## No 1 (2013)

**Year:**2013**Articles:**18**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/33**DOI:**https://doi.org/10.15593/perm.mech/2013.1

On the mechanism of excitation of self-oscillations in gas flow heat supply

#### Abstract

Established mechanisms of excitation of thermoacoustic oscillations associated with the original restructuring of the viscous stress tensor and the dissipation of heat energy when moving gas to local release of heat. In one-dimensional flow of heat dissipation tensor determines the thermal resistance, for which an arbitrary polytropic heat supply found an analytic representation. It is possible to determine the educational negative thermal resistance as a function of the polytropic index, which is a new mechanism of excitation of thermoacoustic oscillations. B.V. Rauschenbach was invited local heat supply area approximated by a surface called the surface of the heat. This surface is a surface of discontinuity of the first kind of hydrodynamic and thermodynamic parameters of the continuum. In this paper, the boundary conditions that must be performed on the surface of the heat. These boundary conditions are a consequence of the surface of the heat fundamental conservation laws: mass, momentum, mass and energy can be considered as the equation for determining the components of the tensor of the dissipation of thermal energy, which is associated with the surface of the heat. For the description of the considered thermal acoustic oscillations found in the form of a mathematical model of distributed dissipative dynamical system. As an example of the application of this mathematical model, we considered the problem of constructing longitudinal thermoacoustic oscillations in a Rijke tube at the location of the heat source directly on its input. Is shown that the excitation of oscillations Rijke phenomenon in this case is N – shaped relationship tension forces of viscous friction along the pipe as a function of the average over the pipe section velocity of heated air.

**PNRPU Mechanics Bulletin**. 2013;(1):7-24

Estimation of the stress-strain state of the revolving long cylinder

#### Abstract

Subject of a study is a difficult stress state of long hollow cylinder which is being bended by its own weight and rotated by turning moment. The stressed state over the cylinder section and the curvature of its longitudinal axis are investigated. The basic assumptions taken in the attention to solve the problem in question are usual common for the mechanics of materials and engineering approximations the hypothesis about the linearity of the physical relationships between the stresses and strains (linear theory of elasticity) and assumption about the small strains. Solution of problem is built on the basis of the differential equation of the elastic flexure of the central axis of cylinder and fundamental relationships between the curvature of this axis, applied loads, strains and stresses in the region of cylinder section. Boundary-value problem for the ordinary differential equation of the fourth order with the appropriate boundary conditions is solved using the method of variations of the arbitrary constants. This made it possible to obtain the exact solution of the task of the long cylinder bending which rotates around the longitudinal axis. This, as a result, made possible to determine the dependence of the curvature of cylinder central axis on the longitudinal coordinate and to find the stress distribution in the of cylinder section. The estimation of the contribution from each of the factors, which act on the subject of study, to the stressed state is executed. Solution of the presented problem made it possible to define equivalent stress in the outlying layers of cylinder as the result of the action of all power factors examined.

**PNRPU Mechanics Bulletin**. 2013;(1):25-38

The use of optimization techniques for numerical analysis of temperature fields in the quartz optical fiber billet

#### Abstract

In this paper a procedure is proposed for determination of heat load under high temperatures. The procedure proposed allows finding temperature of a surface heated by a gas burner flame which is necessary to assure the accuracy of further research of temperature fields and stress-strain state of products. The discussed problem is about determining surface temperature of an optical fiber billet made of quartz glass placed within heating zone of a gas burner and also further research of temperature fields within the billet in course of a production process. Method of determining surface temperature of a billet in a gas burner heating zone includes conducting an experiment to measure temperature in a number of points away from the burner flame and subsequent solving of an optimization problem to calculate the temperature of a surface within heating zone. In the course of experiment surface temperature of a tube was measured by two thermocouples at a distance from the heating zone which was followed by solving an optimization problem. As the target function, sum of squares of deviations of calculated temperature values at given points from the observed experimental values was taken. The developed method has been tested on a series of test problems. The uniqueness of the solution of optimization problem was confirmed by ‘descending’ from a set of different initial values. Temperature fields within billet of a quartz optical fiber heated by a uniformly moving gas burner were examined. To solve a non-stationary heat conduction problem, finite elements method has been used while the optimization problem has been solved by the Golden section method.

**PNRPU Mechanics Bulletin**. 2013;(1):39-51

Very-high-cycle fatigue facture of titanium compressor disks

#### Abstract

In the present paper there is outlined the procedure of stress state calculation for compressor disk of gas-turbine engine during flight-loading cycle (low-cycle fatigue regime – LCF) and under low-amplitude blade vibrations (very high-cycle fatigue – VHCF). The criteria and models of multiaxial fracture are studied in the case of low-cycle fatigue and in the case of very high-cycle fatigue. The model parameters are determined by using the experimental data of uniaxial fatigue tests for various asymmetry coefficients of the cycle. On the basis of the calculated stress state using multiaxial fatigue fracture criteria, the durability estimations for compressor disks are obtained for alternative mechanisms of LCF and VHCF. Two elasticity problems for circular disk are solved for confirmation of the numerical results obtained. In the first problem the centrifugal loading from the blades (analogue of LCF regime) was modeled. In the second problem loading from blade torsion (analogue of VHCF regime) was modeled. The durability estimation also shows that the fatigue fracture in LCF and VHCF regimes can take place for the same period of real time.

**PNRPU Mechanics Bulletin**. 2013;(1):52-67

Theoretical modeling of atomic force microscopy in the study of surfaces with complex nanostructures

#### Abstract

Standard software supplied to decrypt the atomic force scanning (AFM), based mainly on the models using a classical solution of the Hertz contact of two linearly elastic spheres (or a sphere and a flat half, if one of them has infinite radius). In most cases this is enough. However, there are situations where the Hertz solution should be used with great caution. This work is devoted to the theoretical study of these versions. The results of numerical modeling of contact interaction of the probe of an atomic force microscope and the surface with complex nanostructure are presented. Research was conducted for two types of materials: 1) anisotropic elastic medium (tooth enamel), 2) non-linear elastic finite deformable polymer. These are two classes of materials that are fundamentally different in their mechanical behavior. Respectively different theoretical models are needed for the correct interpretation of experimental data. The problem indentation an AFM probe in a transversally isotropic elastic surface is solved for the materials of the first type. Computational dependencies of reaction force in the probe on the depth of indentation and the degree of anisotropy of the material is built. Computer modeling of AFM probe contact interaction with elastomeric nano strands, which can be formed in the polymer at the crack tip, are carried out for the materials of the second type (elastomers). nanostrand is represented as a long longitudinal convexity, lying on a flat elastic surface. The problem was solved in a three-dimensional formulation.

**PNRPU Mechanics Bulletin**. 2013;(1):68-80

Numerical algorithm of solving the problem of large elastic-plastic deformation by FEM

#### Abstract

A numerical algorithm of the investigation of stress-strain state of the elastic–plastic solids with large deformations is described. The left Cauchy-Green tensor and velocity gradient tensor are used as the tensors describing the deformation and deformation rate. The potential of the elastic strain the specific strain energy, which depends on the left Cauchy-Green tensor is introduced. An isotropic material is considered. The state of stress is described by the Cauchy stress tensor. The linearized constitutive equations of elastic deformation are obtained as a function of the derivative of the Truesdell stress rate on the strain rate. The theory of flow and an additive representation of the total deformation rate are used. The von Mises yield criterion is applied. The research algorithm is based on the incremental method. The principle of virtual work in terms of the virtual velocity is used. After linearization the system of linear equations is obtained, where the unknown is the increment of displacement in the current state. The radial return method with an iterative refinement of the current mode of deformation, based on the introduction to the governing equations of additional power voltages is applied. As an example the potential of elastic deformation is considered. The von Mises yield criterion with isotropic hardening is used. The linearized constitutive relations is obtained. The numerical implementation is based on the finite element method. An 8-node brick element is used. Developed algorithm of investigation of large elastoplastic deformations is tested on the solution of the necking of circular bar problem. The results of solutions and comparison with results obtained by other authors is reduced. Also the deformation of a square plate under internal pressure is investigated.

**PNRPU Mechanics Bulletin**. 2013;(1):81-93

Experimental study of Baushinger effect and yield surface at elastoplastic deformation of metals

#### Abstract

This work contains results of the experimental investigation on repeated sign-variable loading at stretching compression of a thin-walled tubular specimen from steel 45 in an automatized computational and experimental installation SN-EVM. Аlso presents the results of the experimental assessment of the Baushinger effect and the influence of various admissions on residual deformation on it. The influence of the admission on residual deformation on definition of secondary offset yield stress was investigated, as well as on initial and subsequent radius function and position of the center of a hypothetical spherical yield surface in stress space in the plastic-flow theory. According to the results of the investigation, the parameter characterizing the Bauschinger and the radius function of the yield surface increase on the increase of admission on residual deformation effect, yet the displacement of the center of the yield surface decreases. It is shown that the parameter characterizing the Bauschinger effect decreases on the increase of the length of the arc of plastic deformation and tends to a stationary value. It is experimentally discovered that for steel 45 the displacement of the center of the yield surface increases on the increase of the length of the arc of plastic deformation. It is also established that the radius of the spherical yield surface lowers temporary, and then increases on the increase of the length of the arc of plastic deformation radius. Thus, in some mathematical models of plastic-flow theory temporary decrease of the radius of the yield surface (contraction of the yield surface) at the beginning of the process of plastic deformation is not associated with the definition of the yield stress for the admission on residual deformation.

**PNRPU Mechanics Bulletin**. 2013;(1):94-105

Optimization of the press tool by production of low-temperature superconductors

#### Abstract

In this article the technique of definition of optimum angles of the technological tool is offered at pressing combined composite preparations. In an optimization basis the pressure of pressing providing the minimum power inputs is necessary. The problem of improvement and optimization of the production technology of low-temperature composite superconductors for achievement of necessary production volumes of release is actual at compliance to quality requirements. Development of theoretical bases and techniques of design of technological processes in relation to the production technology of the low-temperature composite superconductors which are scientifically proving a choice of technological modes and industrial equipment for improvement of quality of low-temperature composite superconductors and providing high technical and economic rates of their production is important. The pressing process is widely used in metal forming.The essence of the pressing process is extrusion of material placed in an enclosed area through the channel formed by the pressing tool. The advantage of process of pressing is the favorable scheme of a tension with prevailing influence of the squeezing tension providing increased plasticity of the pressed material. Therefore process of pressing is widely used when processing by pressure of low-plastic hardly deformed metals and alloys.

**PNRPU Mechanics Bulletin**. 2013;(1):106-120

Convective flows stability at the isothermal diffusion mixing of ternary gas mixtures

#### Abstract

Analysis of the experimental data for the study of mechanical equilibrium instability by the diffusion mixing of ternary gas mixtures in isothermal conditions is carried out within the linear theory of stability. A channel is considered in the form a vertical cylindrical channel of a final height at the boundary conditions implying the absence of components’ transfer through the diffusion channel walls. Predictions of the theory are compared with the experimentally determined location of stability borders for the mixture 0.4722 He + 0.5278 Ar – N2 by a variation for different diameter of the channel. The existence of the intensity maximum of components’ transfer in dependence on the thermodynamic parameters under conditions of the developed convection connects with the mode of disturbances determining the various types of convective flows. Structural formations moving opposite to each other which numbers are defined by the mode of disturbances in the cross-section of the diffusion channel interacts and thereby defines the wave-like change of unstable process intensity. For the examined system first maximum of the unstable process intensity subject to the diameter appearing at d = 6 mm is characterized by the mode of disturbances n = 3 and the critical Rayleigh number R 3 = 972.7. Stabilization of the convective transfer defining by the certain motion type of convective forming i.e. the motion along six channels (three flows and three counterflows) occurs under such conditions. Also, results reveal that increase of the mode of disturbances and the diameter of diffusion channel result in the non-linear increase of the critical Rayleigh number for all components taking part in the transfer. Further, the present results are in good agreement with the experimental data for the system 0.4722 He + 0.5278 Ar – N2.

**PNRPU Mechanics Bulletin**. 2013;(1):121-135

Parameters of process of machining into account temperature models

#### Abstract

In work the basic processes of machining and their influence on quality and durability of cylindrical products, calculated temperature modes are considered at cutting of pipes, parameters of process of cutting are chosen taking into account size and a sign on temperature residual stresses. By machining understand different types of cutting: turning, drilling, milling, grinding, honing, polishing. In this regard, shows how the processes of machining affect the quality of tubular products. The aim is to give a cutting detail desired shape. Also examined parameters of the cutting process, selected to the value and sign of thermal residual stresses. Residual stresses in metal cutting result from uneven plastic deformation and significant heating of the surface layers. Solving the problem of thermoelasticity can determine the stresses induced by heating. Based on the terms of the onset of plastic deformation is not on the surface of the part after passing through the tool, you can optimize the cutting parameters for the treatment of various axisymmetric metal. Thus, knowing the relationship between these parameters and the mechanical properties of the workpiece material and geometry can estimate the stress intensity as a function of the initial temperature and cutting data processing tube. Then based on the results built relationships and dependencies critical cutting parameters for tubular products of different materials and some limiting parameter ratios machining tube. The paper presents the method of determination of the limit relations parameters of the cutting process for axisymmetric products with temperature heating surface due to friction when turning. The method allows to determine the voltage at the surface of the workpiece and the rational selection of the process parameters, depending on the part geometry and the material.

**PNRPU Mechanics Bulletin**. 2013;(1):136-147

Efficiency of master-process ideology for parallel realization of iterative solving procedures for linear algebraic system

#### Abstract

The effectiveness of master-process ideology application for realization of parallel solving procedures for system of linear algebraic equations is considered. Researches were conducted for iterative algorithms of conjugate gradient and Jacobi methods. The sparse format RR(C)U for coefficient matrix was used. Researches were conducted using system of linear algebraic equations computed with method of finite elements for plane boundary elastic problem. “Acceleration” calculated as ratio of successive algorithm execution time to parallel algorithm execution time was used as a criterion of quantity analysis. Analysis was conducted for algebraic systems from 100 to 50000 equations using six core AMD® Phenom II X6 1075T processor based computer. Program realization of iteration algorithms was created using C#. Standard MPI 2.0 was used to provide parallel processes communication. On base of obtained results it is possible to make a conclusion that usage of master-process parallel architecture for conjugate gradient iterative algorithm results in insignificant, from ten to fifteen percents, reduce of acceleration value as regards realization without master-process. This effect is much more lower for Jacobi method. Taking into account structural convenience of master-process parallel architecture the conclusion of possibility of this program architecture solution for considered methods was made.

**PNRPU Mechanics Bulletin**. 2013;(1):148-160

Simulation of the stress-strain state of thin composite coatings based on solutions of the plane problem of strain-gradient elasticity for a layer

#### Abstract

We obtain and investigate the formulation of the problem of strain gradient plane model of elasticity for the composite layer. We used a synthesis of analytical and numerical methods to solve plane problem of gradient elasticity for an infinite layer to allow the distribution of stresses in the plane of ultrathin coatings. We investigate the problem of the impact of surface compressive normal loading on an infinite layer . To solve the problem we use a Fourier integral transform, and the inverse trnsform is computed using a numerical procedure. It is shown that the proposed model can predict the effects of localization interlaminar stresses in the local interphase zones in the coating and consider the influence of non-classical scale factors – the thickness of the coating layers and gradient model parameters. For simulation it is involved the most simple gradient theory of elasticity – applied model of the interphase layer containing single additional physical parameter determining the "gradient" of the environment and the extent of interfacial zones in the border area of the material. This additional physical constant characterizing the contact of dissimilar materials. For the numerical calculations in this paper we use the hypothetical values of the gradient parameter. To solve the problem of practical importance due to the possibility of reliable modelling and optimization of the structure of ultra microstructure protective composite coatings used in the aerospace industry. Also constructed solution can be used to identify additional physical parameters of the gradient theory of elasticity based on a comparison of simulation results and experimental data on the indentation thin layer structures.

**PNRPU Mechanics Bulletin**. 2013;(1):161-181

Combined discrete modeles in the three-dimensional elastic inhomogeneous bodies analysis of complex shape

#### Abstract

Construction procedure of combined discrete models for the stress state of three-dimensional elastic bodies, complex shape design having an inhomogeneous structure to be analyzed has been suggested. Combined models are composed of first-order homogeneous single grid three dimensional finite elements (FE) of cube shape and inhomogeneous (composite) double grid finite elements (DgFE) of rectangular parallelepiped shape. In the vicinity of the body fixing or complex boundary, single grid FE are used, DgFE are for the rest of the body. Construction procedure of DgFE has been shown. To construct DgFE, two nested grids, both fine and coarse ones are used. Area of the DgFE is a basic (fine) descritization into FE of the first order taking into account its inhomogeneous structure and generating the fine grid. On a basic descritization the total potential energy functional of DgFE (using the approximations constructed on a coarse grid) is given in matrix form projected on a coarse grid. Formulas to calculate the stiffness matrix and the nodal forces vector of DgFE are obtained by minimization condition of the functional obtained by nodal displacement of coarse grid. DgFE characteristics are that the inhomogeneous structure is taken into consideration by using the base fine descritization, discrete models of small dimension are formed and the solutions with a specified error generated. Error of the solution varies with ratio steps of coarse and fine grids of DgFE. Advantages of combined discrete models are that they have a small dimension, take into account a complex shape bodies, inhomogeneous structure and generate solutions with a specified error. The example of calculation has been demonstrated.

**PNRPU Mechanics Bulletin**. 2013;(1):182-195

Modeling of the waves on a surface of a cylindrical configuration of magnetic fluid, surrounding a long porous core

#### Abstract

A mathematical model of waves propagation and instability on a surface of a cylindrical column of magnetic fluid of an infinite length, surrounding a coaxial infinite long porous core of the round section is constructed and studied. The conditions are found under which the disturbances of the surface of fluid column become unstable and result in its fragmentation into a chain of connected droplets. The presence of a surface tension is taken into account. The gravity is neglected. The axis of the porous cylinder coincides with the axis of a coaxial solenoid, generating an uniform magnetic field. The problem is solved in a cylindrical coordinate system ( r , θ, z ), in which the fluid column is at rest. The z -axis is directed along the axis of the solenoid. The equations of the motion of magnetic fluid inside and outside of the porous medium and the equations for the magnetic field are written. The boundary conditions for hydrodynamic and magnetic values on discontinuity surfaces are formulated. The disturbances of magnetic field (according to wave propagation) is searched inside and outside of the porous medium, as well as in the air clearance of the solenoid. The full solution of a boundary value problem for hydrodynamic and magnetic values is found. The numerical analysis of the dispersion equation, describing wave propagation on surface is done. The different special cases are considered. The conditions are found under which the disturbances of the surface of fluid column become stable (wave damping) or become unstable (which result to the disturbances growth and the fragmentation of the cylinder into a chain of droplets). It is shown that the size of droplets appearanced in fragmentation process increases with the growth of the magnetic field i.e. magnetic field has a stabilizing influence upon the fragmentation of the fluid column.

**PNRPU Mechanics Bulletin**. 2013;(1):196-209

Evolution of technological stress fields in cylindrical stress applying rod for optical fiber Panda-type during annealing

#### Abstract

Investigated evolution regularities of the technological stress fields in the rod for preform of fiber Panda type during annealing. It is shown that the main reason of reducing the levels of the stress state is the relaxation, due to viscous deformation of silica glass in the temperature range of the glass transition. Implemented mathematical model of technological stresses formation in stress applying rod, on the basis of linear thermoviscoelasticity relations in conditions of annealing. Numerical analysis performed using finite element method (FEM). Identified the quantitative characteristics of stress relaxation in different process conditions of annealing. Defined modes that provide the maximum possible reduction of hazardous tensile normal stress and intensity of the stress tensor. Found that for the rod with uniform doping, the annealing process is more efficient than for the rod doped with regularity close to parabolic.

**PNRPU Mechanics Bulletin**. 2013;(1):210-220

Finite element realization of geometrical immersion method on the basis of Kastilyano's variation principle for two-dimensional problem theory elasticity

#### Abstract

The alternative geometrical immersion method for plane problems theory elasticity, based on a finite elements method in term of stresses within the principle minimal additional work of elastic system is considered. The geometrical immersion method consists in reducing an initial problem for linearly elastic body of any form to iterative sequence of problems theory elasticity on some initial area. Iterative procedure for the solution of the variation equation of the geometrical immersion method and a procedure of formulation of its discrete analog by means of the finite elements method in term of stresses for plane problems theory elasticity in the Cartesian system of coordinates are formulated. The alternative finite element method in terms of stresses function for satisfaction of approximating expressions to the balance equations is used. Practical application of the method on the example of the plane problems solution for the elastic plate with rectangular hole is shown. Rather good compliance of results of stresses fields definition in comparison with a traditional finite elements method in movements is obtained. Practical convergence of iterative procedure of the geometrical immersion method is established. We focus our attention to the problem of the static boundary conditions that are the main for this variation formulation. The way of modification of a flexibility matrix of the system of finite elements and Lagrange’s multipliers method is used.

**PNRPU Mechanics Bulletin**. 2013;(1):221-234

Method of calculating the surface and adhesion energies of elastic bodies

#### Abstract

There are a number of problems for successful solutions to be aware of the surface energy, the energy and the force of adhesion and cohesion. An example is that of the bundle composites. The expression of the energy strength criterion is the total surface energy of the contacting elements or energy of adhesion. These values are determined by methods of the classical theory of elasticity is impossible. Power strength criteria using the limit values of powers – forces of adhesion and cohesion. They are usually determined experimentally, it is not always possible or economically unprofitable. The authors proposed a method for calculating the surface and adhesion energies of elastic bodies in a state of adhesion. It was considered by the authors as follows. Distributed over the boundary of the total surface energy is distributed over its volume free energy change occurring during the formation of the border. The adhesion of the two bodies along the contact surface is formed by a transition layer. In its physical and thermodynamic properties of one body continuously into the properties of another body. The method is based on a variant of the gradient model of continuous elastic medium. It is based on the assumption of many-potential nonlocal interaction infinitely small particles that make up the environment. More to the classic characteristics of the elastic state is calculated by differentiating the known expression bulk density of free energy. It is built on the basis of additional assumptions about the composition of the medium: the elastic part, the phonon and electron (for metals) gases.

**PNRPU Mechanics Bulletin**. 2013;(1):235-259

To the theory of process hydrating of gas bubbles in the conditions of the World ocean

#### Abstract

The theoretical model of migration metane bubbles in the conditions of formation hydrate is offered and constructed in work. Influence of various parameters on process of formation hydrate bubbles is studied. The minimum values of mass expenses of gas and the water necessary for process hydrating are received. Found that the migration of gas bubbles in a vertical channel, there are two modes of the process hydrate formation, depending on the mass expenses of water. If the initial value of the mass expenses greater than the critical, then for a given mass expenses of gas sufficient for hydrate formation, gas bubbles become fully hydrated, thus water temperature in the channel doesn't reach the equilibrium temperature of hydrating. If the initial value of the mass expenses is less critical, then gas bubbles are covered hydration shell, and the water temperature in the channel reaches equilibrium, and the process hydrating is completed. Also found that the increase of the initial mass expenses, the height of ascent of hydrate bubbles grow to a certain critical expenses, and a further increase - falls. The analysis of different depths at which gas sources are located is carried out: 800, 1200 and 1500 m. It is found that if there is a process of partial hydrating, the deeper the source is located, the greater the height of the rising bubbles. If there is a process full of hydrating, the picture is quite different: the higher up the gas source - the greater the height of hydrate formation.

**PNRPU Mechanics Bulletin**. 2013;(1):260-273