The linearization method in the inverse problem of determination of low-permeability zones in an oil-bearing layer

The possibility of determination of low-permeability zones in an oil-bearing layer from bore-pressure measurements is studied. The formulation of the problem on the pressure distribution in the layer is considered in the case when the pressure is invariable across the layer. The problem is reduced to an integral equation. The linearization of the inverse problem of determination of low-permeability zones is considered. An iterative technique is proposed for solution of this problem. The technique involves reduction of the inverse problem to an operator equation.     

Plastic flow of an axisymmetric layer under an end load

A solution is constructed for the problem of the stress state of a thin plastic axisymmetric layer subjected to an end load and compression between rigid slabs. The unknown parameters in the solution are determined from two integral equations. A numerical study is made of the range of compressive loads for which plastic flow occurs. The components of the stresses in the layer are also determined.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 106–110, 1988.     

The inverse problem of determining low-permeability zones in an oil-bearing layer

The article considers the possibility of determining low-permeability zones in an oil-bearing layer from bore-pressure measurements. The problem assumes a constant pressure distribution across the layer and is reduced to an integral equation. Numerical solutions are obtained for the case of an elliptical low-permeability zone; the sensitivity of the measured variables to the location and the size of the impermeable region is analyzed. __________ Translated from Prikladnaya Matematika i Informatika, No. 21, pp. 5–14, 2005.     

The three-dimensional problem of the axisymmetric deformation of an orthotropic spherical layer

A 3D problem of the deformation of an elastic orthotropic spherical layer that is subjected to normal pressure applied to its outer and inner surfaces is analyzed. Asymptotic first-order approximation solutions are obtained for a slightly orthotropic layer for which the elastic moduli in the meridional and circumferential directions have similar values. The solutions that are obtained are used for analyzing the scleral shell under intraocular pressure; however, they can also be used for solving the inverse problem of analyzing the stress–strain state of a human eye during intravitreal injections. The influence that the meridional and circumferential elastic moduli have on the magnitudes of changes in the relative layer thickness and in the length of the anteroposterior eye axis due to elevated intraocular pressure is studied.     

Equilibrium of an elastic layer weakened by plane cracks

The spatial problem of the elastic equilibrium of a layer in whose middle plane there is a system of cracks is considered. The cracks are maintained open under the action of a normal load applied to their edges. The layer faces are compressed between two rigid smooth foundations. The problem is reduced to solving an integral equation of the first kind. The asymptotic methods of “large and small λ” /1/ as well as the method of successive approximations and a variational method are used to construct the solutions of this equation for elliptically and rectangularly shaped cracks in different ranges of variation of the geomtrical parameters.     

Optimization of axisymmetric thermal displacements in a given section of an unbounded layer

We study the problem of optimal control of the distribution of axisymmetric vertical or radial thermal displacements in a given section of an unbounded layer. Using the method of the inverse problem of thermoelasticity we construct the solution of the control problem. For specific cases of heating of the layer we give a numerical analysis of the behavior of the optimal control.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 55–61.     

Homogeneous solutions of the dirichlet problem for an anisotropic layer

A complete system of homogeneous solutions of the Dirichlet problem for an anisotropic layer is constructed. These solutions represent series containing metaharmonic functions of a complex argument which depends on all three coordinates. The solution obtained can be used when considering boundary-value problems of potential theory for a piecewise-homogeneous layer.     

Electroelastic state of an inhomogeneous piezoceramic layer under symmetric loading

A procedure for solving three-dimensional mixed symmetric problems of electroelasticity theory for a piezoceramic layer with an inclusion, which is weakened with a through hole, is proposed. The boundary-value problem is reduced to a system of 12k (k = 1, 2, …) integrodifferential equations. Expressions for stresses characterizing the stress state of the inhomogeneous layer are found. Calculation results for characteristic stresses are presented.     

An effective asymptotic method in the axisymmetric frictionless contact problem for an elastic layer of finite thickness

A brief review of asymptotic methods to deal with frictionless unilateral contact problems for an elastic layer of finite thickness is presented. Under the assumption that the contact radius is small with respect to the layer thickness, an effective asymptotic method is suggested for solving the unilateral contact problem with a priori unknown contact radius. A specific feature of the method is that the construction of an asymptotic approximation is reduced to a linear algebraic system with respect to integral characteristics (polymoments) of the contact pressure. As an example, the sixth‐order asymptotic model has been written out. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonisothermal deformation of an elastoviscoplastic flat heavy layer

We give a solution to the thermoplasticity problem on the slide of a heavy layer from an inclined plane under heating. The reason for the creep effect is the development of a viscoplastic flow due to the dependence of the yield strength of the layer material on temperature. In the framework of large deformation theory we indicate a law of propagation for the elastoplastic boundary, calculate stresses, deformations, and deformation velocities both in the thermoplastic deformation region and the flow region.     

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.     

The propagation of non-stationary waves from a spherical cavity in an acoustic layer

The axisymmetric problem of the propagation of non-stationary waves from a spherical cavity in the plane of an infinite layer filled with an acoustic medium is considered. Using methods of incomplete separation of the variables in the space of Laplace transformation in time, the problem is reduced to an infinite system of linear algebraic equations, the solution of which is investigated in the form of series in exponential functions.     

Boundary-value problem in an infinite layer

We establish necessary and sufficient conditions for a nonlocal two-point boundary-value problem in an infinite layer for the equation $$\frac{{\partial ^2 u(x,t)}}{{\partial t^2 }} + P\left( {\frac{\partial }{{\partial x}}} \right)\frac{{\partial u(x + h_1 ,t)}}{{\partial t}} + Q\left( {\frac{\partial }{{\partial x}}} \right)u(x + h_2 ,t) = 0,$$ whereP(s) andQ(s) are polynomials ins∈? ^m with constant coefficients, to have infinite type and be degenerate.     

Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation

In this paper, we investigate the large-time behavior of solutions to an outflow problem for compressible Navier-Stokes equations. In 2003, Kawashima, Nishibata and Zhu [S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003) 483-500] showed there exists a boundary layer (i.e., stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation. In the present paper, we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation. The proofs are given by an elementary energy method.     

A linearised singularly perturbed convection–diffusion problem with an interior layer

A linear time dependent singularly perturbed convection–diffusion problem is examined. The convective coefficient contains an interior layer (with a hyperbolic tangent profile), which in turn induces an interior layer in the solution. A numerical method consisting of a monotone finite difference operator and a piecewise-uniform Shishkin mesh is constructed and analysed. Neglecting logarithmic factors, first order parameter uniform convergence is established.     

Simulation of current layer dynamics in the magnetogasdynamic interaction with an argon flow

A nonstationary three-dimensional magnetohydrodynamic (MHD) model is used to numerically simulate the formation of a current layer interacting with a transverse magnetic field in a supersonic argon flow. The structural features of the current layer and the characteristics of the process are analyzed at various intensities of the MHD interaction. The problem is solved using the MacCormack method with splitting in spatial coordinates and flux-corrected transport.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

Magnetoelastic vibrations of an electrically conductive elastic layer in a longitudinal magnetic field

Based on the exact solution of a coupled problem of the magnetoelastic vibrations of an elastic electrically conductive layer (for a finite value of its electrical conductance) in a longitudinal magnetic field, we have established that, in the case of such vibrations of a thin plate, one can obtain correct results proceeding from Kirchhoff’s hypothesis and the hypothesis of magnetoelasticity of thin bodies.     

The influence of an external magnetic field on the dynamic stress of an elastic conducting one-sided layer with a longitudinal shear crack

We study the interaction of a magnetoelastic shear wave with a curvilinear tunnel crack in an ideally conducting diamagnetic (resp. paramagnetic) one-sided (resp. two-sided) layer in the presence of an external static magnetic field. The bases of the one-sided layer are free of mechanical load, and the rim of the face is clamped or free. The corresponding linearized boundary-value problem of magnetoelasticity is reduced to a singular integrodifferential equation with subsequent implementation on a computer. We give numerical results that characterize the influence of the size of the preliminary magnetic field, the frequencies of the load, the curvature, and the orientation of the crack on the stress intensity factor. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 96–102.     

Asymptotic solution of the axisymmetric contact problem for an elastic layer of incompressible material

The solution of the axisymmetric contact problem for an elastic layer made of incompressible material and clamped along the base is constructed by regular and singular asymptotic methods.