. Numerical analysis of the developed models has shown good agreement with known experimental data.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):407-429

407-429

### Numerical and experimental research of pure bending of beams made of the titanium ABVT-20 alloy with different properties for tension and compression under creep conditions

#### Abstract

Solution of the problem of pure bending of a beam of rectangular cross section taking into account the difference of properties tension and compression under creep is considered. Program algorithm of mathematical simulation of the stress redistribution process along the height of a beam with allowance for damage accumulation is constructed and implemented. Modeling of creep processes of softening material is based on equations of the kinetic theory of creep and damage. In this paper, Runge-Kutta-Merson numerical integration algorithm for creep damage analysis is presented. The simulation results are compared with the experimental data of pure bending of rectangular section beams from the titanium ABVT-20 alloy under the action of an alternating moment and a prolonged exposure to temperature of 750 °C. A satisfactory agreement between the simulation results and the experimental data was obtained, taking into account the duration of the temperature aging in the creep law.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):430-446

430-446

### The limit diagram under hot sheet metal forming. A review of constitutive models of material, viscous failure criteria and standard tests

#### Abstract

Methods of theoretical analysis and experimental verification of conditions of limit deformation are considered for a reasonable choice of the constitutive equations for mathematical modeling of processes of hot and warm treatment by pressure of sheet metal products with a deep drawing. Attention is focused on the forming limit curve of sheet metal on the plane of the principal strains (one of that corresponds to stretching, and the second can specify stretching or compression), the characteristic of the local state of the material corresponding to the critical growth of strain localization. Localization here is understood as a local thinning of the sheet and corresponds to diffuse form of localization. Other defects (shear bands, crack formation) develop from this limiting state or (formation of folds and wrinkles) are not local and require complete formulation of the problem. The forming limit curve (FLC) defines the conditions of realization of a technological process and can be theoretically predicted depending on the constitutive equations of plasticity, indicator of critical state and initial imperfections. The Marciniak-Kuczyński scheme is considered for getting FLC, where the sample has two zones of homogeneous strains and allows analytical reduction of the problem to the system of several ordinary differential equations solved numerically. The experimental methods assume testing by pressing a punch with a spherical or cylindrical tip into a specimen cut from a sheet. Depending on the depth of the lateral cutouts from the specimen, it can be provided tension or compression of the specimen in the transverse direction in these tests. Both approaches are analyzed as tools for selection and experimental verification of the constitutive model and the limit state indicator. They solve methodological problem of identification of mathematical models on a quite non-standard experiments involving strain localization. With the use of Marciniak-Kuczyński scheme the effect of a number of yield criteria for anisotropic sheet metal, hardening laws, damage accumulation models and criteria of viscous failure on qualitative and quantitative features of the FLC. To do this a proprietary algorithm has been developed. Experimental standard test methods of Hasek, Marciniak and Nakajima were implemented numerically in the software package LS-DYNA. The numerical FLD obtained were compared with theoretical and experimental ones. Possibilities of integration into Marciniak-Kuczyński scheme the dependence on temperature, strain rate and microstructure parameters for each basic rigid-plastic (scleronomous) model were discussed. It is noted this scheme is significantly limited by proportional changes of the main deformations in the sample outside and inside the strain localization zone. It is revealed this scheme is not adapted for determination of limit properties of the metals deformable in the conditions of deformation softening (aluminum, titanium alloys and some steels at temperatures of dynamic recrystallization). For a wider range of material deformation conditions, there is no alternative to the above-mentioned numerical method for predicting FLC. An open and relevant question is the description of the evolution of anisotropic plastic and fracture properties due to the anisotropic damage accumulation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):447-486

447-486

### On the interpretation of rocks elasticity modulus

#### Abstract

We consider the experimental data of the testing of rocks, performed by K. Mogi in his monograph “Experimental rock mechanics” which was published in 2007. Cylindrical samples of rocks were tested according to T. Karman’s scheme: on the first step, hydrostatic pressure was created up to different stress levels, on the second step the axial load was increased at a constant level of reached lateral pressure. During such a complex loading on the second step only the increment of axial strain was measured, which depends of the increment of the axial stress. In origin this dependence is presented in the form of graphs in real scale, which made it possible to convert these graphs into digital format in the form of tabular values. Two rocks are considered: Orikabe Diorite and Nabe-ishi Peridotite. According to the tables obtained, the stress-deformed state of these rocks in the second stage of complex loading for six test programs carried out in the experiment was analyzed. On each implemented loading path, a point is selected that corresponds to the axial stress with the same form of stress state. The latter, as is customary in geomechanics, is characterized by the ratio of the average principal stress to the maximum main stress. Thus, a (calculated) trajectory of proportional loading is distinguished for all stress levels within the elastic range. It is demonstrated that for such calculated loading paths, the experimental value of the increment of axial strain (within the elastic range) is a linear function of the increment of the axial stress. This proves the applicability of the generalized Hooke’s law. As a result, the Young’s modulus and the Poisson’s ratio are determined and shown that they (with respect to rocks) are indeed elastic constants, and not variable quantities, as it is sometimes interpreted.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):487-503

487-503

### The Lagrange multipliers method in covariant formulations of micropolar continuum mechanics theories

#### Abstract

Linear model of micropolar elastic continuum (known also as the Cosserat continuum) is considered. Kinematics and strain measures are discussed. The symmetric small strains tensor, relative microrotation vector and spatial gradient of the total microrotation vector (the wryness tensor) are then employed for a covariant formulation of the micropolar theory. By means of the principle of virtual displacements much simplified by the lack of internal forces and couples contributions to the virtual work and the Lagrange multipliers method the micropolar theory of elasticity is developed. Hemitropic micropolar continuum model is investigated in further details. The paper is to be considered as a universal covariant script of equations of the linear micropolar theory of elasticity derived from the virtual displacements principle.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):504-517

504-517

### Stochastic models of simple controlled systems just-in-time

#### Abstract

We propose a new and simple approach for the mathematical description of a stochastic system that implements the well-known just-in-time principle. This principle (abbreviated JIT ) is also known as a just-in-time manufacturing or Toyota Production System. The models of simple JIT systems are studied in this article in terms of point processes in the reverse time. This approach allows some assumptions about the processes inherent in real systems. Thus, we formulate and solve some, very simple, optimal control problems for a multi-stage just-in-time system and for a system with the bounded intensity. Results are obtained for the objective functions calculated as expected linear or quadratic forms of the deviations of the trajectories from the planned values. The proofs of the statements utilize the martingale technique. Often, just-in-time systems are considered in logistics tasks, and only (or predominantly) deterministic methods are used to describe them. However, it is obvious that stochastic events in such systems and corresponding processes are observed quite often. And it is in such stochastic cases that it is very important to find methods for the optimal management of processes just-in-time. For this description, we propose using martingale methods in this paper. Here, simple approaches for optimal control of stochastic JIT processes are demonstrated. As examples, we consider an extremely simple model of rescheduling and a method of controlling the intensity of the production process, when the probability of implementing a plan is not necessarily equal to one (with the corresponding quadratic loss functional).

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):518-531

518-531

### Couette-Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid with allowance made for heat recovery

#### Abstract

In this paper, we study the steady creeping convective flow of a viscous incompressible fluid in the thin infinite layer. The study of the fluid flow is based on the exact solutions class for the Oberbeck-Boussinesq equations in the Stokes approximation using. The velocity field is described by the Hiemenz exact solution. The temperature field and the pressure field linearly depend on the horizontal (longitudinal) coordinate, it corresponds to the Ostroumov-Birich exact solutions class. The convective motion of a viscous incompressible fluid was induced by tangential stresses on the upper permeable (porous) boundary and thermal source definition at the lower boundary. In addition, the heat exchange according to the Newton-Richmann law takes into account at the upper boundary. The obtained exact solutions describe counterflows in fluids. The stagnant points number in the fluid layer does not exceed three. The formation of counterflows in the fluid is accompanied by sucking and injection of the fluid through the permeable boundary. The larger number of stagnant points presence forms a cellular structure of the streamlines. In addition, the velocity field, which obtained in the solution of the boundary value problem is characterized by localization of the flow near the boundary of the fluid layer (boundary layer). The exact solutions obtained in this paper can be used for the nonlinear Oberbeck-Boussinesq system solving. The Grashof number can take large values, which depends on the geometric anisotropy index for the linearized Oberbeck-Boussinesq system.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):532-548

532-548

### Implicit iterative schemes based on singular decomposition and regularizing algorithms

#### Abstract

A new version of the simple iterations implicit method based on the singular value decomposition is proposed. It is shown that this variant of the simple iterations implicit method can significantly improve the computational stability of the algorithm and at the same time provides a high rate of its convergence. The application of the simple iterations implicit method based on the singular value decomposition for the development of iterative regularization algorithms is considered. The proposed algorithms can be effectively used to solve a wide class of ill-posed and ill-conditioned computational problems.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):549-556

549-556

### Numerical study of the influence of surface defects on the stability of a cylindrical pipe containing fluid

#### Abstract

This paper is concerned with the dynamic behavior of an elastic cylindrical pipe with surface defects interacting with the internal flow of a compressible fluid. A defect in the form of a ring of rectangular cross-section is located on the inner or outer surface of an elastic body and characterized by its own set of physico-mechanical parameters. The behavior of an ideal compressible fluid is described using the potential theory, and the behavior of the pipe is considered in the framework of the linear theory of elasticity. The hydrodynamic pressure exerted by the fluid on the inner surface of the pipe (defect) is determined with the use of the Bernoulli equation. A mathematical formulation of the problem of the elastic body dynamics is based on the variational principle of virtual displacements, and the system of equations for a liquid medium is developed using the Bubnov-Galerkin method. For the numerical implementation of the algorithm, a semi-analytic version of the finite element method is used. The stability of the system is estimated based on the results of computation and analysis of complex eigenvalues for a coupled system of equations. Verification of the model is carried out for the case of an ideal pipe by comparing the obtained results with the known experimental and numerical data. The effect of the geometric and physicomechanical parameters of the defect on the critical fluid velocity responsible for the loss of stability is studied for a cylindrical pipe clamped at both ends. It is shown that defects reduce the boundary of hydroelastic stability. It has been found that the defect located on the outer surface of the pipe exerts a greater impact on the system stability than it does when located on the wetted surface of the pipe.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):557-573

557-573

### Investigation of deflection of the CNT/G composite by molecular dynamics simulation

#### Abstract

Graphene and nanomaterials based on graphene have been using in the field of biomedicine as a material for biosensorics. The main components in biosensors are sensors, which must be flexible, scalable, sensitive and reliable. The deformation of the material changes its electrical resistance, therefore the study of the mechanical properties of composites, consisting of nanotubes and graphene, is the urgent task. Currently, active development of methods for the synthesis of composites consisting of graphene and parallel to it oriented nanotubes have been carrying. However, papers on the investigation of the optical and electronic properties of this composition was carried out not enough, and papers on the investigation of the mechanical properties of composites have not been found. The aim of this work is a theoretical investigation of the depending the bending force on the transverse displacement of atom in center of the composite material consisting of graphene and parallel to it (8, 0) zigzag nanotubes. The choice of a nanotube (8, 0) for research in this work is due to the minimum diameter of the nanotubes that make up the composite of this type. The stability of the composite was estimated by calculating the value of enthalpy and is characterized by a negative value of enthalpy. It was established that enthalpies do not change depending on the distance between the axes, along which the nanotubes belonging to the composites are oriented. Composite material was retained on both edges by support in the absence of a substrate. The search for the equilibrium state of the structure was determined by the molecular mechanics method using the Brenner energy potential within the framework of the molecular dynamics method. Mathematical modeling of the action of the needle of the atomic force microscope was carried out using the single-layer armchair carbon nanotube. The interaction between the armchair nanotube and the composite is carried out by means of the van der Waals forces.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):574-585

574-585

### A construction algorithm for full parametric analytical solutions in the basic mixed problem of elastostatics for the simply connected body

#### Abstract

Using analytical solutions to analyze the state of the bodies at the research and engineering calculations provides computing resources. We propose a methodology for structuring full parametric solutions to the problems of mathematical physics, including the basic mixed problem of elastostatic. The tool is a relatively new energy method of boundary states based on computer algebra. The method is based on the concept of state of the medium, isomorphism of Hilbert spaces of internal and boundary states of the body. The method is self-sufficient in the sense that, in principle, does not require comparison of the solution of test problems with those constructed by other methods. For inclusion in the solution in an explicit form of the medium constants we recommend saving computing resources method of boundary states with perturbations in which the direct method is combined with approach to A. Poincare. To explicitly include in the decision parameters the boundary conditions we suggested the technology of the reference solutions. Its effectiveness is demonstrated on a concrete example the basic mixed problem of elastostatic. The object of research is a limited simply connected body whose boundary is divided into three sections. At each site held individual method of parameterization of the points of the border: polar, cylindrical, spherical coordinate systems. The calculations are made using the computer algebra of the system “Mathematica” and demonstrated the effectiveness of the developed methodology to achieve this goal. The sequence of steps leading to guaranteed achievement of goal is described. The decision of a concrete task is made. Its results are presented in explicit analytical form containing all the parameters of the boundary value problem of elasticity theory and illustrated graphically after calculation by the analytic solution for a concrete set of parameter values.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(3):586-598

586-598