Mechanics Of Continuous Media Research Articles

О развитии приближенных методов исследования процессов в блочно структурированной среде

We need to consider the model of the medium as close as possible to the natural one, to apply a mathematical apparatus that adequately and reliably describes the processes and phenomena occurring in the studied medium of a complex structure in the process of creating modern systems for monitoring the geophysical medium, regulating the quality of the environment. Such a possibility is provided by methods having a topological basis, in particular, a differential factorization method. However, with strict adherence to the algorithm, the numerical evaluation of the parameters under study requires considerable time. Therefore, it is necessary to identify those moments where it is possible to proceed to an approximate solution without compromising the accuracy of the result obtained. A method is proposed for constructing approximate solutions of systems of integral equations arising in the investigation of boundary problems of the mechanics of a solid deformed body and the mechanics of continuous media by a differential factorization method for media of complex structure. The justification of the proposed conclusions for the problems posed both in Cartesian and curvilinear coordinate systems is fulfilled. As an example, the application of approximate methods in problems of assessing the quality of the aquatic environment or the atmosphere is considered. The results can be used for express forecast of the ecological state of the environment, in systems of integrated geoecological monitoring.

Size-Dependent Thermal Boundary Resistance and Thermal Conductivity in Si/Ge Core–Shell Nanowires

Heat transfer through interfaces in nanostructures is becoming ever more important in functional nanodevices. The existence of interface between two dissimilar materials overheats nanoelectronics and impacts heat transfer greatly. It is a challenge how to modulate interface to tailor the thermal transport properties of nanodevices from the perspective of atomic level. In this brief, we consider how size and interface strain affect thermal boundary resistance (TBR) as well as thermal conductivity (TC) of Si/Ge core–shell nanowires (CSNWs) using a thermal kinetic method in terms of atomic-bond-relaxation correlation mechanism and continuum medium mechanics. We propose a theoretical model to pursue the underlying mechanism on the TBR and TC that determined on the core or shell thickness, surface roughness, and interface mismatch. Our approach provides a useful guidance to the theoretical design and experimental control of epitaxial growth in the radial CSNWs for practice applications.

The method of dynamic operating modes investigation in electrohydraulic drive systems with improved accuracy of positioning executive elements

The article describes the method of dynamic operating modes investigation in electrohydraulic drive systems with improved accuracy of positioning output element. The method is the evolution of the fundamental positions of the mechanics of continuous media with moving boundaries as applied to the research of non-stationary processes accompanying the operation of hydraulic drive systems with piston hydraulic machines. The method is based on generalized modeling (technical, physical, mathematical and computer), takes into account the peculiarities of mutual influence of electric and hydraulic machines during their joint work as part of the electrohydraulic drive, which raises the level and adequacy of actuators simulation, as well as the reliability of the assessment of their technical condition. The method extends the field of research, improves the accuracy of the calculation of the positioning of the executive elements, taking into account the different dynamic modes of the drives under study.

Simulation of cylindrical shell collapse with considering plasticity and fracture of metals

In this paper, in one-dimensional cylindrical formulation, the equations of mechanics of continuous medium supplemented by equations of plasticity and fracture models are numerically solved. The results of the calculation were verified on the data obtained by interferometric measurement of the free-surface velocity profile in the problem of high-speed collision of metallic samples. The model is applied to the problem of cylindrical metal shell collapse.

Influence of bed material entrainment and non-Newtonian rheology on turbulent geophysical flows dynamics. Numerical study

This paper deals with the mathematical and numerical modeling of the propagation stage of geophysical gravity-driven flows, such as snow avalanches, mudflows, and rapid landslides. New mathematical models are presented which are based on full, not-depth-averaged equations of mechanics of continuous media. The models account for three important issues: non-Newtonian rheology of the moving material, entrainment of the bed material by the flow, and turbulence. The main objective is to investigate the effect of these three factors on the flow dynamics and on the value of the entrainment rate. To exclude the influence of many other factors, e.g., the complicated slope topography, only the motion down a long uniform slope with a constant inclination angle is studied numerically. Moreover, the entire flow from the front to the rear area was not modeled, but only its middle part where the flow is approximately uniform in length. One of the qualitative results is that in motion along homogeneous slope the mass entrainment increases the flow velocity and depth while the entrainment rate at large time tends to become constant which depends on the physical properties of the flow and the underlying material but not on the current values of the flow velocity and depth.

New Model for Solving Mixed Integral Equation of the First Kind with Generalized Potential Kernel

New technique model is used to solve the mixed integral equation (\textbf{MIE}) of the first kind, with a position kernel contains a generalized potential function multiplying by a continuous function and continuous kernel in time, in the space $L_{2} (\Omega )\times C[0,T],\, 0\leq T<1$, $\Omega$ is the domain of integration and $T$ is the time. The integral equation arises in the treatment of various semi-symmetric contact problems, in one, two, and three dimensions, with mixed boundary conditions in the mechanics of continuous media. The solution of the \textbf{MIE }when the kernel of position takes the potential function form, elliptic function form, Carleman function and logarithmic kernel are discussed and obtain in this work. Moreover, many special cases are derived.

Numerical solution of the Navier-Stokes equations using the Path Tubes method

The present work addresses an extension of the Path Tubes method for solving the time-dependent Navier-Stokes equations for an incompressible Newtonian fluid. The approach used is a physically intuitive methodology whose formulation is based on the theoretical foundations of the mechanics of continuous media. This version of the Path Tubes method draws on a semi-Lagrangian time discretization that employs the Reynolds’s transport theorem and a localization strategy. This time discretization can be seen as a transformation that acts on the Navier-Stokes equations, transforming this classical nonlinear model into linear partial differential equations of the (essentially) parabolic type.The result is an implicit semi-Lagrangian algorithm that allows the use of classical schemes for spatial discretization such as central-difference formulas, without the need to use upwind techniques or high-order corrections for time derivatives. The Path Tubes method was implemented through parallel computing. For this, we use a computer equipped with shared-memory multiprocessors and the OpenMP software. After intensive numerical tests and using different values of Reynolds numbers typical of advection-dominated flows, the proposed method proved to be accurate and able to work with coarse grids.

Role of particle collisions in shock wave interaction with a dense spherical layer of a gas suspension

The problem of interaction of an expanding spherical shock wave with a layer of particles is considered within the framework of the model of mechanics of continuous media with due allowance for granular pressure in the dense gas suspension. The influence of particle collisions on the shock wave expansion process is analyzed. Generation of collisional pressure and formation of shock wave structures in the gas suspension are found to be the governing factors of motion of the cloud of particles at the initial stage.

Formula omitted]-model of micro- and macrosystems

A mathematical model (referred as Ψ-model for convenience) has been developed, which allows describing certain class of micro- and macrosystems. Ψ-model is based on quantum mechanics and classical mechanics of continuous media. Ψ-model describes micro- and macrosystems, in which vector field of velocities of probability flows, charge, mass has specific spiral structure. The field of velocities has spiral structure on concentric spherical surfaces. The velocity field is not defined and has a characteristic property on the poles of sphere and on the axis and tends to zero at infinity. The behavior of Ψ-model can be described in the general case with time-dependent periodic singular solution of the Schrödinger equation.The goal of this paper is to choose a particular probability flux in the continuity equation which we solve in this paper and deduce from it the solution of the Schrödinger equation. For example, in the frame of approach the problem with modified Coulomb potential was considered.

Winter School on Continuous Media Mechanics

PREFACEWinter 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 studiesAll 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 Mechanicsacademician of RAS V.P. Matveenko

Method for recognizing wave dynamics damage in high-speed milling cutter

Under the influence of a high-speed, interrupted-cutting impact load, a great difference is existed among the internal load propagation of a milling cutter. Furthermore, the cutter damage caused by partial particle severe vibration has restricted the improvement of a high-speed milling energy efficiency; thus, the essence of wave dynamics damage in milling cutter remains has yet to be revealed. In this paper, through the relation between the systematic whole vibration and the particle motion, the dynamic response of milling cutter’s particle to cutting force load can be solved by the particle motion differential equation which is constructed with a one-dimensional string dynamic system. A combination of Newton’s second law and the constitutive equation of milling cutter material establishes the wave dynamics equation of milling cutter components. An approach for solving the wave front position and wave velocity of milling cutter’s stress wave is proposed, and the propagation path of transient cutting force to the milling cutter is communicated. The attenuation model of stress wave reflection is established to provide a method for revealing the stress wave transmission and distribution in milling cutter. The constitutive relation of milling cutter components under the impact load is obtained by split Hopkinson pressure bar experiment. A force connection method is adopted to make the trans-scale correlation analysis between continuum medium mechanics and molecular dynamics, thereby revealing the wave dynamics damage characteristics of a high-speed milling cutter. The results show that the potential damage position and types of milling cutter can be distinguished by the above method.

The inverse distance weighted interpolation applied to a particular form of the path tubes Method: Theory and computation for advection in incompressible flow

A simplified version of the inverse distance weighted interpolation formula was mathematically analyzed and employed in the context of a semi-Lagrangian scheme for the equation of advection. The scheme used is a particular formulation of Path Tubes method, a physically intuitive conservative method whose formulation is based on the theoretical foundations of the mechanics of continuous media, which uses the so-called Reynolds's transport theorem to establish judiciously its main property written on the basis of a conservative integral equation. The resulting algorithm is a five-point explicit semi-Lagrangian scheme for incompressible flow. From rigorous deductions based on the von Neumann stability analysis, it was proven that this explicit method is unconditionally stable. The present algorithm was evaluated using test problems that are prototypes of more sophisticated mathematical models commonly employed in the prediction of intensity variations of weather fronts. In addition, our methodology was compared against two other methods available in the literature. The proposed explicit semi-Lagrangian scheme was able to work with long time steps and proved to be accurate, non-oscillatory and non-diffusive.

NUMERICAL MODELING OF THE PROCESS OF FILLING THE CCM MOLD WITH METAL

Mathematical modeling of the liquid melt flow in the mold of the continuous casting plant is still poorly investigated. Analytic solutions of melt flow in the general case are a complex mathematical problem. Nevertheless, for some cases, exact solutions have been found. Such analytical solutions serve as a means of verifying the results of numerical methods of solution. The purpose of this work is the use of the numerical method proposed by Professor V.I. Odinokov, based on the finite-difference representation of the original system of equations. The method has been successfully used in the mechanics of continuous media, in the lithium manufacturing for the mathematical modeling of the strained deformed state of shell molds of cast models, as well as in other technological works, which indicates its universality. In the present work, hydrodynamic and thermal flows of liquid metal during the steel casting into the rectangular section mold of a continuous cast steel have become objects of research, and the result is a spatial mathematical model describing the flows of liquid metal in the mold. To simulate the processes that take place during filling, the software complex “Odyssey” is used. The basis of theoretical calculations includes the fundamental equations of hydrodynamics, the equations of mathematical physics (the equation of heat conductivity with regard to mass-transfer) and the approved numerical method. The solution of the system of differential equations formulated in this work was carried out numerically. The investigated area was divided into elements of finite dimensions, for each element the resulting system of equations was recorded in the difference form. The solution result is the metal flow velocity fields and the temperature fields in the mold volume. To solve the obtained system of algebraic equations, the numerical schemes and calculation algorithms were developed. Based on the developed numerical schemes and algorithms a computational program was compiled in Fortran-4. The mathematical model allows to vary the geometric dimensions of the crystal-mash and sectional metal exit holes from the submerged nozzle, and may also help to understand the scheme of the cast metal movement that affects the heat sink walls of the mold, and to find the optimal parameters of the output of liquid metal from the submerged nozzle at different modes of casting. An example of calculating the casting of steel into a rectangular mold with a height of 100 cm and cross-sectional dimensions of 2000.40 cm is given. The casting was carried out from a submerged nozzle symmetrically on both sides in a horizontal plane. The result of the solution is presented in the graphical form. The motion of liquid metal flows in different sections of the mold is shown. The areas of the circular flow of metal are revealed, as well as the areas in the mold volume, where the vortex motion of the liquid metal is observed, their magnitudes and intensity are determined. The presented temperature field indicates the presence of a local area with a high temperature at the wall of the mold, which is explained by the directed flow of hot metal emerging from the hole in the submerged nozzle.

Numerical research of the perforation of targets by ice impactors at different temperatures

In the present work, the process of interaction of ice impactors of spherical shape with targets of aluminum alloy and asbotextolite has been numerically investigated. The initial temperature of the impactors was set from 0 to 500° C. The speed of the impactors was set in the range from 500 m/s to 1500 m/s. The behavior of the medium was described from the general positions of the mechanics of continuous media and corresponded to modern concepts of the destruction of solids under load. The material was considered as isotropic, elastoplastic, compressible, porous medium. The equation of state is chosen in the Walsh form. The problem was solved in a two-dimensional formulation for the case of axial symmetry. The calculations were carried out using the Lagrangian technique of computer simulation, which takes into account the fragmentation of the material in shear and shear failure. The current configurations of the impactor-target and the velocity of the leading fragments are obtained when penetrating the targets.

Application of the Path Tubes Method to the Navier-Stokes Equations

This work deals with an extension of the Path Tubes method for the solution of the time-dependent Navier-Stokes equations for an incompressible Newtonian fluid. Departing from a physically intuitive methodology based on the theoretical basis of the mechanics of continuous media, a robust numerical technique is obtained. This version of the Path Tubes method draws on a semi-Lagrangian time-discretization that employs the Reynolds’ transport theorem, and a localization approach, to establish an implicit semi-Lagrangian algorithm that allows the use of classical schemes for spatial discretization, such as central-difference formulas, without the need to use upwind techniques, or high-order corrections for time derivatives. Some of the extensive numerical tests are shown herein, in particular for Reynolds’ numbers typical of advection dominated flows. The tests show the method is accurate, even for coarse grids.

О модели изотермической акустики для двухкомпонентной среды

A mathematical model describing the processes of isothermal acoustics in a heterogeneous medium with two components separated by a common boundary is studied. One of the components is an elastic body, and the other one is a poroelastic medium (for example, it may be a liquid-saturated soil). The poroelastic medium is permeated with a system of pores filled with viscous weakly compressible liquid. The differential equations of the model describing the motion of an elastic body and the joint motion of a solid skeleton and liquid in the pores are based on the classical laws of continuous medium mechanics and adequately reflect the physical processes. However, these equations contain rapidly oscillating coefficients that depend on a small parameter equal to the ratio of the mean pore size to the size of the region under study. The existence of such coefficients prevents the use of the model for carrying out numerical calculations. The generalized solution of the initial boundary-value problem is given; and the theorem for existence and uniqueness of the generalized solution is presented together with its a priori estimates. For performing the homogenization procedure, the standard assumption about the periodicity of the pore space and solid skeleton is adopted. The obtained a priori estimates and the N. Nguetseng's two-scale convergence method were used as a basis for deriving the averaged equations and the initial boundary conditions (that is, the limit equations with the small parameter tending to zero). Different limiting modes depending on the continuous medium parameters are obtained. An averaged model for a special case that does not contain rapidly oscillating coefficients and can be used for numerical calculations is presented.

The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

Abstract As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the \true trajectories" of the physical systems represent stationary points of the corresponding functionals. It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are Euler-Lagrange equations of an appropriately formulated variational principle. There are several well established geometrical theories providing a general description of variational problems of different kinds. One of the most universal and comprehensive is the calculus of variations on fibred manifolds and their jet prolongations. Among others, it includes a complete general solution of the so-called strong inverse variational problem allowing one not only to decide whether a concrete equation of motion can be obtained from a variational principle, but also to construct a corresponding variational functional. Moreover, conservation laws can be derived from symmetries of the Lagrangian defining this functional, or directly from symmetries of the equations. In this paper we apply the variational theory on jet bundles to tackle some fundamental problems of physics, namely the questions on existence of a Lagrangian and the problem of conservation laws. The aim is to demonstrate that the methods are universal, and easily applicable to distinct physical disciplines: from classical mechanics, through special relativity, waves, classical electrodynamics, to quantum mechanics.

AN ANTHOLOGY OF THE DISTINGUISHED ACHIEVEMENTS IN SCIENCE AND TECHNIQUE. PART 30: PORTRAIT OF THE KHARKOV MATHEMATICIAN, MECHANICAL ENGINEER AND CYBERNETICIST VLADIMIR LOGVINOVICH RVACHEV

Purpose. Description of basic scientific achievements, features of personality and way of life of the known Kharkov mathematician, mechanical engineer and cyberneticist, academician of NAS of Ukraine Rvachev V.L .in the short form is presented. Methodology. Existent scientific approaches for treatment and systematization of mathematical knowledges, modern achievements in area of methods of direct solution of linear (nonlinear) boundary problems of mechanics and mathematical physics with the scope terms of different types for the physical bodies of difficult geometrical form. Methods of historical method at research of development in society of analytical geometry, applied mathematics, classical mechanics and mathematical physics. Results. Short information is resulted about the basic creative and vital stages, and also fundamental scientific achievements of the indicated scientist-mathematician the scientific legacy of which entered in the treasure-house of world mathematical science. Are some personal qualities of this prominent soviet Ukrainian mathematician of the 20-th century, forming scientific school on the mathematical method of R−functions and leaving about itself kind memory for thankful students and descendants. Originality. First taught in 1970-th at the known mathematician of contemporaneity of Rvachev V.L. to bases of the applied mathematics and theory of R−functions by a scientist-electrophysicist from the Kharkov polytechnic institute presented for the wide circle of readers a short scientifically-historical essay about this large scientist-teacher, being based on his scientific labours, published biobibliographic materials and flashbacks of his devoted students-followers about him. Practical value. Scientific popularization of the special physical and mathematical knowledges and distinguished scientific achievements of the known Kharkov scientist-mathematician Rvachev V.L. in area of the applied mathematics, classical mechanics of continuous media, mathematical physics and technical cybernetics.

EXPERIMENTAL-COMPUTATIONAL ANALYSIS OF HIGH-SPEED INTERACTION OF A SOLID WITH WATER BARRIERS

The stress−strain state of projectiles is investigated when a high-speed projectile enters water and interacts with barriers protected by a water layer. Experimental investigations are carried out using a high-speed ballistic setup. Calculations are performed within the framework of the mechanics of continuous media for an elastoplastic model of a solid with allowance for fracture and a hydrodynamic water model. Depending on the projectile speed, different stress−strain state regimes are observed: from little deformed one at a speed of about 1 km/s to a fractured one at a speed of about 2 km/s. The calculation technique allows the distances in water to be determined at which a metal barrier can be punched in the inertial model.

An analytical model of coal wellbore stability based on block limit equilibrium considering irregular distribution of cleats

An analytical model of coal wellbore stability is established based on the block limit equilibrium method, taking the irregular distribution of cleats into account. Because coal seam is cut into discontinuous blocks with different shapes and sizes, block slip can be treated as the symbol of coal wellbore instability. According to the geometric characteristic of blocks, blocks surrounding the wellbore can be divided into 3 categorizes: type I, type II and type III. To evaluate the block slip, the stress distribution near the wellbore is determined based on the continuous medium mechanics and the fracture mechanics. Using Mohr–Coulomb criterion and limit equilibrium method, the critical conditions of block slip are derived by defining the failure coefficient, including type I, type II and type III. The proposed model is verified by comparison of the obtained results from the proposed analytical method with the discrete element method as well as the actual drilling. It can be proved that the proposed model can predict accurately block slip. Additionally, the effect of drilling fluid pressures on the wellbore stability is specially investigated. The results show that with the increasing of drilling fluid pressure, the degree of wellbore collapse may be more severe. Meanwhile, the analysis method of reasonable drilling fluid pressure for coal drilling is proposed based on the interval theory, combining the slip area of blocks. Finally, the parametric study is focused on investigating the effects of the internal friction angle of face cleats, the cohesion of face cleats and the tensile strength of butt cleats on the failure coefficients of blocks.