Pressure of an elastic half space on a rigid base with rectangular hole in the case of a liquid bridge between them

A model of contact between an elastic half space and a rigid base with a shallow surface rectangular hole is proposed. The hole contains an incompressible liquid and gas. The liquid occupies the middle part of the hole and forms a capillary bridge between the opposite surfaces. The remaining volume of the hole is filled with gas under a constant pressure. The liquid completely wets the surfaces of the bodies. The pressure drop at the liquid–gas interface caused by the surface tension is defined by the Laplace formula. The corresponding plane contact problem for the elastic half space is essentially nonlinear because the pressure of the liquid and the length of the capillary in the contact-boundary conditions are not known in advance and depend on the external load. The problem is reduced to a system of three equations (a singular integral equation for the function of height of the hole and two transcendental equations for the length of the capillary and the height of the meniscus). An analytic-numerical procedure for the solution of these equations is proposed. Dependences of the length of the capillary and the pressure drop at the liquid–gas interface on the external load, volume of liquid, and its surface tension are analyzed. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 150–156, January–March, 2008.     

Limiting state of a multilayered nonlinearly elastic long cylindrical shell under the action of nonuniform external pressure

The buckling of a long multilayered nonlinearly elastic shell made of different materials and subject to the action of external pressure is investigated. The load is not hydrostatic and greatly varies in value and direction. Neglecting the effect of end fastening of the shell, the problem is reduced to an analysis of the loss of load-carrying ability of a ring of unit width separated from the shell. The solution is based on a variational method of mixed type formulated for heterogeneous nonlinearly elastic bodies, taking into account the geometrical nonlinearity, in a combination with the Rayleigh–Ritz method. The initial analysis is reduced to solving the Cauchy problem for a nonlinear ordinary differential equation resolved for the derivative. Numerically, using the Runge–Kutta method, the effect of the number of layers and of the parameter of nonuniformity of the external pressure on the critical buckling force is revealed. The urgency and importance of the problem are connected with the research of reserves in the saving of materials with a simultaneous possibility of increasing the load-carrying ability of a structure.     

A dynamical (time-harmonic) axisymmetric stress field in a finitely prestretched multilayered slab on a rigid foundation

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the dynamical (time-harmonic) axisymmetric stress field in a finitely prestretched multilayered slab resting on a rigid foundation is studied. It is assumed that the slab consists of two-layer packets. The elasticity of layer materials is described by the Treloar potential. It is assumed that the material of the lower layer in the packets is more rigid than that of the upper one. Numerical results are presented for the cases where the number of layers (packets) in the slab is 2 (1), 4 (2), or 6 (3). These results concern the normal stresses acting on the interface between the layers of the first, upper packet and on the interface between the first and second packets. The influence of the number, prestretch level, and thickness of the layers on relation ships between the stresses and the frequency of the external force is analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 667–680, September–October, 2006.     

A mathematical model of mechanical contact of bodies across a thin inhomogeneous layer

We obtain the generalized conditions of mechanical contact of bodies that model a thin inhomogeneous isotropic layer situated between them. The technique is based on the application of averaging of the equilibrium equations for the layer and the assumption of a cubic distribution of displacements over its thickness. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 107–109.     

Near-surface buckling in stability of a system consisting of a moderately rigid substrate, a viscoelastic bond layer, and an elastic covering layer

Within the frame work of the three-dimensional linearized theory of stability of deformable bodies (TLTSDB), the near-surface buckling instability of a system consisting of a half-plane (substrate), a viscoelastic bond layer, and an elastic covering layer is suggested. The equations of the TLTSDB are obtained from the three-dimensional geometrically non linear equations of viscoelasticity theory by using the boundary-form perturbation technique. By employing the Laplace transform, a method for solving the problem is developed. It is supposed that the covering layer has an insignificant initial imperfection. The stability of the system is considered lost if the imperfection starts to increase and grows indefinitely. Numerical results for the critical compressive force and the critical time are presented. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 4, pp. 517–530, July–August, 2006.     

Heat transfer in sliding of a plane-parallel layer over a base

An analytic solution of the thermal problem of friction for a plane-parallel layer–base tribosystem under conditions of incomplete thermal contact between contacting bodies is obtained. Asymptotics of the obtained solution for small and large values of time are determined. For the materials of a cermet layer–iron base friction pair, we investigate the influence of the thermal conductivity coefficient of the contact on the temperature distribution and intensity of a heat fluxes.     

Local near-surface buckling of a system consisting of an elastic (viscoelastic) substrate,a viscoelastic (elastic) bond layer,and an elastic (viscoelastic) covering layer

Within the framework of a piecewise homogenous body model and with the use of a three-dimensional linearized theory of stability (TLTS), the local near-surface buckling of a material system consisting of a viscoelastic (elastic) half-plane, an elastic (viscoelastic) bond layer, and a viscoelastic (elastic) covering layer is investigated. A plane-strain state is considered, and it is assumed that the near-surface buckling instability is caused by the evolution of a local initial curving (imperfection) of the elastic layer with time or with an external compressive force at fixed instants of time. The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary-form perturbation technique. A method for solving the problems considered by employing the Laplace and Fourier transformations is developed. It is supposed that the aforementioned elastic layer has an insignificant initial local imperfection, and the stability is lost if this imperfection starts to grow infinitely. Numerical results on the critical compressive force and the critical time are presented. The influence of rheological parameters of the viscoelastic materials on the critical time is investigated. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operator. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 771–788, November–December, 2007.     

Wave propagation in an elastic layer with voids located in a liquid

We obtain the dispersion equations that describe the propagation of waves in an elastic layer with voids locted between two liquid half-spaces. We study certain limiting cases corresponding to the absence of voids or liquid. We obtain the roots of the dispersion equations for both dissipative and nondissipative systems. It is shown that the relation of the real part of the phase velocity to the wave number in a dissipative system is qualitatively similar to the corresponding relation for the real value of the phase velocity in the case when dissipation is absent. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 90–96.     

Effects of the magnetic field on direct contact melting transport processes during rotation

Contact melting heat transfer occurs via relative motion between the heating source and a phase change material (PCM) during melting in various applications. In this study, we investigated the physics of the close contact melting process generated by rotation and when subjected to an applied magnetic field. We transformed the physical model comprising the three-dimensional mass, momentum, and energy equations of the liquid melt layer in the cylindrical coordinate system, including the effects of the Lorentz forces and coupled with an interfacial energy jump condition, into a set of nonlinear similarity equations. Various characteristic dimensionless variables were identified, including an external force parameter σ, which defines the relationship between the external load on the PCM and the centrifugal force due to rotation, and a magnetic field parameter M. Numerical results were obtained and we systematically studied and interpreted the effects of various dimensionless variables on the contact melting and heat transfer processes during rotation, including the structures of the flow and thermal fields, melt layer thickness, and the melting and heat transfer rates. In particular, our results demonstrate that the melting and heat transfer rates increase while the liquid melt film becomes thinner as the external force parameter σ increases. By contrast, an increase in the magnetic field parameter M decreases the melting and heat transfer rates, while yielding relatively thicker melt layers.     

Propagation of torsional waves in a prestretched compound hollow circular cylinder

The propagation of torsional waves in a prestressed compound (bi-layered) hollow circular cylinder is in vestigated within the frame work of a piecewise homogeneous body model, with the use of a three-dimensional linerized theory of elastic waves in initially stressed bodies. The elasticity relations for components of the compound cylinder are obtained from the Murnaghan potential. Numerical investigations are performed for bronze and steel. According to the results obtained, the effect of variations in the geometric (the ratio between the thickness of the cylinder and its inner radius) and mechanical parameters on the dispersion curves are analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 1, pp. 103–116, January–February, 2008.     

The development of slugging in two-layer hydraulic flows

In this paper, we develop a hydraulic theory to describe theoccurrence and structure of slugging in a confined two-layergas–liquid flow generated by prescribed, constant, upstreamvolumetric flow rates in each layer. A linearized theory forthe uniform flow is established, after which we use bifurcationtheory to study fully non-linear periodic travelling wave structures.We find that a two-parameter family of such travelling wavesolutions exists. Under given conditions, the volumetric flowrate constraint provides a relation between these two parameters.To select a unique periodic travelling wave solution, we requirea further relation. We first investigate the conjecture thatthe periodic travelling wave solution selected in the initialvalue problem has the same wavelength as the linearly most temporallyunstable mode. To do this, we solve the initial value problemnumerically on a periodic domain. We find that the separationof the liquid slugs that form is much longer than the wavelengthof the most unstable temporal mode. We then develop a differentconjecture based on the convective instability of the long ‘tails’of the available periodic travelling wave solutions, which leadsto a better understanding of the wavelength selection process.     

Determination of the temperatures and heat flows for a conical slide support taking account of frictional heating

We consider an axisymmetric problem of heat conduction taking account of frictional heating in a conetorus pair that models the functioning of a conical support. The bodies are pressed together and are rotating about a common axis. Heat is generated in the region of contact of the bodies due to frictional forces. Outside the region of contact there is heat exchange with the surrounding medium. The thermal contact between the two bodies is nonideal. The problem is reduced to a system of integral equations whose solution is constructed by the method of successive approximations. We give the results of numerical studies of the temperature distribution and heat flows from the geometric and thermophysical parameters of the body. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 19–27.     

The thermally stressed state of a thickwalled hollow composite cylinder

We propose an approach to the determination of the temperature, stress, and displacement, fields in laminated hollow composite cylinders. The cylinders are studied in three-dimensional formulation; the material of each lamina has the properties of thermal and elastic orthotropy. We study the influence of the heat flow passing through the end surfaces and nonuniform heating of the cylinder to its thermally stressed state. We exhibit peculiarities in the stress and displacement distributions caused both by anisotropy of the mechanical and thermophysical properties and by the nature of the thermal actions applied. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 14–18     

The limiting state of a multilayer nonlinearly elastic eccentric ring

The limiting state of a multilayer eccentric ring made of a nonlinearly elastic material and subjected to a uniform external pressure is investigated. The topicality and importance of the problem are connected with the search for reserves of savings in materials, with a simultaneous in crease in the load-carrying capacity of structures. Since rings often must have walls of varying thickness, their critical buckling force is determined as a function of a parameter considering this fact. In solving the problem, the geometric nonlinearity is also taken into account. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 761–770, November–December, 2007.     

A theorem of instability by linear approximation for a one-dimensional non-linearly elastic body

On the basis of and in a development of the ideas and results of A.A. Movchan (Sr.), that extend to continuous bodies the definitions and main fundamental theorems of Lyapunov on stability and instability, a criterion for instability of the equilibrium position of a one-dimensional non-linearly elastic body subject to potential external forces is established. For the specified simplest type of continuous elastic system (which possesses, however, a number of fundamental properties of continuous elastic systems including unboundedness of the operator of linear approximation and discreteness of its spectrum) a theorem of instability by linear approximation is stated and proved. The method of proof is a version of Persidskii's sector method.     

Postbuckling of a shear-deformable anisotropic laminated cylindrical shell under external pressure in thermal environments

A postbuckling analysis is presented for a shear-deformable anisotropic laminated cylindrical shell of finite length subjected to external pressure in thermal environments. The material properties are expressed as linear functions of temperature. The governing equations are based on Reddy’s higher-order shear-deformation shell theory with the von Karman-Donnell-type kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. The boundary-layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling region, and the initial geometric imperfections of the shell, is extended to the case of shear-deformable anisotropic laminated cylindrical shells under lateral or hydrostatic pressure in thermal environments. The singular perturbation technique is employed to determine the interactive buckling loads and postbuckling equilibrium paths. The results obtained show that the variation in temperature, layer setting, and the geometric parameters of such shells have a significant influence on their buckling load and postbuckling behavior. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 789–822, November–December, 2007.     

Reflected-refracted and head sh waves formed on a thin layer that is in welded contact with two elastic half-spaces

The reflection and refraction of SH waves by an elastic layer sandwiched between two elastic half-spaces are studied by using the numerical simulation on the basis of contour integration in the complex plane of the horizontal component of the slowness vector. The propagation of body, channel, head, and screened body waves are simulated in time and spectral domains. The wave fields associated with the propagation in the layer have strong attenuation, provided that the wave length is smaller than the value of the thickness of the layer. The stationary wave field of such waves is of resonance nature. Moreover, the maximum of the modulus of the spectral function is shifted to higher frequences as the epicentric distance increases. Thus, the attenuation of such waves depends on spectral characteristics of a source-receiver system. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1997, pp. 91–120. Translated by T. N. Surkova.     

Heat conduction of shells reinforced by fibers with constant and variable cross sections

We propose a model for heat conduction of a spatially reinforced medium and present its generalization to the case of a polyreinforced layer. We consider the heat-conduction equations for fibrous shells and construct a procedure for reduction of a three-dimensional problem of heat conduction to a two-dimensional one. Analytic solutions of a stationary problem of heat conduction are found for thin conic shells of revolution for various structures of reinforcement, and a graphical comparison of the corresponding results is performed. We study one of the approaches to rational reinforcement of thin shells, according to which the thermal “transparency” of a shell in the transverse direction is taken as a criterion of rational design. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 132–150, April–June, 1998.     

Temperature field of a layer of a fibrous composite material under conditions of external thermal radiation

On the basis of an expression for the scattering phase function of an individual fiber, relations modeling radiative properties of fibrous composites are obtained. We calculate the temperature state of a layer of such a material under condition of external thermal radiation. Dependences of temperature distributions on the volume fraction of fibers in the composite, their reflectance, and size are investigated. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 185–193, January–March, 2008.