Table of contents

PREFACE

Winter Schools on Continuum Mechanics are held biennially by the Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences beginning with 2017. The papers in this issue of the journal were selected on the basis of their bearing on the materials of the XX Winter School held in 2017.

About 350 people from 30 cities of 7 countries (Russia, Austria, Germany, Poland, Israel, India, Japan) took part in the work of the XX-th Winter School on Continuum Mechanics. Russian participants represented 47 organizations in 23 cities (Perm, St.-Petersburg, Moscow, Yekaterinburg, Krasnoyarsk, Novosibirsk, Kazan, Chelyabinsk, Samara Rostov-on-Don, etc.) The scope of the problems addressed at the Conference is extremely wide and includes the following topics:

– computational continuum mechanics

– coupled problems of solid mechanics

– physics and mechanics of meso- and nano-structured systems

– convection, hydrodynamic stability and turbulence

– hydrodynamics of multiphase media

– hydrodynamics of non-Newtonian liquids and liquids with special characteristics

– interdisciplinary studies

All these topics are among the central themes of modern continuum mechanics, as evidenced by the increasing number of publications in the leading Russian and foreign journals During the panel sessions a great number of papers on the fundamental and applied problems of continuum mechanics were presented. Of particular interest were the invited papers of the well-known specialists in different areas of continuum mechanics (full members of RAS, corresponding members of RAS, doctors of sciences), and the presentations of representatives of industries manufacturing hi-tech products.

Chairman of the XX Winter School On Continuum Mechanics

academician of RAS V.P. Matveenko

All papers published in this volume of IOP Conference Series: Materials Science and Engineering have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

In this work, we experimentally investigate the mechanism of jet formation on liquid surface near the edge of a vibrating plate that is partially immersed in the liquid. Under the conditions of resonant bending vibrations, the vibrating plate excites capillary waves in the form of Faraday ripples on the surface of the liquid layer wetting the plate surface. Two-dimensional capillary waves also appear on the curved surface of the liquid near the edges of the vibrating plate. The vibrations of the plate area generate hydrodynamic pressure on the liquid surface, which initiates surface eddy flows. At a certain position of the vibrating plate in the liquid, capillary oscillations in the form of standing waves appear along the boundary of the wetting layer of the liquid, directly under the free edge of the plate. The vibrations of the plate edge modulate the standing waves in the transverse direction, which results in the periodic variation of the surface curvature of the wetting liquid layer, from negative to positive values. The inertia forces, periodically varying with the frequency of the plate vibrations, combined with the Laplace pressure, in the excited standing capillary waves on the surface flow under the plate edge, initiates the periodic ejection of particles of the liquid forming a jet.

In this paper we consider the geometrically nonlinear problem of determining the equilibrium position of a sandwich plate consisting of two external carrier layers and located between transversely soft core, connected with carrier layer by means of adhesive joint. We investigate the generalized statement of the problem. For its numerical implementation we offer a two-layer iterative process and investigate the convergence of the method. Numerical experiments are carried out for the model problem.

The transfer matrix method for the theory of elastic wave scattering in anisotropic layered medium is developed. The cases of the six-beam diffraction are considered. The method of the transfer matrix exp(Wz) computation is based on the use of polynomials of principal minors in combination with scaling of matrix W. This method, in contrast to the Lagrange-Sylvester and others known polynomial approaches does not require solution of eigenvalue problem. Evaluations of the scaling parameter and relative truncation error of the method are made. Some features of the conversions of shear waves and longitudinal wave are demonstrated by calculations of the diffraction on a crystalline silicon layer. It is shown that the conversion SH wave into SV waves and SV wave into SH waves are equivalent.

The fitting of a looped belt on two pulleys with different radii is considered. A geometrically nonlinear model with account for tension and transverse shear is applied for modeling the belt. The pulleys are considered rigid bodies, and the belt-pulley contact is assumed frictionless. The problem has an axis of symmetry, therefore the boundary value problem is formulated and solved for a half of the belt. The considered part consists of three segments, two contact segments and a free span segment between them. The introduction of a dimensionless material coordinate at all segments leads to a system of ordinary differential equations of fifteenth order. The nonlinear boundary value problem for this system and boundary conditions is solved numerically with the shooting method and the finite difference method. As a result, the belt shape including the rotation angle, the forces, moments and contact pressure are determined. The contact pressure increases near the end point of contact areas, however no concentrated contact forces occur.

In work the technique of calculation of elements of three-dimensional reinforced concrete substructures located in a soil, interacting with each other through rubber linings is realized. To describe the interaction of deformable structures with the ground, special "semi-infinite" finite elements are used. A technique has been implemented that allows one to describe the contact interaction of three-dimensional structures by means of a special contact finite element with specific properties. The obtained numerical results are compared with the experimental data, their good agreement is noted.

In the work for different cases of nozzles of layers of the super-magnetic pendulum, exact solutions in the stresses are obtained, and estimates of the increase in its specific energy are given. Dependences of tensile stresses in a tape are obtained when winding it into a super-flywheel, an increase in the specific kinetic energy in the super flywheel is observed with a decrease in the relative radius of the inner hole.

A pair of magnetizable solid particles embedded in a cylinder made of high-elasticity material is considered as a model of a mesoscopic structure element of a magnetorheological elastomer. An applied magnetic field induces ponderomotive interaction of the particles making them to move relative to one another so as to balance the counteracting magnetic and elastic forces. In a certain parameter range, the system exhibits bistability due to which under the increase / decrease of the field, the interparticle distance changes in a hysteretic manner. This behavior has a significant effect on the ability of the mesoscopic element to resist external load. Using the developed two-particle model prone to the magnetomechanical hysteresis, we extend it to the case of a virtually macroscopic sample presenting the latter as a superposition of such elements with distributed interparticle distances. In spite of its simplicity, this scheme in a generally correct way describes the field-induced changes of the internal structure and elastic modulus of the magnetorheological composites.

Nanopowder uniform and uniaxial cold compaction processes are simulated by 2D granular dynamics method. The interaction of particles in addition to wide-known contact laws involves the dispersion forces of attraction and possibility of interparticle solid bridges formation, which have a large importance for nanopowders. Different model systems are investigated: monosized systems with particle diameter of 10, 20 and 30 nm; bidisperse systems with different content of small (diameter is 10 nm) and large (30 nm) particles; polydisperse systems corresponding to the log-normal size distribution law with different width. Non-monotone dependence of compact density on powder content is revealed in bidisperse systems. The deviations of compact density in polydisperse systems from the density of corresponding monosized system are found to be minor, less than 1 per cent.

This paper is devoted to the analysis of the dynamic behavior of cylindrical shells, containing an internal annular layer of ideal fluid and subject to the external supersonic gas flow. The aerodynamic pressure is calculated based on the quasi-static aerodynamic theory. The behavior of the compressible fluid is described in terms of the perturbation velocity potential. A mathematical formulation of the problem is developed based on the classical theory of shells and virtual displacement principle. A solution of the problem involves computation of complex eigenvalues of the coupled system of equations. The paper presents the results of numerical experiments, which were performed to estimate the influence of the fluid flow velocity on the value of the static pressure in the unperturbed gas flow for shells, interacting with fluid layers of different thicknesses. The numerical simulation shows that a reduction of the fluid layer thickness and increase of the fluid velocity produce a stabilizing effect by virtue of increasing the threshold of aerodynamic stability. However, an essential reduction of the layer thickness can lead, depending on the preset combinations of boundary conditions, to a considerable growth of the stability threshold or to the onset of instability.

A new exact solution for layered convection of a viscous incompressible fluid is found in this paper. A fluid flow in an infinite layer is considered. Convection in the fluid is induced by tangential stresses specified on the upper non-deformable boundary. Temperature corrections are given on the both boundaries of the fluid layer. The analysis of hydrodynamic fields allows us to state the presence of two stagnant points in the flow of a fluid. It is shown that, in the case of thermocapillary convection in a fluid, only one stagnation point can exist.

Numerical simulation of the helical flow in a cylindrical channel with diverter was carried out using open-source software OpenFOAM Extend 4.0. The velocity, vorticity and helicity density distributions were analyzed. It was shown that azimithal contribution of helicity is negative near the wall and positive in the center. In opposite axial helicity contribution is negative in the center and positive near the wall. Analysis of helicity of non-axisymmetric part of the flow showed that it has substantial values near the diverter but than rapidly decreases with y (axial coordinate) and further downstream it can be neglected. Dependencies of integrated values of azimuthal H_ϕ and axial H_y contributions of helicity density on y show a remarkable quantitative similarity. It was found that integral values of H_ϕ and H_y are negative for all y. Magnitudes of H_ϕ and H_y decrease after the diverter up to y ≈ 70 mm and after that monotonically increase. The flow behind the diverter is characterized by substantial amount of helicity and can be used as a helicity generator.

This work is aimed at developing a thermodynamic approach to describing the propagation of fatigue cracks in metals. An attempt is made to explain the change in the character of heat dissipation at different stages of crack propagation: the nucleation, the Paris regime, the critical growth. The studies were conducted on two metal alloy: 304 AISE stainless steel and titanium alloy VT1-0. The investigation of the fatigue crack propagation was carried out on flat samples with stress concentrators. The stress concentrator was the triangular side notch. To monitor the dissipated thermal energy it was used method of infrared thermography and the contact heat flux sensor based on the Seebeck effect. Also the registration system of the acoustic emission was used for more exactly description of the fatigue crack propagation. Analysis of acoustic emission data on the basis of cluster analysis made it possible to classify various mechanisms of the damage process. A correlation was found between the integral dissipated thermal energy and the total energy of acoustic emission during the propagation of a fatigue crack. The joint application of these techniques has made it possible to reveal the moment's activation of failure mechanisms and their relationship to the dissipated heat flux.

The paper deals with the problem of determining the mechanical macroparameters of the porous material in case of knowing the information about it's structure. Fabric tensor and porosity was used to describe structure of the material. Experimental study presented. In research two-component liquid polyurethane plastics of cold curing Lasilcast (Lc-12) was used. Then samples was scanned on computer tomography. Resulting data was analyzed. Regular subvolume was cut out after analyses. Then mechanical tests was performed. As a result we get information about fabric tensor, porosity, Young's modulus and Poisson ratio of the sample. In the abstract presented results for some samples. Taking into account the law of porosity variation, we considered the problem of evaluating the mechanical macro parameters depending on the nature of the porous structure. To evaluate the macroparameters, we built the dependence of the Young's modules and Poisson ratio of the material on the rotation angle α and the pore ellipticity parameter λ. The sensitivity of the deformations to the elastic constants was also estimated.

The present study includes experimental investigations of rheological properties of the epoxy composite matrix based on epoxy resin L and hardener EPH 161. It is shown that the temperature dependences of the oligomer viscosity and the hardener can be satisfactorily described by the Arrhenius equation. Dependence of epoxy resin viscosity on temperature and volume concentration of the hardener at the initial stage of the polymerization process is investigated. It is shown that the experimental results can be generalized with one universal dependence of concentration and temperature.

Our research is related to the employment of photoplethysmography (PPG) and laser Doppler flowmetry (LDF) techniques (measuring the blood volume and flux, respectively) for the peripheral vascular system. We derive the governing equations of the wave dynamics for the case of extremely inhomogeneous parameters. We argue for the conjecture that the blood-vascular system as a wave-conducting medium should be nearly reflection-free. With the reflectionlessness condition, one can find the general solution to the governing equation and, on the basis of this solution, analyse the relationships between PPG- and LDF-signals.

Convection in a fluid layer with a free surface in a cylindrical container non-homogeneously heated from below was studied experimentally. The heater has a circular form and was placed in the center of the vessel. Such a system in the presents of rotation is very promising for studying the nature of tropical cyclones. In current paper we considered the influence of different geometric parameters on the dynamics of convection flows. For it two experimental setups of different sizes were used. Measurements were done for fluids with different values of Prandtl number and heating powers. It was shown that the structure and intensity of mean flow producing by a horizontal temperature gradient are defined by Rayleigh number Ra. The basic flow leads to unstable temperature stratification over the heating area and the formation of a system of secondary flows. The formation of secondary flows depends on characteristics of thermal boundary layer and is described by local Rayleigh number Ra_δ.

Within the framework of self consistent dynamic problems, the impact of dislocations and point defects on the spatial localization of nonlinear acoustic waves propagating in materials has been studied.

To date, polymer compositions based on polyethylene or PVC is widely used as insulating materials. These materials processing conjugate with a number of problems during selection of the rational extrusion regimes. To minimize the time and cost when determining the technological regime uses mathematical modeling techniques. The paper discusses heat and mass transfer processes in the extruder screw channel, output adapter and the cable head. During the study were determined coefficients for three rheological models based on obtained viscosity vs. shear rate experimental data. Also a comparative analysis of this viscosimetric laws application possibility for studying polymer melt flow during its processing on the extrusion equipment was held. As a result of numerical study the temperature, viscosity and shear rate fields in the extruder screw channel and forming tool were obtained.

The process of coextrusion consists in a simultaneous creation of all necessary insulating layers of different polymers in the channel of a special forming tool. The main focus of this study is the analysis of technological, geometrical and rheological characteristics on the values of the layer's thickness. In this paper are considered three geometries of cable head on the three–dimensional and two–dimensional representation. The mathematical models of separate and joint flow of polymer melts have been implemented by the finite element method in Ansys software package. The velocity fields, temperature, pressure in the cross–sections of the channel and by the length have been obtained. The influence of some thickness characteristics of insulation layers has been identified.

Evaluation of the mechanical state of a structure or its components in the process of operation based on detection of internal damages (damage detection) becomes especially important in such rapidly developing spheres of production as machine building, aerospace industry, etc. One of the most important features of these industries is the application of new types of materials among which polymer based composite materials occupy a significant position. Hence, they must have sufficient operational rigidity and strength. However, defects of various kinds may arise during the manufacture. Delamination is the most common defect in structures made from composite materials and represents a phenomenon that involves the complex fracture of layers and interlayer compounds. Among the reasons of delamination occurrence are: disposition of anti-adhesive lubricants, films; insufficient content of binder, high content of volatile elements; violation of the molding regime; poor quality of anti-adhesive coating on the surface of the tooling. One of the effective methods for analyzing the influence of defects is numerical simulation. With the help of numerical methods, it is possible to track the evolution of various parameters when the defect size and quantity change. In the paper, a multilayered plate of an equally resistant carbon fiber reinforced plastic was considered, with a thickness of each layer equal to 0.2 mm. Various static loading cases are studied: uniaxial tension, three and four-point bending. For each type of loading, a numerical calculation of the stress-strain state was performed for healthy and delaminated plates, with different number and size of the defects. Contact interaction between adjacent surfaces in the zone of delamination was taken into account.

In this paper, the problems containing two or more small parameters are investigated. In the analysis of such problems, a diagrammatic method is used to visualize the structure of perturbed flow regions, and to estimate the similarity parameters and mathematical models depending on the limiting transitions. The interaction between the laminar flows and transonic flows is considered in the context of the problem containing a small parameter inverse to the Reynolds number, the Mach number other than 1 and the pressure perturbation amplitude of the initiating viscous-inviscid interaction processes. The results of the flow analysis in a laminar boundary layer under the assumption of discontinuity of boundary conditions, for example, discontinuity of the surface velocity, are presented. For such problems a diagram of possible embedded perturbed flow regions is plotted. In addition, the results of studying the nonstationary processes of interaction between the flow in the boundary layer and the external supersonic flow are discussed.

The characteristics of an oscillating vortex flow of liquid metal were studied experimentally using an ultrasound Doppler velocimeter (UDV). The flow was generated by a local alternating magnetic field induced in a rectangular thin cell filled with gallium eutectic. The influence of medium temperature change and stirring on the UDV measurements was considered. The best set of parameters providing the reliability of long-term measurements were determined. The non-monotonic behavior of dependence of the local kinetic energy on the external alternating magnetic field intensity was found.

This paper presents the results of numerical simulation of the Soret-induced convection of ternary mixture in the rectangular cavity elongated in horizontal direction in gravity field. The cavity has rigid impermeable boundaries. It is heated from the bellow and undergoes translational linearly polarized vibrations of finite amplitude and frequency in the horizontal direction. The problem is solved by finite difference method in the framework of full unsteady non-linear approach. The procedure of diagonalization of the molecular diffusion coefficient matrix is applied, allowing to eliminate cross-diffusion components in the equations and to reduce the number of the governing parameters. The calculations are performed for model ternary mixture with positive separation ratios of the components. The data on the vibration effect on temporal evolution of instantaneous and average fields and integral characteristics of the flow and heat and mass transfer at different levels of gravity are obtained.

Flows and deformations of free surface of isothermal liquid bridge under the influence of axial vibrations of finite amplitude and frequency in microgravity conditions are studied numerically by Volume of Fluid method. Investigation is carried out in the framework of full non-average approach. Numerical data on instantaneous and average velocity fields and instantaneous and average shapes of free surface of fluid at different vibration frequencies and amplitudes are obtained.

The paper deals with the investigation of critical perturbations structure in a horizontal layer of fluid with temperature inversion of density. The lower boundary of the layer is rigid and upper boundary is free and undeformable. The constant vertical heat flux is imposed at both boundaries. The thermocapillarity, evaporation and radiation are neglected. The temperature dependence of the density is assumed to be quadratic. The coordinate of the density inversion point which shows the position of the plane of maximal density in the fluid layer in the conductive state is used for the description of density inversion effect. The influence of the location of density inversion point on structure of critical perturbations is studied. It is shown that, depending on the location of the inversion point, velocity profile of longwave critical perturbations can have two-floor or three-floor structure. Stability to finite-wavelength perturbations is studied for the entire range of their existence. The asymptotic formulas for critical values of the Rayleigh number and wave number are obtained, the structure of finite-wavelength critical perturbations .is determined.

The natural porous media (e.g. soil, sand, peat etc.) usually are partially saturated by groundwater. The saturation of soil depends on hydrostatic pressure which is linearly increased with depth. Often some gases (e.g. nitrogen, oxygen, carbon dioxide, methane etc.) are dissolved into the groundwater. The solubility of gases is very small because of that two assumptions is applied: I. The concentration of gas is equal to solubility, II. Solubility depends only on pressure (for isothermal systems). In this way some part of dissolved gas transfers from the solution to the bubble phase. The gas bubbles are immovably trapped in a porous matrix by surface-tension forces and the dominant mechanism of transport of gas mass becomes the diffusion of gas molecules through the liquid. If the value of water content is small then the transport of gas becomes slow and gas accumulates into bubble phase. The presence of bubble phase additionally decreases the water content and slows down the transport. As result the significant mass of gas should be accumulated into the massif of porous media. We derive the transport equations and find the solution which is demonstrated the accumulation of gases. The influence of saturation, porosity and filtration velocity to accumulation process is investigated and discussed.

The origin of the mechanisms of blood flow oscillations at low frequencies is discussed. It is known that even isolated arteriole demonstrates oscillations with the frequency close to 0.1 Hz, which is caused by the synchronous activity of myocyte cells. On the other hand, oscillations with close frequency are found in the heart rate, which are associated with quite different mechanism. The main purpose of this work is to study phase coherence of the blood flow oscillations in the peripheral vessels under basal and perturbed conditions. Local heating which locally influences the microvascular tone, as one of currently elucidated in sufficient detail physiological test, was chosen. During such provocation blood flow though the small vessels significantly increases because of vasodilation induced by the local synthesis of nitric oxide. In the first part of the paper microvascular response to the local test is quantified in healthy and pathological conditions of diabetes mellitus type 1. It is obtained that regardless of the pathology, subjects with high basal perfusion had lower reserve for vasodilation, which can be caused by the low elasticity of microvascular structure. Further synchronization of pulsations of the heated and undisturbed skin was evaluated on the base of wavelet phase coherency analysis. Being highly synchronised in basal conditions 0.1 Hz pulsations became more independent during heating, especially during NO-mediated vasodilation.

A two-dimensional version of the soil deformation method based on the particle method is implemented in this paper. For describing the behavior of a continuous medium, the previously used two-parameter "modified" Lennard-Jones interaction potential was chosen. A number of model problems describing the loss of stability of the soil embankment in the field of gravity and the destruction of the soil under the action of the liquid phase are solved.

The effect of the universal acid-base indicator on the pattern formation and mass transfer in a two-layer system composed of two reactive miscible liquids in a vertical Hele-Shaw cell is studied experimentally. The reaction we study is a neutralization one. It turns out that the presence of the indicator leads to a change in the spatio-temporal characteristics of the system and even in the mass transfer mechanism near the reaction front—from diffusive to convective. The conditions, where the universal indicator does not affect the reaction and can be used as a visualizing mean, are reported.

In this paper, with the aim of providing passive control of structure vibrations a new approach has been proposed for selecting optimal parameters of external electric shunt circuits connected to piezoelectric elements located on the surface of the structure. The approach is based on the mathematical formulation of the natural vibration problem. The results of solution of this problem are the complex eigenfrequencies, the real part of which represents the vibration frequency and the imaginary part corresponds to the damping ratio, characterizing the rate of damping. A criterion of search for optimal parameters of the external passive shunt circuits, which can provide the system with desired dissipative properties, has been derived based on the analysis of responses of the real and imaginary parts of different complex eigenfrequencies to changes in the values of the parameters of the electric circuit. The efficiency of this approach has been verified in the context of natural vibration problem of rigidly clamped plate and semi-cylindrical shell, which is solved for series-connected and parallel –connected external resonance (consisting of resistive and inductive elements) R-L circuits. It has been shown that at lower (more energy-intensive) frequencies, a series-connected external circuit has the advantage of providing lower values of the circuit parameters, which renders it more attractive in terms of practical applications.

The eddy current flowmeter for liquid metal passing through the cylindrical channel is studied experimentally and numerically. The alternating magnetic field is created by an annular-shape coil and resulting field is measured by two symmetric coils. The dependence of the signal on the magnetic field frequency was obtained for a solid duralumin cylinder. The phase shift and amplitude ratio linearly depends on the flowrate in a wide range of parameters. The workability of the method was tested on a liquid sodium, flowing along a cylindrical channel. The dependencies for liquid sodium and solid duralumin cylinder differ due to the existence of a non-uniform velocity profile along the radius of the fluid. This necessitates an additional calibration procedure on the liquid metal in addition to research on the solid conductor. The developed flowmeter is used to measure the flowrate in the sodium loop.

A study of the aeroacoustic characteristics of the PNRPU jet rig is considered. The design of the jet rig is described. The velocity of the jet along the axis is measured at a certain distance from the nozzle exit section. The comparison of the velocity with a semi-empirical model is performed. A good agreement of the velocities is observed. The noise of the jet is measured for different velocities in the direction of 30 and 90 degrees. The obtained spectra are compared with the results of measurements of the jet rig in the anechoic chamber AC-2 TsAGI. Localization of the noise sources of the jet with the Bruel & Kjaer microphone array is also carried out. The investigations are carried out in a new anechoic chamber at PNRPU.

For well-stirred multiphase fluid systems the mean interface area per unit volume, or "specific interface area" S_V, is a significant characteristic of the system state. In particular, it is important for the dynamics of systems of immiscible liquids experiencing interfacial boiling. We estimate the value of parameter S_V as a function of the heat influx ${\dot{Q}}_{V}$ to the system or the average system overheat 〈Θ〉 above the interfacial boiling point. The derived results can be reformulated for the case of an endothermic chemical reaction between two liquid reagents with the gaseous form of one of the reaction products. The final results are restricted to the case of thin layers, where the potential gravitational energy of bubbles leaving the contact interface is small compared to their surface tension energy.

The inequalities which must be satisfied the characteristics of elastic state of the materials of contacting bodies at their adhesion (coalescence) and its absence (antiadhesion) were obtained. These are the result of the analysis of adhesion phenomena and its absence. The analysis is made on the basis of a special variant of a nonlocal theory of elasticity. Its main hypothesis is infinitely small particles of the continuous elastic medium interact with each other at finite distances with the help of many-particle potential forces. The results of using criterial inequalities were confirmed by known experimental data.

The layered convective flow of a viscous incompressible fluid is considered with the specified velocities at the bottom of an infinite layer. A new exact stationary and nonstationary solution of the Oberbeck-Boussinesq system is presented. The account of fluid velocity at the bottom is characterized by the presence of two stagnant points, this being indicative of the nonmonotonic kinetic energy profile with two local extrema.

In this study, the models describing the behavior of shape memory alloys, ferromagnetic materials and polymers have been constructed, using a formalized approach to develop the constitutive equations for complex media under large deformations. The kinematic and constitutive equations, satisfying the principles of thermodynamics and objectivity, have been derived. The application of the Galerkin procedure to the systems of equations of solid mechanics allowed us to obtain the Lagrange variational equation and variational formulation of the magnetostatics problems. These relations have been tested in the context of the problems of finite deformation in shape memory alloys and ferromagnetic materials during forward and reverse martensitic transformations and in shape memory polymers during forward and reverse relaxation transitions from a highly elastic to a glassy state.

A three-dimensional model for deformation of metal matrix composite with aluminum matrix and silicon carbide reinforcement particles is developed. The model accounts for an internal structure of the composite, as well as rheology of its constituents. The model is further used in numerical simulations in order to study the evolution of stress-strain state parameters in a randomly-chosen composite microstructure fragment under uniaxial tension and compression loading on micro- and macroscale. The parameters include the stress stiffness coefficient, the Lode-Nadai coefficient and equivalent (von Mises) strain. It is found that local deformation regions and internal tensile stress concentration regions appear in the material of composite matrix. Adhering to a phenomenological damage theory, a damage development is computed in the matrix metal. We present damage fields and damage distributions for uniaxial tension and compression.

It is generally believed that helicity can play a significant role in turbulent systems, e.g. supporting the generation of large-scale magnetic fields, but its exact contribution is not clearly understood. For example there are well-known examples of large scale dynamos produced by a flow which is pointwise non-helical. In any case a break of mirror symmetry seems to be always at the heart of the dynamo mechanism. A fruitful framework to analyze such processes is the use of helical mode decomposition. In pure hydrodynamics such framework has proved its availability in study of the processes responsible for helicity cascades. It has also been used in the analysis of MHD helical mode interactions. The present work deals with the kinematic dynamo problem, solving the induction equation within the framework of helical Fourier modes decomposition. We show that the simplest modes configuration leading to an unstable solution has the form of a tetrahedron. Then the dynamo is produced by only two scales flow. We find necessary conditions for such dynamo action, not certainly related to flow helicity. The results help to understand generic dynamo flows like the one studied by G.O. Roberts (1972).

The effect of inhomogeneous temperature distribution at boundary on convection in an infinite horizontal plane layer is investigated. Direct numerical simulation of the compressible non-isothermal flow in a cubic cell with rigid horizontal walls is performed under periodic vertical boundary conditions. Laminar and weakly non-linear flow regimes are determined. The heated area on the cell bottom obeys regular or fractal distributions. The intensities of heat flux through the layer are compared for different heterogeneous distributions of heating elements at a specific temperature gradient in the case when the area of heated surface remains constant. Fractal geometry shows the appearance of the multiscale structure of the flow and the enhancement of heat transfer.

Localized heating in the rotating layer of fluid leads to the formation of intensive cyclonic vortex. Cyclonic vortex becomes unstable at low values of viscosity and fast rotation of the experimental model. The instability of the vortex is tightly connected with a structure of the radial inflow. For moderate values of rotational Reynolds number Re the radial flows consist of several branches which transport angular momentum to the center of the model. When Re exceeds critical value (about 23) radial inflow changes its structure and appears as one wide branch which does not reach the center. As a result of strong anisotropy of radial inflow the cyclonic vortex is formed at some distance from the center. Further increase of Re leads to chaotic state with several vortices which appears at different locations near the periphery of the heating area. The map of regimes with stable and unstable vortices is presented.

The structure of the convective flow of molten magnesium in a metallothermic titanium reduction reactor has been studied numerically in a three-dimensional non-stationary formulation with conjugated heat transfer between liquid magnesium and solids (steel walls of the cavity and titanium block). A nonuniform computational mesh with a total of 3.7 million grid points was used. The Large Eddy Simulation technique was applied to take into account the turbulence in the liquid phase. The instantaneous and average characteristics of the process and the velocity and temperature pulsation fields are analyzed. The simulations have been performed for three specific heating regimes: with furnace heaters operating at full power, with furnace heaters switched on at the bottom of the vessel only, and with switched-off furnace heaters. It is shown that the localization of the cooling zone can completely reorganize the structure of the large-scale flow. Therefore, by changing heating regimes, it is possible to influence the flow structure for the purpose of creating the most favorable conditions for the reaction. It is also shown that the presence of the titanium block strongly affects the flow structure.

The problem of determining the strength of engineering structures, considering the effects of the non-local fracture in the area of stress concentrators is a great scientific and industrial interest. This work is aimed on modification of the classical theory of critical distance that is known as a method of failure prediction based on linear-elastic analysis in case of elasto-plastic material behaviour to improve the accuracy of estimation of lifetime of notched components. Accounting plasticity has been implemented with the use of the Simplified Johnson-Cook model. Mechanical tests were carried out using a 300 kN electromechanical testing machine Shimadzu AG-X Plus. The cylindrical un-notched specimens and specimens with stress concentrators of titanium alloy Grade2 were tested under tensile loading with different grippers travel speed, which ensured several orders of strain rate. The results of elasto-plastic analyses of stress distributions near a wide variety of notches are presented. The results showed that the use of the modification of the TCD based on elasto-plastic analysis gives us estimates falling within an error interval of ±5-10%, that more accurate predictions than the linear elastic TCD solution. The use of an improved description of the stress-strain state at the notch tip allows introducing the critical distances as a material parameter.

Paper is devoted to a numerical justification of the recent a posteriori error estimate for Reissner-Mindlin plates. This majorant provides a reliable control of accuracy of any conforming approximate solution of the problem including solutions obtained with commercial software for mechanical engineering. The estimate is developed on the basis of the functional approach and is applicable to several types of boundary conditions. To verify the approach, numerical examples with mesh refinements are provided.

A convective flow of liquid sodium generated nearby a hot round in the upper solid end face of a vertical cylinder has been studied experimentally and numerically. A developed turbulent flow is observed in the upper part of the cylinder. Strong velocity pulsations penetrate in the bulk of the metal up to a distance of about the diameter of the cylinder. Mean velocity fields reveal a toroidal vortex, which is localized in a narrow upper zone. Numerical simulations were done for two types of thermal boundary conditions (BCs): fixed temperature and fixed homogeneous heat flux on both heat exchangers. Experimental values of time-averaged velocity and temperature in the vortex are in good agreement with numerical data. The size and the intensity of the vortex weakly depend on BCs. The whole bulk of the metal is not involved in the motion. The temperature field depends much more on the BCs. Under fixed heat fluxes the temperature pulsations become much stronger and penetrate essentially deeper in the liquid metal, though the flow is slightly stronger under fixed boundary temperature. The considered flow is supposed to be a simplified model of the liquid magnesium flow in a reactor of metallothermic titanium reduction.

The formation and dynamics of such an elementary object of fluid dynamics as a vortex ring was investigated. Intense turbulent vortex rings created by a piston generator were considered. For qualitative determination of vortex ring properties, the law of motion for the generator's piston was experimentally established. This law was used in the numerical simulation, which was performed for the axisymmetric problem formulation with the use of ANSYS Fluent software. The obtained results correspond to the self-similar law of vortex ring motion and experimental results.

Among various methods that allow controlling quasi-static vertical displacements of structures, the hydrostatic leveling method remains relevant. A multi-point hydrostatic leveling systems allows controlling the vertical displacement field with the required spatial resolution. It is assumed that the liquid level in the each measuring vessel has the same absolute elevation. However, it is influenced by various external factors, which are difficult to eliminate when implementing the method in real constructions. Consequently, it is necessary to assess the influence of these factors and develop methods aimed at their reduction or compensation. A mathematical model, describing the viscous incompressible fluid motion located in a multipoint level with absolutely rigid walls, is presented in the paper. The experiments performed with two point levels of different lengths, as well as analytical estimates of other authors, made it possible to estimate the degree of confidence of the model and the boundaries of applicability. The influence of non-uniform density of a liquid on the liquid level in a 4-point hydrostatic level of different topologies is numerically estimated using the model. An estimation of transient processes in the level, caused by an air pressure surge over one of the measuring vessels is carried out.

In this paper, based on some example problems it was demonstrated that in examining the possibilities of smart structure applications, the matter of considerable researchers' concern is the problem of location of piezoelectric elements in the structure to allow effective realization of its smart functions in the framework of the specified strategy of structure control and target purposes (vibration damping, defectoscopy, etc.) The numerical and experimental investigations have shown that for structures with the elements made of piezoelectric materials, it is more convenient to use as a parameter, specifying the best location of the piezoelectric element for damping the vibrations at the prescribed frequency, the coefficient of electromechanical coupling, which is evaluated by the values of eigenfrequencies of the structure in the short-circuit and open-circuit regimes. The values of eigenfrequencies of vibrations are evaluated by solving the problem of natural vibrations of electromechanical systems by the finite element method using the applied ANSYS package. The investigation were conducted for a thin-walled aluminum shell in the form of half-cylinder.