Problems of a cut in a three-dimensional elastic wedge

The Ritz variational method is applied to problems of a crack (a cut) in the middle half-plane of a three-dimensional elastic wedge. The faces of the elastic wedge are either stress-free (Problem A) or are under conditions of sliding or rigid clamping (Problems B and C respectively). The crack is open and is under a specified normal load. Each of the problems reduces to an operator integrodifferential equation in relation to the jump in normal displacement in the crack area. The method selected makes it possible to calculate the stress intensity factor at a relatively small distance from the edge of the wedge to the cut area. Numerical and asymptotic solutions [Pozharskii DA. An elliptical crack in an elastic three-dimensional wedge. Izv. Ross Akad. Nauk. MTT 1993;(6):105–12] for an elliptical crack are compared. In the second part of the paper the case of a cut reaching the edge of the wedge at one point is considered. This models a V-shaped crack whose apex has reached the edge of the wedge, giving a new singular point in the solution of boundary-value problems for equations of elastic equilibrium. The asymptotic form of the normal displacements and stress in the vicinity of the crack tip is investigated. Here, the method employed in [Babeshko VA, Glushkov YeV, Zinchenko ZhF. The dynamics of Inhomogeneous Linearly Elastic Media. Moscow: Nauka; 1989] and [Glushkov YeV, Glushkova NV. Singularities of the elastic stress field in the vicinity of the tip of a V-shaped three-dimensional crack. Izv. Ross Akad. Nauk. MTT 1992;(4):82–6] to find the operator spectrum is refined. The new basis function system selected enables the elements of an infinite-dimensional matrix to be expressed as converging series. The asymptotic form of the normal stress outside a V-shaped cut, which is identical with the asymptotic form of the contact pressure in the contact problem for an elastic wedge of half the aperture angle, is determined, when the contact area supplements the cut area up to the face of the wedge.     

Transmission of sound in a wedge-shaped layer of fluid with a rigid bottom

The theory of generalized rays is applied to the problem of determining the exact shape of the pressure pulse received at very large ranges, in the case of a sound source in a 3° perfect wedge of fluid. The time variation of the pressure at the point source is assumed to be given by a Heaviside unit function H(t). It is found that horizontal refraction and backscattering have a pronounced effect on the overall appearance of the received pulse: at very large down–slope (normal to the apex line) ranges, the shape of the received pulse does not change much, while, at very large cross–slope (parallel to the apex line) ranges, it changes drastically. Thus, in a 3° perfect wedge of fluid, long–range down–slope propagation (exclusively affected by backscattering) seems to be nondispersive, but long–range cross–slope propagation (affected by horizontal refraction and backscattering) seems to be highly dispersive. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The motion of a screw dislocation in a wedge-shaped region

The problem of antiplane deformation for a wedge-shaped region containing a uniformly moving screw dislocation is considered. A general solution of the problem is obtained using Laplace and Kontorovich-Lebedev integral transformations. It is shown during the solution that the method is suitable for a wedge apex angle greater than n. The limiting case, when the wedge-shaped region is a half-plane, is also considered. For this case the solution can be simplified considerably.     

Scattering of fluid loaded elastic plate waves at the vertex of a wedge of arbitrary angle, I: analytic solution

This article is the first part of an investigation into thescattering of fluid coupled structural waves by an angular discontinuityat the junction of two plates of different material properties.These two thin elastic plates are semi-infinite in extent thereforeforming the faces of an infinite wedge, the interior of whichcontains a compressible fluid. A plane unattenuated structuralwave is incident along the lower face of the wedge and is scatteredat the apex. The edges of the elastic plates may be joined ina variety of different ways, for example, they may be pin-jointedto an external structure or welded to each other. In the formercase, the plates will experience only the usual flexural vibrationswhereas in the latter case longitudinal (in-plane) disturbanceswill be generated and will propagate away from the wedge apex. An exact explicit solution is sought in terms of a Sommerfeldintegral representation for the fluid velocity potential. Thispermits the boundary-value problem to be recast as a functionaldifference equation which is easily solved in terms of the Maliuzhinetsspecial function (Maliuzhinets, Soviet Phys. Dokl. 3 1958).The chosen ansatz for the solution is of a different form fromthat used previously by the authors for the less complicatedmembrane wedge problem. The new ansatz has several analyticand numerical advantages which enable the reflection and transmissioncoefficients to be expressed explicitly in a compact form thatis ideal for computation. In the second part of this study a full numerical investigationof the reflection and transmission coefficients will be presentedfor a variety of interesting parameter ranges and edge conditions.     

Static frictional indentation of an elastic half-plane by a rigid unsymmetrical punch

Static rigid 2-D indentation of a linearly elastic half-plane in the presence of Coulomb friction which reverses its sign along the contact length is studied. The solution approach lies within the context of the mathematical theory of elastic contact mechanics. A rigid punch, having an unsymmetrical profile with respect to its apex and no concave regions, both slides over and indents slowly the surface of the deformable body. Both a normal and a tangential force may, therefore, be exerted on the punch. In such a situation, depending upon the punch profile and the relative magnitudes of the two external forces, a point in the contact zone may exist at which the surface friction changes direction. Moreover, this point of sign reversal may not coincide, in general, with the indentor's apex. This position and the positions of the contact zone edges can be determined only by first constructing a solution form containing the three problem's unspecified lengths, and then solving numerically a system of non-linear equations containing integrals not available in closed form.The mathematical procedure used to construct the solution deals with the Navier-Cauchy partial differential equations (plane-strain elastostatic field equations) supplied with boundary conditions of a mixed type. We succeed in formulating a second-kind Cauchy singular integral equation and solving it exactly by analytic-function theory methods.Representative numerical results are presented for two indentor profiles of practical interest—the parabola and the wedge.     

Limiting behaviour of the occupation of wedges by complex Brownian motion

Summary We prove a theorem which gives the lim inf behaviour ast tends to 0 for the amount of time a complex Brownian motion spends in a wedge with apex at the origin. The result is then shown to hold uniformly for all wedges a.s..     

用“调和函数”表示的压电介质平面问题的通解

本文首先从压电介质平面问题基本方程出发，引入一个位移函数，导出其通解，然后利用推广的Almansi定理，将通解化为简单形式，即通过三个“调和函数”来表示所有的物理量，其次，推导了楔形体顶端受集中力和点电荷作用时有限形式的解，退化可得半无限平面直线边界受集中力和点电荷的解。     

Stability of transonic shock for supersonic flow past a wedge

When steady supersonic flow hits a slim wedge, there may appear an oblique transonic shock attached to the vertex of the wedge, if the downstream pressure is rather large. This paper studies stability in certain weighted partial Hölder spaces of the oblique transonic shock attached to the vertex of a wedge, which is against steady supersonic flows, under perturbations of the upstream flow and the profile of the wedge. We show that under reasonable conditions on the upcoming supersonic flow and the slope of the wedge, such transonic shocks are structural stable. Mathematically, we solve an elliptic–hyperbolic mixed type in an unbounded domain, and the flow field is proved to be C¹. Copyright © 2015 John Wiley & Sons, Ltd.     

Global Existence of a Shock for the Supersonic Flow Past a Curved Wedge

This note is devoted to the study of the global existence of a shock wave for the supersonic flow past a curved wedge. When the curved wedge is a small perturbation of a straight wedge and the angle of the wedge is less than some critical value, we show that a shock attached at the wedge will exist globally.     

Irradiation of a cone by a magnetic dipole

Th electromagnetic field produced by a magnetic dipole in thepresence of a perfectly conducting cone of arbitrary cross-sectionis determined. The solution is used to find out how a currenton the cone travelling towards the apex is reflected. Some valuesof the reflection coefficient are calculated. In particular,it is shown that there is a sort of resonance with the reflectionincreasing significantly as the cone approaches a plane.     

MHD boundary-layer flow of a micropolar fluid past a wedge with constant wall heat flux

The steady laminar magnetohydrodynamic (MHD) boundary-layer flow past a wedge with constant surface heat flux immersed in an incompressible micropolar fluid in the presence of a variable magnetic field is investigated in this paper. The governing partial differential equations are transformed into a system of ordinary differential equations using similarity variables, and then they are solved numerically by means of an implicit finite-difference scheme known as the Keller-box method. Numerical results show that micropolar fluids display drag reduction and consequently reduce the heat transfer rate at the surface, compared to the Newtonian fluids. The opposite trends are observed for the effects of the magnetic field on the fluid flow and heat transfer characteristics.     

Electromagnetic fields in the presence of an infinite dielectric wedge: the phased line source excitation case

Electromagnetic fields, excited by an electric phased line sourcein the presence of an infinite dielectric wedge, are determinedby application of the Kontorovich–Lebedev transform. TheMaxwell's equations together with the conditions of continuityof the tangential field components at the material interfacesare formulated as a vector boundary-value problem. By representingthe field components as Kontorovich–Lebedev integrals,the problem is reduced to a system of singular integral equationsfor the unknown spectral functions. We construct numerical solutionsto those equations that permit fields evaluation for valuesof the wedge refractive index, not necessarily close to unity,and for arbitrary positioned source and observer. Numericalresults showing the influence of a wedge presence on the directivityof a phased line source are presented and verified through finite-differencefrequency-domain simulations.     

Boundary value problems involving Herschel-Bulkley material

The behaviour of a Herschel-Bulkley material near the apex ofa semi-infinite plate is studied. The plate is immersed in thecentre of a longitudinal strip the sides of which are subjectedto equal and opposite velocities. The problem is consideredin the hodograph plane, where it is possible to obtain a solution.Transforming this solution back to the real plane yields anexact expansion of the velocity field near the apex of the plate.The leading term of the velocity field is checked by using aninvariant integral which involves the energy-momentum tensorof the system. A reciprocal theorem is also derived, wherebythe coefficient of the singular strain rate field at the plateedge is related to finite integrals in the hodograph plane.A check of this method is effected by comparison with the solutionof the above problem.     

Brownian motion in a wedge with oblique reflection

This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties:

• (i) The state space is an infinite two-dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion.
• (ii) The process reflects instantaneously at the boundary of the wedge, the angle of reflection being constant along each side.
• (iii) The amount of time that the process spends at the comer of the wedge is zero (i.e., the set of times for which the process is at the comer has Lebesgue measure zero).

Hereafter, let ξ be the angle of the wedge (0 < ξ < 2π), let θ₁ and θ₂ be the angles of reflection on the two sides of the wedge, measured from the inward normals, the positive angles being toward the corner (-½π < θ₁, θ₂ ½π), and set α = (θ₁ + θ₂)/ξ. The question of existence and uniqueness is recast as a submartingale problem in the style used by Stroock and Varadhan (Diffusion processes with boundary conditions, Comm. Pure Appl. Math. 24, 1971, pp. 147-225), for diffusions on smooth domains with smooth boundary conditions. It is shown that no solution exists if α ≧ 2. In this case, there is a unique continuous strong Markov process satisfying (i)-(ii) above; it reaches the corner of the wedge almost surely and it remains there. If α < 2, however, then there is a unique continuous strong Markov process statisfying (i)-(iii). It is shown that starting away from the corner this process does not reach the corner of the wedge if α ≦ 0, and does reach the corner if 0 < α < 2. The general theory of multi-dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner. For some values of α, the process arises from diffusion approximations to storage systems and queueing networks. (i) The state space is an infinite two-dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the wedge, and the angle of reflection being constant along each side. (iii) The amount of time that the process spends at the corner of the wedge is zero (i.e., the set of times for which the process is at the corner has Lebesgue measure zero).     

Analytic Study of Magnetohydrodynamic Flow and Boundary Layer Control Over a Wedge

A genuine variational principle developed by Gyarmati, in the field of thermodynamics of irreversible processes unifying the theoretical requirements of technical, environmental and biological sciences is employed to study the effects of uniform suction and injection on MHD flow adjacent to an isothermal wedge with pressure gradient in the presence of a transverse magnetic field. The velocity distribution inside the boundary layer has been considered as a simple polynomial function and the variational principle is formulated. The Euler-Lagrange equation is reduced to a simple polynomial equation in terms of momentum boundary layer thickness. The velocity profiles, displacement thickness and the coefficient of skin friction are calculated for various values of wedge angle parameter m, magnetic parameter ξ and suction/injection parameter H. The present results are compared with known available results and the comparison is found to be satisfactory. The present study establishes high accuracy of results obtained by this variational technique.     

Singularity of the behavior of an electroelastic field in a piecewise homogeneous piezoelectric wedge

With the help of a local solution, the behavior of an electroelastic field in the vicinity of border of the contact surface of a composite piezoelectric wedge is investigated. Numerical results showing the relation between the degree of singularity of stresses and the electroelastic properties of joined materials and joint geometry are presented. In the plane of openings of homogeneous wedges, the limit curves separating the region of low stresses from the region of concentrated stresses at the border of contact surface are constructed. Some special cases are considered.     

Supersonic flow past a concave double wedge

The supersonic flow past a concave double wedge is discussed. Because of the interaction between the outer shock attached at the head of the wedge and the inner shock issued from the concave corner, there is a rarefaction wave issued from the intersection of the outer and inner shock. The rarefaction wave is reflected by the outer shock and the wedge infinitely, while the outer shock is also bent due to interaction. The global existence of the solution is proved under the assumptions that the outer shock is weak and the difference of two slopes of the double wedge is small. Meanwhile, a rigorous proof of the asymptotic behavior of the global solution is given. The property is often ap plied to numerical computation. Project partially supported by the National Natural Science Foundation of China and Doctoral Programme Foundation of IHEC.     

Molecular destruction of polymers at the apex of a major crack

We employed infrared spectroscopy and infrared spectroscopy of deflected total internal reflection to study the rupture of polymer macromolecular chains (polyethylene and polypropylene) at the apex of a growing crack. It was shown that the break concentration at the crack surface is of the same order of magnitude as the total concentration of chemical bonds in the polymer; molecular rupture occurs not only at the apex itself but also at some distance from it. The dependence of the break concentration on the distance from the crack apex was measured. The observed patterns were related to the experimentally determined external-load distribution function for individual chemical bonds near the crack apex.I. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 4, pp. 621–625, July–August, 1972.     

Influence of temperature-dependent viscosity and thermal radiation on MHD forced convection over a non-isothermal wedge

An analysis has been carried out to study heat transfer characteristics of an incompressible Newtonian electrically conducting and heat generating/absorbing fluid having temperature-dependent viscosity over a non-isothermal wedge in the presence of thermal radiation. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The wedge surface is assumed to be permeable so as to allow for possible wall suction or injection. The effects of viscous dissipation, Joule heating, stress work and thermal radiation are included in the model. The governing differential equations are derived and transformed using a non-similarity transformation. The transformed equations are solved numerically by applying a fifth-order Runge-Kutta-Fehlberg scheme with shooting technique. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results for the velocity and temperature profiles for a prescribed magnetic field parameter as well as the development of the local skin-friction coefficient and local Nusselt number with the magnetic field and radiation parameters are presented graphically and in tabulated form to elucidate the influence of the various physical parameters.     

The problem of an inclusion in a three-dimensional elastic wedge

Fundamental solutions for a three-dimensional wedge are used to investigate problems of a thin, rigid, elliptic inclusion in a wedge. A regular asymptotic form is employed which has previously been used in contact problems for a wedge [1] and in problems of a crack in a wedge [2] in the case of an elliptic shape of the contact region or crack. The method is effective in the case of an inclusion which is sufficiently distant from an edge of the wedge when the known exact solution for the space [3] can be taken as the zeroth approximation. A numerical analysis and comparison of different characteristics of wedge problems is carried out.