Plane steady vibration of an elastic wedge

The problem of plane steady vibration of an elastic wedge of arbitrary angle (less than 180 degrees) subject to harmonic normal and shearing tractions on its faces is reduced to a system of singular integral equations by the superposition of two half-plane solutions. The integral equations have kernels with Cauchy singularities of a non-translation type, except for the 90 degree wedge. The locations of these singularity lines are shown graphically as a function of wedge angle.     

Solutions for the anisotropic elastic wedge at critical wedge angles

The classical solution for an isotropic elastic wedge loaded by uniform tractions on the sides of the wedge becomes infinite everywhere in the wedge when the wedge angle 2 equals , 2 or 2_* where tan 2_* = 2_*. When the wedge is loaded by a concentrated couple at the wedge apex the solution also becomes infinite at 2 = 2_*. A similar situation occurs when the wedge is anisotropic except that 2_* is governed by a different equation and depends on material properties. Solutions which do not become infinite everywhere in the wedge are available for isotropic elastic wedges. In this paper we present solutions for the anisotropic elastic wedge at critical wedge angles. The main feature of the solutions obtained here is that they are in a real form even though Stroh's complex formalism is employed.     

Plane elasticity problem for a multi-wedge system with a thin wedge

The paper presents a method for studying a system of elastic wedges containing a thin wedge with the angle Θ₀, which may be arbitrary small. An analysis shows that the considered problem, involving 2-D vectors of tractions and displacements, cannot be solved by straight-forward extension of the method previously worked out by the authors for analogous scalar problems. The difficulty arises because of the disclosed feature of the dependences between the Mellin transformed displacements and tractions at the boundaries of a thin wedge: they are linearly dependent when their Taylor’s expansions in Θ₀ are represented by the first terms only. The difficulty is removed by using the consequences of the linear dependence and by an appropriate re-arrangement of variables. Then simple physical models, simulating the influence of a thin wedge on a multi-wedge system, become available. The models cover the cases of a very rigid and very compliant thin wedge and also intermediate cases. The ranges of the models applicability are studied analytically and illustrated by numerical results.     

An incompressible rubber wedge under the tension of a line load at its tip

An analysis is made of the stress distribution and the deformation pattern near the apex of an incompressible rubber wedge under the tension of a concentrated force. The asymptotic method for the tip stress field is developed. The problem is treated as a plane strain case. Finally, the FEM numerical results agree with the theoretical ones.     

Numerical method and models for anti-plane strain of a system with a thin elastic wedge

The paper presents a general method to find asymptotics for a (multi-)wedge system containing a thin wedge. It employs separation of the symmetric and anti-symmetric parts of the boundary displacements and tractions of the wedge. The method is applicable when the angle of the thin wedge turns to zero. A physical interpretation of the derived equations is obtained by using power expansions of non-polynomial functions, which appear after the Mellin transform. We establish that the first term in the expansion of the symmetric part corresponds to shear, while the first term of the anti-symmetric part describes deflection of the wedge axis. Numerical experiments, performed by using a code developed on the basis of the theory, show that using only the first terms of the expansions insignificantly influence accuracy: the approximate results coincide with the exact values of roots to the third significant digit even for the wedge angle of 30°.     

Surface shape of a magnetic liquid drop near the edge of a magnetic wedge

The two-dimensional problem of the shape of the free surface of a magnetic fluid in a gravity field, a uniform external magnetic field and the nonuniform field of a magnetized metal wedge is considered. The results of numerically calculating the shape of the free surface of a magnetic liquid drop retained on an inclined plane by the field of a magnetizing wedge are presented. The changes in the shape of the free surface of an infinite volume of magnetic liquid near the edge of a wedge with increase in the external field are investigated. It is shown that for a certain critical field some of the magnetic liquid separates and adheres to the edge of the wedge. Experimental data on the determination of the maximum cross-sectional area of a drop retained by the magnetic field of a wedge and the critical rise of the magnetic liquid relative to the level outside the field are presented. The experimental and theoretical results are in agreement.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 113–119, July–August, 1992.The authors wish to thank V. V. Gogosov for useful discussions and his interest in the work.     

Local nonsimilarity solution for the impact of the buoyancy force on heat and mass transfer in a flow over a porous wedge with a heat source in the presence of suction/injection

Combined heat and mass transfer in free, forced and mixed convection flows along a porous wedge with internal heat generation in the presence of uniform suction or injection is investigated. The boundary-layer analysis is formulated in terms of the combined thermal and solute buoyancy effect. The flow field characteristics are analyzed using the Runge-Kutta-Gill method, the shooting method, and the local nonsimilarity method. Due to the effect of the buoyancy force, power law of temperature and concentration, and suction/injection on the wall of the wedge, the flow field is locally nonsimilar. Numerical calculations up to third-order level of truncation are carried out for different values of dimensionless parameters as a special case. The effects of the buoyancy force, suction, heat generation, and variable wall temperature and concentration on the dimensionless velocity, temperature, and concentration profiles are studied. The results obtained are found to be in good agreement with previously published works.     

Flow of power-law fluids over a moving wedge surface with wall mass injection

The boundary layer problem of a power-law fluid flow with fluid injection on a wedge whose surface is moving with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. The free stream velocity, the injection velocity at the surface, moving velocity of the wedge surface, the wedge angle and the power law index of non-Newtonian fluid are assumed variables. The fourth order Runge–Kutta method modified by Gill is used to solve the non-dimensional boundary layer equations for non-Newtonian flow field. Without fluid injection, for every angle of wedge β, a limiting value for velocity ratio λ _cr (velocity of the wedge surface/velocity of the uniform flow) is found for each power-law index n. The value of λ _cr increases with the increasing wedge angle β. The value of wedge angle also restricts the physical characteristics of the fluid to be used. The effects of the different parameters on velocity profile and on skin friction are studied and the drag reduction is discussed. In case of C = 2.5 and velocity ratio λ = 0.2 for wedge angle β = 0.5 with the fluid with power law-index n = 0.5, 48.8% drag reduction is obtained.     

Antiplane electro-mechanical field of a piezoelectric finite wedge under shear loading and at fixed-grounded boundary conditions

This paper presents the general solutions of antiplane electro-mechanical field solutions for a piezoelectric finite wedge subjected to a pair of concentrated forces and free charges. The boundary conditions on the circular segment are considered as fixed and grounded. Employing the finite Mellin transform method, the stress and electrical displacement at all fields of the piezoelectric finite wedge are derived analytically. In addition, the singularity orders and intensity factors of stress and electrical displacement can also be obtained. These parameters can be applied to examine the fracture behavior of the wedge structure. After being reduced to the problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.     

Singularities of physical fields at the apex of a piezoceramic wedge (plane strain)

A uniform piezoceramic wedge is studied. Singularities of coupled physical fields at the apex of the wedge are investigated with homogeneous boundary conditions on its faces. The problem is solved by using complex representations of the components of the electroelastic field for two basic types of mechanical and electrical boundary conditions. The results of computations are presented. Sumy State University, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 89–92, March, 1999.     

The wedge subjected to general tractions: A paradox resolved

A wedge subjected to tractions in proportion tor ⁿ (n≥0), is considered. The stresses in the solutions of the classical theory of elasticity become infinite when the angle of the wedge is ρ or 2ρ. The paradox has been resolved by Dempsey^[4] and T.C.T. Ting^[5] whenn=0. The purpose of this paper is to resolve the paradox whenn>0.     

Paradox solution on elastic wedge dissimilar materials

According to the Hellinger-Reissner variational principle and introducing proper transformation of variables , the problem on elastic wedge dissimilar materials can be led to Hamiltonian system, so the solution of the problem can be got by employing the separation of variables method and symplectic eigenfunction expansion under symplectic space, which consists of original variables and their dual variables . The eigenvalue - 1 is a special one of all symplectic eigenvalue for Hamiltonian system in polar coordinate . In general, the eigenvalue - 1 is a single eigenvalue, and the classical solution of an elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is got directly by solving the eigenfunction vector for eigenvalue - 1. But the eigenvalue - 1 becomes a double eigenvalue when the vertex angles and modulus of the materials satisfy certain definite relationships and the classical solution for the stress distribution becomes infinite at this moment, that is, the para     

极坐标哈密顿体系约当型与弹性楔的佯谬解

讨论了极坐标弹性平面哈密顿体系的当型,并通过约当型的求解,直接给出了相关弹性楔体佯谬问题的解,从理论上阐明了经典弹性力学中某些佯谬问题的出现是由于其对应的是哈密顿体系中特殊的约当型解,同时也很自然地为该类问题提供了一个通用,有效的求解方法。     

The plane self-similar anisotropic and angularly inhomogeneous wedge under power law tractions and the asymptotic analysis of the stress field

In this study the generally anisotropic and angularly inhomogeneous wedge, under power law tractions of order n of the radial coordinate r at its external faces is considered. At first, using variable separable relations in the equilibrium equations, the strain–stress relations and the strain compatibility equation, a differential system of equations is constructed and investigated. Decoupling this system, an ordinary differential equation is derived and the stress and displacement fields may be determined. The proposed procedure is also applied to the elastostatic problem of an isotropic and angularly inhomogeneous wedge. In the sequel William's asymptotic analysis in the case of angular inhomogeneity is examined. Finally, applications for the case of an angularly inhomogeneous wedge-shape dam and for the asymptotic procedure in an isotropic wedge with angularly varying shear modulus, are made.     

MPS方法数值模拟楔形体入水问题

入水问题是船舶海洋工程中典型的流动问题。当船舶在恶劣海况中航行或海洋平台遭遇恶劣天气时,结构物和水体之间往往会出现剧烈的砰击作用。砰击发生时,伴随着结构物湿表面的变化、自由液面的翻卷和破碎等强非线性现象。本文采用本课题组自主开发的基于移动粒子半隐式法MPS(Moving Particle Semi-Implicit Method)的求解器MLParticle-SJTU对二维楔形体入水问题进行了数值模拟。选取斜升角为30°的楔形体模拟其入水过程,研究了不同粒子布置方式对于计算结果如垂向水动力和自由面变化的影响,并与MLM砰击模型(Modified Logvinovich Model)的结果进行了比较,吻合较好。在此基础上,进一步研究了不同斜升角对计算结果(垂向水动力和自由面变化)的影响,其中流动分离前的垂向水动力与MLM结果相近,表明了MPS方法能有效地模拟入水问题。     

Saint-Venants principle and the plane elastic wedge

The stress field due to self-equilibrating loading on the inner or outer arc of a plane strain elastic wedge sector is affected by two agencies: a geometric effect of increasing or decreasing area, and decay as anticipated by Saint-Venants principle (SVP) . When the load is applied to the inner arc the two effects act in concert ; however, when the load is applied to the outer arc the two effects act in opposition and for a wedge angle in excess of the half-space, 2α > π, for the symmetric case, and for 2α > 1.43π for the asymmetric case, the geometric effect is dominant over Saint-Venant decay and stress level increases as one moves away from the outer arc, confirming the inapplicability of SVP. This is additional to previously reported difficulties at these angle when a self-equilibrated load on the inner arc decays at the same rate as does a concentrated moment, and is explained in terms of the interaction of a near-field geometric effect and a far-field stress interference effect at a traction-free edge. For wedge angle 2α = 2π the unique Modes I and II inverse square root stress singularities at the crack tip, which are at the heart of Linear Elastic Fracture Mechanics (LEFM) , can be attributed to this inapplicability for just one symmetric and one asymmetric eigenmode.     

Mixed convection flow over a vertical wedge embedded in a highly porous medium

 The steady mixed convection flow over a vertical wedge with a magnetic field embedded in a porous medium has been investigated. The effects of the permeability of the medium, surface mass transfer and viscous dissipation on the flow and temperature fields have been included in the analysis. The coupled nonlinear partial differential equations governing the flow field have been solved numerically using the Keller box method. The skin friction and heat transfer are found to increase with the parameters characterizing the permeability of the medium, buoyancy force, magnetic field and pressure gradient. However the effect of the permeability and magnetic field on the heat transfer is very small. The heat transfer increases with the Prandtl number, but the skin friction decreases. The buoyancy force which assists the forced convection flow causes an overshoot in the velocity profiles. Both the skin friction and heat transfer increase with suction and the effect of injection is just the reverse. Received on 21 May 1999     

Stress analysis of a wedge disclination dipole interacting with a circular nanoinhomogeneity

This paper investigates the interaction between a wedge disclination dipole and a circular nanoinhomogeneity embedded in an infinite matrix. The Gurtin–Murdoch surface/interface elasticity model is applied to account for the interface stress effect of the nanoinhomogeneity. A closed form solution for the stress fields inside the inhomogeneity and matrix is derived with the complex variable method of Muskhelishvili. The influences of the interfacial and bulk material properties as well as the geometric parameters on the material force of the wedge disclination dipole are systematically discussed. It is found that the interface stress effect may influence the material force of the wedge disclination dipole significantly.     

The wedge subjected to tractions: a paradox resolved

The classical two-dimensional solution provided by Lévy for the stress distribution in an elastic wedge, loaded by a uniform pressure on one face, becomes infinite when the opening angle 2 of the wedge satisfies the equation tan 2 = 2. Such pathological behavior prompted the investigation in this paper of the stresses and displacements that are induced by tractions of O(r ^–) as r0. The key point is to choose an Airy stress function which generates stresses capable of accommodating unrestricted loading. Fortunately conditions can be derived which pre-determine the form of the necessary Airy stress function. The results show that inhomogeneous boundary conditions can induce stresses of O(r ^–), O(r ^– ln r), or O(r ^– ln² r) as r0, depending on which conditions are satisfied. The stress function used by Williams is sufficient only if the induced stress and displacement behavior is of the power type. The wedge loaded by uniform antisymmetric shear tractions is shown in this paper to exhibit stresses of O(ln r) as r0 for the half-plane or crack geometry. At the critical opening angle 2, uniform antisymmetric normal and symmetric shear tractions also induce the above type of stress singularity. No anticipating such stresses, Lévy used an insufficiently general Airy stress function that led to the observed pathological behavior at 2.     

An old elasticity problem in a unilateral setting

A limiting case of the Michell problem involving an elastic wedge is the unbounded solid with a semi-infinite cut, the tip of which is subjected to a concentrated force. For the limiting case, the classical solution leads to overlapping of material whenever the component of the force along the axis of symmetry is directed away from the cut, and the problem must be solved anew using unilateral boundary conditions. The required mathematics is simple, and the subject is suitable for classroom discussion. Two examples are solved explicitly, and additional exercise problems are suggested.