Asymptotic solution of the problem of the action of a stamp on an elastic layer lying on the surface of a compressible fluid of infinite depth

This paper considers a two-dimensional linear unsteady problem of rigid-stamp indentation on an elastic layer of finite thickness lying on the surface of a compressible fluid of infinite depth. The Lamé equations holds for the elastic layer, and the wave equation for the fluid velocity potential. Using the Laplace and Fourier transforms, the problem is reduced to determining the contact stresses under the stamp from the solution of an integral equation of the first kind, whose kernel has a logarithmic singularity. An asymptotic solution of the problem is constructed for large times of interaction. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 131–142, March–April, 2008.     

Laminated shells with debonding between laminas in temperature field

A heat-conduction problem is formulated for laminated plates and shells with a heat-conducting layer and debonding between laminas. The approach consists in analyzing how the layer thickness changes in the process of debonding of laminas and deformation of plates and shells. The three-dimensional thermoelastic and heat-conduction equations are expanded into polynomial Legendre series in thickness. The first-order, Timoshenko’s, and Kirchhoff-Love equations are examined. A numerical example of laminated shells with a heat-conducting layer is considered Published in Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 135–141, July 2006.     

Effect of a thin coating on the pressure distribution in contact problems involving frictional heat generation

A thermoelastic problem for a layer of finite thickness one of whose surfaces is subjected to the action of normal pressure and heat flux is studied. A relationship among vertical displacements of the surface of the layer, the surface temperature, and the disturbing factors is obtained. Corresponding relations are obtained for a layer of small thickness. An axisymmetric contact problem for a rigid heat-conducting base whose surface is coated with a thin elastic layer is studied as an example. Franko L'vov State University, L'vov 290602. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 1, pp. 110–118, January–February, 1998.     

Mode III fracture problem of a cracked FGPM surface layer bonded to a cracked FGPM substrate

This work deals with the mode III fracture problem of a cracked functionally graded piezoelectric surface layer bonded to a cracked functionally graded piezoelectric substrate. The cracks are normal to the interface and the electro-elastic material properties are assumed to be varied along the crack direction. Potential and flux types of boundary condition are assigned on the edge of the surface layer. The problem under the assumptions of impermeable and permeable cracks can be formulated to the standard singular integral equations, which are solved by using the Gauss–Chebyshev technique. The effects of the boundary conditions, the material properties and crack interaction on the stress and electric displacement intensity factors are discussed.     

Dynamics of a vapor shell around a heated particle in a liquid

The dynamics and heat and mass exchange of a vapor bubble containing a heated particle is studied in relation to the problem of vapor explosions. It is shown that the process involves two stages: dynamic stage and thermal stage. The dynamic stage is characterized by pressure fluctuations and a rapid increase in the thickness of the vapor layer. It is shown that the simplifying assumptions of the constancy of assumptions of constant heat conductivity of the vapor and linear temperature profile in the vapor layer lead to qualitatively incorrect results. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 69–78, July–August, 2007     

Influence of intense surface cooling on hypersonic viscous flow past a delta wing

The influence of intense surface cooling on the parameters of a laminar boundary layer flow on a thin delta wing in a hypersonic viscous perfect-gas stream is studied for the strong viscous-inviscid interaction regime. The effect of the power-law shape of the wing cross-section and the wing thickness to boundary-layer displacement thickness ratio on the local and total aerodynamic characteristics is numerically investigated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 57–64, July–August, 1998.     

Two-dimensional problem of the impact of a vertical wall on a layer of a partially aerated liquid

The two-dimensional unsteady problem of the impact of a vertical wall on a layer of a liquid which is mixed with air near the wall and does not contain air bubbles away from the wall is solved in a linear approximation. The gas-liquid mixture is modeled by a homogeneous, ideal, and weakly compressible medium with a reduced sound velocity dependent on the air concentration in the gas-liquid mixture. Outside the gas-liquid layer, the liquid is considered ideal and incompressible. During the initial stage of the impact, the liquid flow and the hydrodynamic pressure are determined using the linear theory of the potential motion of an inhomogeneous liquid. The dependence of the amplitude of the impact pressure along the wall on the air concentration in the gas-liquid layer and on the thickness of this layer is investigated. For a small relative thickness of the layer, the thin-layer approximation is used. It is shown that the solution of the original problem tends to the approximate solution as the thickness of the layer decreases. It is shown that the presence of the gas-liquid layer leads to wall pressure oscillations. Estimates are obtained for the pressure amplitude and the oscillation period. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 34–46, September–October, 2006.     

One formulation of the problem of elastoplastic separation

The problem of the beginning of motion of a cut in a plane under symmetric external loading is considered. The material lying on the cut continuation forms a layer (interaction layer). A transition to a plastic state within the layer is assumed to be possible. The behavior of the layer is described by an ideally elastoplastic model, and the plane outside the layer is assumed to be linearly elastic. A system of boundary integral equations for determining the stress-strain state is derived. Based on this system, a discrete model of separation of the layer material is constructed under the assumption of a constant stress-strain state in the element of the interaction layer. The distribution of stresses in the pre-fracture zone is determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 187–195, July–August, 2009     

Discrete-continuum model of symmetric separation of a material

A problem of the beginning of motion of a finite-width cut in a linearly elastic plane under the action of symmetric external loading is formulated. The material on the way of cut propagation forms a layer (interaction layer). The stress-strain state of the material is postulated to be homogeneous across this layer. A system of integral boundary equations is obtained for determining the stress-strain state. Based on this system of equations, a discrete model of separation of the layer material is constructed under the assumption of a constant stress-strain state in an element of the interaction layer. The stress distribution in the pre-fracture zone is determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 134–140, January–February, 2009.     

A mesomechanical fracture model for an orthotropic material with different tensile and compressive strengths

The paper addresses a fracture problem for an orthotropic cracked plate made of a material with different tensile and compressive strengths and subjected to biaxial loading. The problem is solved using a micromechanical fracture model proposed earlier by the authors. It is assumed that the fracture of the material in the fracture process zones at the crack front is described by the Gol’denblat–Kopnov failure criterion. Strength curves for an orthotropic cracked plate with different strength and fracture-toughness parameters are plotted     

Method of characteristics in electroelastic analysis of a layer subject to dynamic mechanical loading

The paper proposes an analytic procedure based on the method of characteristics to study the nonstationary thickness vibration of a piezoelectric layer polarized across the thickness and subjected to dynamic mechanical loading. The problem is solved for a suddenly applied harmonic mechanical load. The dynamic electroelastic state of the layer is analyzed Translated from Prikladnaya Mekhanika, Vol. 45, No. 1, pp. 82–89, January 2009.     

Reflection and refraction of a planar acoustic wave in an anisotropic inhomogeneous layer

The problem of reflection and refraction of a planar acoustic wave by an inhomogeneous elastic layer whose material possesses general-type anisotropy is considered. The equations of motion of the elastic layer are reduced to a system of ordinary differential equations. The boundary-value problem for this system is solved by two methods: by reduction to problems with initial conditions and by the method of power series. Analytical expressions that describe acoustic fields outside the layer are obtained. Calculation results of the transmission factor for transversely isotropic layers inhomogeneous in thickness are presented. Tula State University, Tula 300600. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 179–184, September–October, 1999.     

The Slow Erosion Limit in a Model of Granular Flow

We study a 2 × 2 system of balance laws that describes the evolution of a granular material (avalanche) flowing downhill. The original model was proposed by Hadeler and Kuttler (Granul Matter 2:9–18, 1999). The Cauchy problem for this system has been studied by the authors in recent papers (Amadori and Shen in Commun Partial Differ Equ 34:1003–1040, 2009; Shen in J Math Anal Appl 339:828–838, 2008). In this paper, we first consider an initial-boundary value problem. The boundary condition is given by the flow of the incoming material. For this problem we prove the global existence of BV solutions for a suitable class of data, with bounded but possibly large total variations. We then study the “slow erosion (or deposition) limit”. We show that, if the thickness of the moving layer remains small, then the profile of the standing layer depends only on the total mass of the avalanche flowing downhill, not on the time-law describing the rate at which the material slides down. More precisely, in the limit as the thickness of the moving layer tends to zero, the slope of the mountain is provided by an entropy solution to a scalar integro-differential conservation law.     

Transverse cracking of orthotropic composite laminates: a fracture mechanics approach

This research is concerned with the fracture mechanics of a laminated composite medium, which contains a central layer sandwiched by two outer layers. There is a periodic array of cracks in the central layer along the central axis of the medium. Fourier transform is used to reduce the problem to the solution of a system of dual integral equations, which are solved by the singular integral equation technique. Rigorous fracture mechanics analysis, which exactly satisfies all boundary conditions of the problem, is conducted. Numerical solutions for the crack tip field and the stress in the medium are obtained for various values such as crack length, crack spacing and layer thickness. Results are also given for the reduction of the equivalent Young’s modulus of the laminate due to multiple cracking. The cases of axial extension and residual temperature change of the composite medium are accounted for.     

Dynamic contact angle for etting of a surface by a van der waals fluid

We consider the creeping motion of a thin layer of a nonvolatile viscous fluid spreading due to capillary forces over a rigid surface covered by a thin homogeneous film (microfilm). The influence of van der Waals forces on the asymptotic slope of the free boundary of the layer is studied in the region of large thickness, where capillary forces dominate. A solution of the problem of the slope angle is obtained for the entire possible range of the microfilm thickness. In the limit of small thickness of the microfilm, this solution is in agreement with the well-known solution of the problem of the dynamics of wetting of a dry surface in the presence of a precursory film and van der Waals forces. The role of the condition at the end of the precursory film is studied. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences. Tyumen' 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika. Vol. 41, No. 4, pp. 101–105, July–August. 2000.     

Mathematical modeling of two problems of wave dynamics in heterogeneous media

Two problems of heterogeneous media mechanics are investigated in the paper. The first one, concerned with the shock wave/dust layer interaction, is solved within the framework of the equilibrium model of heterogeneous media mechanics. The second problem deals with the simulation of a Riemann problem for a stratified layer of solid particles.This paper is based on work that was presented at the 20th International Colloquium on the Dynamics of Explosions and Reactive Systems, Montreal, Canada, July 31–August 5, 2005     

Interaction of an external supersonic flow with the subsonic layer near a wavy wall

A solution of the problem of supersonic flow past a wavy wall with an adjacent subsonic layer is obtained. The solution is a generalization of the well-known solutions [1] of the linear problem of purely subsonic and purely supersonic flow past a wavy wall and goes over into these solutions in the limiting cases of infinite and zero wall-layer thickness, respectively. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 97–103, January–February, 1997.     

Effect of contact interaction of the crack faces for a crack under harmonic loading

This paper is concerned with dynamic problems in fracture mechanics for elastic solids having cracks with contacting faces. The contact problem for a penny-shaped crack with a nonzero initial opening under normally incident harmonic wave is solved by the method of boundary integral equations. The solutions are compared with those that neglect the contact interaction of the crack faces. Results are presented for different values of the initial crack opening Presented at the 6th International Conference on Modern Practice in Stress and Vibration Analysis (Bath, United Kingdom, September 5–7, 2006). Published in Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 125–131, July 2007.     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Kinetics of charge-exchange interaction of dense flows

The nonlinear problem of charge exchange between an ion flow and neutral particles is considered. An exact solution of the equations of charge-exchange interaction in plane geometry is found. Parameters determining the effectiveness of interpenetration of dense flows and the structure of the layer of intense interaction are obtained. Institute of Laser Physics, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 11–19, March–April, 2000.