Continuation of some solutions in potential theory and their applications to elastic contact and crack problems

The following mixed boundary value problem is considered: arbitrary tangential displacements are prescribed inside a circle, while the tangential and normal stresses outside the circle are zero. In this case, a direct and simple formula is derived for the tangential displacements outside the circle in terms of prescribed displacements inside, thus making the tangential displacement known all over the boundary. The original problem is no longer mixed, and the complete solution becomes readily available. Another solution for the case, when arbitrary tangential displacements are prescribed outside a circle, is derived in a similar manner. The reciprocal theorem is used to derive the continuation formulae for the tangential stresses inside and outside a circle. Application of these results to contact and crack problems is demonstrated.     

A refined model for three-point bending of sandwich panels 2. Transverse stresses

According to the relationships derived in [1], transverse normal and tangential stresses in a sandwich panel have been analyzed. Asymptotic formulas for the stress concentration area in the vicinity of point forces are derived. Analytical estimates of a normal stress at the central and end sections of the panel are deduced. The Saint-Venant effect of the degeneration of a panel of finite length into an infinite strip is studied. For the estimation of the concentration of the transverse tangential stress, the possibility of a superposition of the solution of the slippage problem of the face layers and the classical solution allowing for shear is substantiated. It is shown that the local Reissner-type effects are specified by reducing the concentration of the tangential stress in the face layers along the longitudinal coordinate and transition to the steady tangential stress state in the filler layer. The concentration coefficients of the tangential stress are derived as functions of the dimensional parameters of the panel section.Institute of Polymer Mechanics. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 66–93, January–February, 1998.     

Special relativistic effects revealed in the Riemann problem for three-dimensional relativistic Euler equations

We consider the Riemann problem of three-dimensional relativistic Euler equations with two discontinuous initial states separated by a planar hypersurface. Based on the detailed analysis on the Riemann solutions, special relativistic effects are revealed, which are the variations of limiting relative normal velocities and intermediate states and thus the smooth transition of wave patterns when the tangential velocities in the initial states are suitably varied. While in the corresponding non-relativistic fluid, these special relativistic effects will not occur.     

The stress state of planar surface of a nanometer-sized elastic body under periodic loading

The paper is concerned with the model of an elastic body in the form of a half-plane whose boundary is subjected to periodic loading. It is assumed that there exists an additional surface stress, which is characteristic of nanometer-sized bodies and which obeys the laws of surface elasticity theory. With the use of the boundary properties of analytical functions and the Goursat-Kolosov complex potentials, the boundary value problem in its general setting with an arbitrary load is reduced to a hypersingular integral equation with respect to the derivative of the surface stress. For a periodic load, the solution of this equation is obtained in the form of a Fourier series. The effect of the surface stress upon the stress state of the boundary of the half-plane is examined with independent action of periodically distributed tangential and normal loads. In particular, the size effect was discovered, which is manifested in the dependence of stresses versus the period of loading within several dozens of nanometers. Normal loads are shown to be responsible for tangential stresses on the boundary, which are zero in the classical solution.     

Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.     

Retracted: Unsteady flow and heat transfer on a rotating disk in a porous medium

This article has been retracted. See retraction notice DOI: 10.1002/mma.850 . An unsteady flow and heat transfer in a porous medium of a viscous incompressible fluid over a rotating disk in an otherwise ambient fluid are studied. The unsteadiness in the flow field is caused by the angular velocity of the disk which varies with time. The new self‐similar solution of the Navier–Stokes and energy equations is obtained numerically. The solution obtained here is not only the solution of the Navier–Stokes equations, but also of the boundary layer equations. Also, for a simple scaling factor, it represents the solution of the flow and heat transfer in the forward stagnation‐point region of a rotating sphere or over a rotating cone. The asymptotic behaviour of the solution for a large porosity or for a large independent variable is also examined. The surface shear stresses in the radial and tangential directions and the surface heat transfer increase as the acceleration parameter increases. Also, the surface shear stress in the radial direction and the surface heat transfer decrease with increasing porosity, but the surface shear stress in the tangential direction increases. Copyright © 2006 John Wiley & Sons, Ltd.     

The equilibrium shapes of non-uniform vortices in the ocean

The problem of the equilibrium shapes of rotating vortices in a stratified ocean, which is in a statically stable state, is considered within the framework of the model of an ideal incompressible fluid. An equilibrium shape is a surface on which the pressures in the vortex and in the ocean are equal, in which case, on this surface the normal components of the velocities of the media are equal to zero, but a discontinuity of the tangential components is allowed. The case of stratification of the media along the local vertical is considered. The external medium (the ocean) may consist of several layers, differing sharply in density.     

Modeling of three dimensional thermocapillary flows with evaporation at the interface based on the solutions of a special type of the convection equations

Theoretical and numerical study of the convection processes, which are accompanied by evaporation/condensation, in the framework of new non-standard problem is largely motivated by new physical experiments. One of the principal questions is to understand the character and to evaluate the degree of influence of particular factors or their combined action on the structure of the joint flows of liquid and gas-vapor mixture. The flow topology is determined by four main mechanisms: natural and thermocapillary convection, tangential stresses and mass transfer due to evaporation at the interface. The mathematical modeling of the fluid flows in an infinite channel with a rectangular cross section is carried out on the basis of the solution of a special type of the convection equations. The effects of thermodiffusion and diffusive thermal conductivity in the gas phase and evaporation at the thermocapillary interface are taken into consideration. Numerical investigations are performed for the liquid – gas (ethanol – nitrogen) system under normal and low gravity. The fluid flows are characterized as translational and progressively rotational motions and can be realized in various forms.     

On perturbations associated with the creation of lift acting on a body in a transonic stream of a dissipative gas PMM vol. 31, no. 6, 1967, pp. 1035–1049

The flow pattern of a viscous imcompressible fluid past a finite body is well known; an approximate solution of the related problem can, for example, be found in the book by Landau and Lifshits [1]. Finn [2] made a rigorous and exhaustive study of plane-parallel flows. No fundamental difficulties arise in passing from the motion of an incompressible fluid to a transonic flow of a compressible gas, however the velocity field is different, when the velocity of particles becomes critical at infinity.

The pattern of a sonic flow past a body of circular cross-section was investigated in paper [3]. This paper deals with perturbations associated with the creation of lift acting on an arbitrary body in a three-dimensional flow. When solving this problem it is necessary to consider not only the external stream, but also the laminar vortex trail because of the velocity vector transverse components becoming infinitely great, if functions defining these are formally extended into the trail area. This difficulty arises in investigations of three-dimensional flows only. The solution defining perturbation damping in an axisymmetric sonic stream of a dissipative gas has in its first approximation one singular point only, and does not contain any other singularities along the axis of symmetry [3].

The external stream pattern is essentially formed by the action of normal viscous stresses and the longitudinal component of the heat flux vector, while the distribution of gas parameters in the laminar trail is defined by tangential stresses. The conjunction of solutions valid for each of these areas makes the closure of the problem, and the determination of all necessary parameters possible.     

Resultant forces and moments in mixed-mixed problems of the theory of elasticity

A problem is called mixed-mixed, when both normal and tangential displacements are prescribed on a part of the boundary, while the normal and tangential stresses are prescribed at the rest of the boundary. Exact closed form expressions have been derived for the resultant normal and tangential forces, tilting moment and torque, directly through the prescribed displacements, thus eliminating the need for determination of stresses. The problem solved treats a transversely isotropic elastic half-space, with arbitrary normal and tangential displacements prescribed inside a circle, and the rest of the boundary being stress-free. The interaction between an arbitrary force inside the half-space and a bonded punch is considered as an example. No similar result has ever been reported, even in the case of isotropy.     

Deformation of a composite elastic plane weakened by a periodic system of the arbitrarily loaded slits

The paper deals with the problems of periodic system of cuts distributed along the boundary of a bond connecting two elastic half-planes and acted upon by nonperiodic loads. In one problem it is assumed that the cuts are open, with normal and tangential stresses applied to their edges, while in another problem the edges touch each other and are loaded by tangential stresses. The method of solution is based on the simultaneous use of the discrete Fourier transformation and the theory of boundary value problems for automorphous analytic functions. The solutions are otained in quadratures. Other classes of problems to which the proposed methods can be applied, are described.

Generally speaking, in the case of irregular loads, the solution is usually based on the theory of representation of the symmetry groups /1,2/, and in the case of certain types of symmetry, particularly the translational, on the discrete Fourier transforms /3– 6/. However the objects of transformation may be different in one and the same problem, and their choice affects significantly the solvability of the boundary value problem for the transformed quantities in the cell of periods. Below two problems of the theory of cracks are solved in quadratures to illustrate the effective simultaneous use of the discrete Fourier transformation and the Muskhelishvili method.     

Flow of a micropolar fluid through a circular cylinder subject to longitudinal and torsional oscillations

The internal flow of a micropolar fluid inside a circular cylinder which is subject to longitudinal and torsional oscillations is investigated. Analytical expressions of the fluid velocity and micro-rotation are obtained. Explicit expressions of the shear stresses and drag force acting at the wall of the cylinder are derived as well. A numerical analysis followed to examine the effect of the micropolar fluid on the two components of the velocity field through graphical curves. In addition, the magnitude of the tangential drag is computed and compared with the case of a classical fluid.     

Flow of a Maxwell fluid between two side walls induced by a constantly accelerating plate

The unsteady flow of a Maxwell fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate is studied. Exact solutions for the velocity field are established by means of the Fourier sine transforms. The adequate tangential stresses are also determined. The similar solutions for a Newtonian fluid are obtained as limiting cases of our solutions. In the absence of the side walls, the similar solutions for the unsteady flow over an infinite flat plate are recovered. Finally, for comparison, the velocity field in the middle of the channel and the shear stresses at the bottom wall and on the side walls are plotted for different values of the material constants.      

Problem of the Motion of an Elastic Medium Formed at the Solidification Front

The following self-similar problem is considered. At the initial instant of time, a phase transformation front starts moving at constant velocity from a certain plane (which will be called a wall or a piston, depending on whether it is assumed to be fixed or movable); at this front, an elastic medium is formed as a result of solidification from a medium without tangential stresses. On the wall, boundary conditions are defined for the components of velocity, stress, or strain. Behind the solidification front, plane nonlinear elastic waves can propagate in the medium formed, provided that the velocities of these waves are less than the velocity of the front. The medium formed is assumed to be incompressible, weakly nonlinear, and with low anisotropy. Under these assumptions, the solution of the self-similar problem is described qualitatively for arbitrary parameters appearing in the statement of the problem. The study is based on the authors’ previous investigation of solidification fronts whose structure is described by the Kelvin–Voigt model of a viscoelastic medium.     

External circular crack under arbitrary shear loading

An exact closed form solution in terms of elementary functions has been obtained to the governing integral equation of an external circular crack in a transversely isotropic elastic body. The crack is subjected to arbitrary tangential loading applied antisymmetrically to its faces. The recently discovered method of continuity solutions was used here. The solution to the governing integral equation gives the direct relationship between the tangential displacements of the crack faces and the applied loading. Now a complete solution to the problem, with formulae for the field of all stresses and displacements, is possible.     

Vibration of a stamp partially adhering to an elastic medium : PMM, vol. 42, no. 6, 1978, pp. 1085–1092

There is examined the problem of vibration of a stamp of arbitrary planform occupying a space Ω and vibrating harmonically in an elastic medium with plane boundaries. It is assumed that the elastic medium is a packet of layers with parallel boundaries, at rest in the stiff or elastic half-space. Contact of three kinds is realized under the stamp: rigid adhesion in the domain Ω₁, friction-free contact in domain Ω₂, there are no tangential contact stresses, and “film” contact without normal force in domain Ω₃ (there are no normal contact stresses, only tangential stresses are present.). It is assumed that the boundaries of all the domains have twice continuously differentiable curvature and Ω = Ω₁ Ω₂ Ω₃.

The problem under consideration assumes the presence of a static load pressing the stamp to the layer and hindering the formation of a separation zone. Moreover, a dynamic load, harmonic in time, acts on the stamp causing dynamical stresses which are of the greatest interest since the solution of the static problem is obtained as a particular case of the dynamic problem for ω = 0 (ω is the frequency of vibration). The general solution is constructed in the form of a sum of static and dynamic solutions.

A uniqueness theorem is established for the integral equation of the problem mentioned and for the case of axisymmetric vibration of a circular stamp partially coupled rigidly to the layer, partially making friction-free contact, the problem is reduced to an effectively solvable system of integral equations of the second kind, which reduce easily to a Fredholm system.

These results are an extension of the method elucidated in [1], where by the approach in [1] must be altered qualitatively to obtain them.     

Micropolar fluid flow through the membrane composed of impermeable cylindrical particles coated by porous layer under the effect of magnetic field

In the present work, the magnetohydrodynamic flow of a micropolar fluid through the membrane composed of impermeable cylindrical particles coated by porous layer is considered. The flow of a fluid is taken parallel to an axis of cylinder and a uniform magnetic field is applied in transverse direction of the flow. The problem is solved by using the cell model technique for the flow through assemblage of cylindrical particles. The solution of the problem has been obtained by using no-slip condition, continuity of velocity and stresses at interfaces along with Happle's no-couple stress condition as the boundary conditions. The expressions for the linear velocity, micro-rotational velocity, flow rate and hydrodynamic permeability of the membrane are achieved in this work. The obtained solution for velocities is used to plot the graph against various transport parameters such as, Hartmann number, coupling parameter, porosity, scaling parameter etc. The effect of these transport parameters on the flow velocity, micro-rotational velocity, and the hydrodynamic permeability of the membrane have been presented and discussed in this work.     

Geometric surface evolution with tangential contribution

Surface processing tools based on Partial Differential Equations (PDEs) are useful in a variety of applications in computer graphics, digital animation, computer aided modelling, and computer vision. In this work, we deal with computational issues arising from the discretization of geometric PDE models for the evolution of surfaces, considering both normal and tangential velocities. The evolution of the surface is formulated in a Lagrangian framework. We propose several strategies for tangential velocities, yielding uniform redistribution of mesh points along the evolving family of surfaces, preventing computational instabilities and increasing the mesh regularity. Numerical schemes based on finite co-volume approximation in space will be considered. Finally, we describe how this framework may be employed in applications such as mesh regularization, morphing, and features preserving surface smoothing.     

stresses and reversible deformations during shear flow of polymer solutions and melts

The influence of the accumulated elastic energy on the relationships between tangential stresses, the first difference between the normal stresses and the reversible deformations during isothermal shearing steady-state flow of polymer solutions and melts, is analyzed. It is shown that the reversible deformation in the non-Newtonian flow region is related to the tangential and normal stresses by Lodge's formula, if the thixotropic disruption of the structural flow units is accompanied by the dissipation of the elastic energy accumulated in them; the conservation of the elastic energy accumulated during the flow causes exceeding of the reversible deformation values as compared with the values calculated by Lodge's formula.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 886–895, September–October, 1972.     

Distribution of self-balanced stresses in a stratified composite material with warped structures

We use the model of a linear, piecewise homogeneous elastic body to study the distribution of selfbalanced normal and tangential stresses for a horizontal deformation of a stratified composite material with warped structures which is under the action at infinity of uniformly distributed normal stresses directed along the stratification.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 83–89, 1987.