Diffraction of a plane electromagnetic wave obliquely incident upon the edge of a wedge with a dielectric coating

The diffraction of a plane electromagnetic wave obliquely incident upon the edge of a coated wedge is considered. The generalized impedance boundary conditions (GIBC's) on the wedge's faces are used to simulate the effect of the coatings. To insure the well-posedness of the problem, special contact conditions (CC's) on the edge are additionally imposed. By using Sommerfeld integrals, the problem is reduced to a system of coupled functional equations, which is solved by the perturbation method. It is shown that, for a certain range of the angles of oblique incidence, the solution can be represented in the form of convergent series that are Neumann series for linear equations with contracting operators. Nonuniform asymptotics of the wave field for regions outside a neighborhood of the edge of the wedge are constructed. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 72–95. Translated by L. G. Vardapetyan and M. A. Lyalinov.     

Diffraction on a semi-infinite screen of a plane nonstationary wave the amplitude of which increases linearly along the front set

An exact analytic solution of the two-variables nonstationary problem of diffraction on an ideal half-infinite screen is obtained by the Smirnov-Sobolev method. The source of the field is an incident plane acoustic wave with a δ-function profile. The wave amplitude is a linear function increasing along the front set. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 138–152.     

Wedging of an elastic wedge by a rigid plate under conditions of contact with detaching

We consider a problem of wedging of an elastic wedge by a rigid plate along an edge crack that is located on the axis of symmetry of the wedge and reaches its vertex. The detachment of the crack faces from the surfaces of the plate is taken into account. Using the Wiener–Hopf method, we obtain an analytic solution of the problem. The size of the detachment zone, the stress intensity factor, the distribution of stresses on the line of continuation of the crack and in the contact domain, and circular displacements of the crack faces are determined.     

Plane contact problem of the indentation of a stamp into an elastic wedge

We consider the contact interaction of a stamp with rectilinear base and an elastic wedge. One of the wedge faces is fixed, and the stamp edge touches the wedge vertex. Using the Wiener–Hopf method, we have obtained an exact solution of this problem. We have also determined the stress distributions in the contact region and on the wedge fixed face as well as the displacements of its free boundary.     

Diffraction of a Plane Electromagnetic Wave Obliquely Incident on the Edge of an Impedance Wedge: the Uniqueness of a Classical Solution

The problem on the diffraction of an electromagnetic plane wave on an impedance wedge is considered. The uniqueness of a classical solution is proved. Bibliography: 9 titles.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Stress recovery procedure for solving boundary value problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the variational Lagrange equation, is described. The vectors of the forces reduced to the mesh points are constructed for the known stiffness matrices of the elements using the displacements at the mesh points found from the solution of the problem. On the other hand, these mesh point forces are determined in terms of the unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The stresses at the mesh points are determined for the values of these forces found on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations, which relate the forces, stresses and the normal to the surface. The special features of the use of the stress recovery procedure are demonstrated for a plane problem in the linear theory of elasticity.     

The transient retardation of a rectilinear viscoplastic flow when the loading stresses are abruptly removed

The development of a viscoplastic flow in a solid layer of an elastoviscoplastic material on an inclined plane is considered when loading stresses act on its free surface. It is shown that the elastoplastic boundary starts its motion from the rigid inclined plane and, propagating through the elastic core, it can reach the free surface of the layer. An exact solution is obtained for the dynamic problem of the retardation of developed viscoplastic flow after the loading stresses are abruptly removed. The possibility of writing the equation of motion for the unloading wave in terms of the displacements is pointed out. It reduces to an inhomogeneous wave equation where the velocity of the unloading wave is found to be equal to the velocity of the equivoluminal elastic wave. Reflection of the unloading wave from a rigid boundary in the form of an inclined plane is also considered.     

Study of the stability in the problem on flowing around a wedge. The case of strong wave

We consider the flow of an inviscid nonheatconducting gas in the thermodynamical equilibrium state around a plane infinite wedge and study the stationary solution to this problem, the so-called strong shock wave; the flow behind the shock front is subsonic.We find a solution to a mixed problem for a linear analog of the initial problem, prove that the solution trace on the shock wave is the superposition of direct and reflected waves, and, the main point, justify the Lyapunov asymptotical stability of the strong shock wave provided that the angle at the wedge vertex is small, the uniform Lopatinsky condition is fulfilled, the initial data have a compact support, and the solvability conditions take place if needed (their number depends on the class in which the generalized solution is found).     

The limiting absorption principle in the problem of a transparent wedge

The problem of diffraction of a plane wave by a transparent wedge is considered (the wave numbers inside the wedge and outside it are different). Absorption in a medium is assumed to occur. The existence of a solution satisfying the limiting absorption principle is proved. The existence theorem and the fact that the limit passage can be performed are proved for the corresponding system of equations. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 138–148.     

压电基体中部分脱开的刚性导体椭圆夹杂分析

通过利用八维Stroh公式以及共形映射、解析延拓和奇点分析技术,获得了对一压电基体中已部分脱开的刚性导体椭圆夹杂二维问题的闭合形式全场解答。也推导了一些新的恒等式和求和式,通过这些恒等式及求和式可获得沿界面应力和电位移分布以及刚性夹杂转动的实形式表示。正如所预料的,在脱开界面的端部应力及电位移显现出与在压电材料Griffith界面裂纹的研究中所发现的相似的奇异行为。最后也给出了几个算例以展示所得到解答的一般性以及各种载荷条件、几何参数和机电常数等对界面处应力及电位移分布的影响。     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

Dynamic contact between a spherical inclusion and a matrix upon incidence of an elastic wave

Using the boundary element method, we have studied the dynamic displacements and stresses in an infinite elastic matrix with a spherical elastic inclusion, caused by the propagation of an elastic wave. The original problem has been reduced to a system of boundary integral equations for the contact displacements and tractions on the interface between the inclusion and matrix. Based on the numerical solution of these equations, we have analyzed the influence of the direction of wave propagation and frequency on the important physical parameters, depending on the elastic characteristics of composite constituents.     

Shortwave scattering by a diffraction echelette-grating

The two-dimensional problem of the scattering of a plane wave by a periodic perfectly conducting grating, an echelette with a right angle, is considered in the case of a high-frequency approximation (the wavelength is assumed to be small as compared with the period of the grating). The situation where the incident plane wave glides along one of the faces of a wedge is discussed. A ray-optical solution of the problem (a shortwave asymptotic result) is derived by the method of summing multiple diffracted fields, which is well known in the geometric theory of diffraction. The main result of this paper consists of obtaining simple formulas for the efficiency e_n of a diffraction order with maximal value of e_n, derived in the shortwave approximation. Numerical results are presented, and important optical properties obtained by asymptotic analysis are described. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 109–135. Translated by V. V. Zalipaev.     

Diffraction of a monochromatic plane shear wave on a reinforced cylindrical cavity in an elastic half-space

This article examines a boundary-value problem concerning the diffraction of a monochromatic plane shear wave on a reinforced cylindrical cavity in an elastic half-space. It is assumed that longitudinal shear stresses are absent and that the normal displacements over the entire boundary are specified. Through the use of a special form of the Lamé representation in cylindrical coordinates, the problem is reduced to the determination of scalar functions which satisfy the Helmholtz equation. The coefficients of the Fourier expansions of these functions in the angular coordinate are written as the sum of Fourier and Weber integrals. The densities of these integrals are determined exactly. A specific example is examined.Translated from Dinamicheskie Sistemy, No. 5 pp. 42–49, 1986.     

Diffraction of a plane wave on a slippery elastic wedge

Diffraction of a plane elastic wave on a slippery wedge is considered; by a slippery wedge we mean a wedge in which the tangent tension and the normal component of the displacement vector are equal to zero on its surface. It is known that one can construct an explicit solution of this problem. The Sommerfeld representation of this solution is found in construct the paper. Bibliography: 6 titles.     

压电梁的多项式解（Ⅱ）——典型问题解析解

从正交各向异性压电介质平面问题,对于材料3个特征根互不相等情况下,以3个拟调和位移函数表达位移、电势、应力和电位移的通解出发,利用调和多项式的显式表达式,结合试凑法,给出了平面压电梁的若干典型问题的解析解,包括悬臂压电梁自由端作用横向集中力和点电荷,悬臂压电梁表面作用线性电势和均布载荷,以及两端简支压电梁作用均布载荷等的解析解．     

Diffraction of a non-stationary linearly inhomogeneous acoustic wave which slides off a semi-infinite soft screen

An exact analytical solution of a new non-stationary scalar diffraction problem is obtained and analysed. A plane acoustic wave with a profile in the form of a delta function propagates along a semi-infinite soft screen. The wave amplitude varies linearly along the wave front. After reaching the end of the screen it “slides” off the screen, generating a diffraction field. A special modification of the Smirnov–Sobolev method is used to find this field. The solution is obtained in the form of an elementary function. It is shown that the sliding wave excites a travelling perturbation that is unlimited along the length of the screen. A similar phenomenon obviously also occurs when elastic waves slide from a cut (crack), which must be taken into account, in particular, in fracture theory.     

Scattering of the plane wave by a transparent wedge

In the paper it is proved that the problem of scattering of the plane wave by a transparent wedge has a unique solution, provided that the radiation condition should be meant in the following form: if one subtracts from the solution the incident wave and all reflected and refracted waves, then the remainder satisfies the radiation condition in integral form. The problem is scalar, the velocities of the wave inside and outside the wedge are not equal, the wave process is described by the classical Helmholtz equations, and the conjugation boundary condition is satisfied on the sides of the wedge. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 5–18.     

Stresses of a rotating circular disk of variable thickness carrying a current and bearing a coaxial viscoelastic coating

In the present work we consider a circular elastic disk (conductor) of variable thickness under the influence of a steady coaxial current and bearing a coaxial viscoelastic coating (insulator). In both conductor and insulator there exist a heat source generation. As a first step, the solution of purely elastic conductor and insulator is obtained. Then the problem of model with viscoelastic coating is solved using the correspondence principle and Ilyushin’s approximation method. A numerical example is given and the results are discussed in order to investigate the influences of time, temperature, and rotation on the stresses and displacements.