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.     

The wedge subjected to tractions: a paradox re-examined

The classical two-dimensional solution for the stress distribution in an elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2 satisfies the equation tan 235-1. This paradox was resolved recently by Dempsey who obtained a solution which is bounded at 235-2. However, for not equal but very close to 235-3, the classical solution can still be very large as approaches 235-4. In this paper we re-examine the paradox. We obtain a solution which remains bounded as approaches 235-5 and reproduces Dempsey's solution in the limit 235-6. At 235-7 the stress distribution contains a (ln r) term for general loadings. The (ln r) term disappears under a special loading and the stresses are bounded for all r. Moreover, the solution is not unique. In other words, for the wedge angle 235-8 under a special loading, infinitely many solutions exist for which the stresses are bounded for all r. We also obtain solutions which are bounded and approach Dempsey's solutions when 2= and 2. Again, under a special loading infinitely many solutions exist for which the stresses are bounded for all r. Care has been exercised in this paper to present the solutions in a form in which the terms r ^- and ln r are replaced by R ^-gl and ln R where R=r/r ₀is the dimensionless radial distance and r ₀ is an arbitrary constant having the dimension of length.     

Mode I crack tips propagating at different speeds under differential surface tractions

By application of the theory of complex functions, mode I crack tips propagating at different speeds under differential surface tractions were researched. Analytical solutions are attained by the approaches of self-similar functions. The problems considered can be facilely transformed into Riemann–Hilbert problems and their closed solutions are obtained rather straightforward by this method.     

The critical angle of the anisotropic elastic wedge subject to uniform tractions

The classical solution for an isotropic elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2₀ satisfies the equation tan 2₀ = 2₀. This is the critical wedge angle which also renders infinite solutions for other types of loadings. In this paper, we study the associated problem for the anisotropic elastic wedge. We first present uniform stress solutions which are possible for symmetric loadings. For antisymmetric loadings, a uniform stress solution is in general not possible and we present a non-uniform stress solution in which the stress depends on but not on r. The non-uniform stress solution breaks down at a critical angle. We present an equation for the critical angle which depends on the elastic constants. The Stroh formalism is employed in the analysis. An integral representation of the solution is obtained by using new identities which are derived in the paper.     

裂纹面受荷载作用的应力强度因子的计算

基于比例边界有限元法计算了裂纹面有荷载作用情况下裂纹尖端的应力强度因子,给出了有限介质裂纹面作用荷载的比例边界有限元方程的基本求解过程.对于随径向坐标任意变化的一类面荷载的积分能够显式计算,不需要引入额外的近似;并将计算结果与解析解和数值结果进行对比,结果表明比例边界有限元法在计算裂纹面作用荷载时的应力强度因子是有效且精确的.此外,该方法可方便地处理各向异性材料裂纹问题,本文给出了正交各向异性矩形盘裂纹面受均布荷载情况的应力强度因子.     

纳米流体在伸/缩楔体上的MHD流动数值研究

对纳米流体在伸/缩楔体上的磁流体(MHD)流动进行了数值研究。首先,通过相似变换将控制偏微分方程转化为非线性常微分方程组;然后,利用Matlab软件,借助打靶法,结合四阶五常龙格库塔迭代方案进行数值求解;最后,详细讨论了各控制参数对无量纲速度、温度、浓度、表面摩擦系数、局部Nusselt数和局部Sherwood数的影响。结果表明,楔体在拉伸情况下只有唯一解,理论上不会出现边界层分离;而在一定收缩强度范围内存在双解,边界层流动在壁面处可能会出现边界层分离,壁面抽吸会使边界层分离推迟;楔体在拉伸情况下,磁场参数对表面摩擦系数的影响较大,对局部Nusselt数和局部Sherwood数的影响较小。     

The application of asymptotic solutions to contact problems characterised by logarithmic singularities

We give the contact pressure distribution near a contacting wedge having a slightly rounded form adjacent to a discontinuity in surface profile. It is shown that, well away from the rounding the pressure is logarithmic in form, just as it is near the apex of a sharp wedge. This pair of solutions may then be used to ‘patch in’ a roundness correction relevant to any punch having a discontinuous gradient. Further, it is noted that the multiplier on the logarithm term is pre-determined by the change in gradient. This process is applied to a finite, slightly blunt wedge, where the exact answer is known, and to a wheel having a worn flat. The agreement with the exact solution in the former case is seen to be very good.     

The curved bar subjected to an arbitrarily distributed loading: Another paradox in the two-dimensional theory of elasticity

The problem of the curved bar subjected to an arbitrarily distributed loading on the surfacesr=a andr=b is solved by using the method of complex functions and expanding the boundary conditions atr=a andr=b into Fourier series. Then another paradox in the two-dimensional theory of elasticity is discovered, i. e., the classical solution becomes infinite when the curved bar is subjected to a uniform loading or when the angle included between the two ends of the curved bar 2 is equal to 2 and the curved bar is subjected to a sine or cosine loading. In this paper the paradox is resolved successfully and the solutions for the paradox are obtained. Moreover, the modified classical solution which remains bounded as 2 approaches 2 is provided.     

The more general displacement solutions for the plane elasticity problems

In this paper,the author obtains the more general displacement solutions for theisotropic plane elasticity problems.The general solution obtained in ref.[1 ]is merelythe particular case of this paper,In comparison with ref.[1],the general solutions ofthis paper contain more arbitrary constants.Thus they may satisfy more boundaryconditions.     

Dynamic propagation problems concerning surfaces of asymmetrical mode III crack subjected to moving loads

With the theory of complex functions, dynamic propagation problems concerning surfaces of asymmetrical mode III crack subjected to moving loads are investigated. General representations of analytical solutions are obtained with self-similar functions. The problems can be easily converted into Riemann-Hilbert problems using this technique. Analytical solutions to stress, displacement and dynamic stress intensity factor under constant and unit-step moving loads on the surfaces of asymmetrical extension crack, respectively, are obtained. By applying these solutions, together with the superposition principle, solutions of discretionarily intricate problems can be found. Project supported by the Post-Doctoral Science Foundation of China (No. 2005038199) and the Natural Science Foundation of Heilongjiang Province of China (No. ZJG04-08)     

A directed rod subjected to internal constraints

In this paper, the field equations for a directed rod are given. The form of the reaction forces with respect to a general internal constraint imposed on the rod is also derived, based on the assumption that the reaction forces do no work in any deformation compatible with the constraint. Finally, the rod subjected to both special and fundamental internal constraints is treated.     

Local non-similarity solution to impact of chemical reaction on MHD mixed convection heat and mass transfer flow over porous wedge in the presence of suction / injection

Combined heat and mass transfer on free, forced, and mixed convection flow along a porous wedge with magnetic effect in the presence of chemical reaction is investigated. The flow field characteristics are analyzed by the Runge-Kutta-Gill scheme with the shooting method as well as the local non-similarity method up to the third level of truncation, which are used to reduce the governing partial differential equations into nine ordinary differential equations. The governing boundary layer equations are converted to a dimensionless form by Falkner-Skan transformations. Because of the effect of suction/injection on the wall of the wedge with buoyancy force and variable wall temperature, the flow field is locally non-similar. Numerical calculations up to the third order level of truncation are carried out as a special case for different values of dimensionless parameters. Effects of the magnetic field strength in the presence of chemical reaction with variable wall temperature and concentration on the dimensionless velocity, temperature and concentration profiles are shown graphically.     

Analytical methods for perfect wedge diffraction: A review

The subject of diffraction of waves by sharp boundaries has been studied intensively for well over a century, initiated by groundbreaking mathematicians and physicists including Sommerfeld, Macdonald and Poincaré. The significance of such canonical diffraction models, and their analytical solutions, was recognised much more broadly thanks to Keller, who introduced a geometrical theory of diffraction (GTD) in the middle of the last century, and other important mathematicians such as Fock and Babich. This has led to a very wide variety of approaches to be developed in order to tackle such two and three dimensional diffraction problems, with the purpose of obtaining elegant and compact analytic solutions capable of easy numerical evaluation.The purpose of this review article is to showcase the disparate mathematical techniques that have been proposed. For ease of exposition, mathematical brevity, and for the broadest interest to the reader, all approaches are aimed at one canonical model, namely diffraction of a monochromatic scalar plane wave by a two-dimensional wedge with perfect Dirichlet or Neumann boundaries. The first three approaches offered are those most commonly used today in diffraction theory, although not necessarily in the context of wedge diffraction. These are the Sommerfeld–Malyuzhinets method, the Wiener–Hopf technique, and the Kontorovich–Lebedev transform approach. Then follows three less well-known and somewhat novel methods, which would be of interest even to specialists in the field, namely the embedding method, a random walk approach, and the technique of functionally-invariant solutions.Having offered the exact solution of this problem in a variety of forms, a numerical comparison between the exact solution and several powerful approximations such as GTD is performed and critically assessed.     

受r^n分布载荷的楔：佯谬的解决

对表面受与ｒｎ（ｎ≥０）成正比的分布载荷的楔，当楔顶角２α与ｎ之间满足一定关系时，经典解为无穷大，这是一个佯谬．本文采用复变函数方法，研究了这个佯谬的所有情形，并发现存在二次佯谬，即对某些特定的（ｎ，α），佯谬解仍为无穷大，对此本文也予以解决     

The Lur'e-Vorovich Method in Mixed Problems of Bending of Cylindrical Bodies

The mixed problem of bending of a transtropic thick plate and a short cylinder is solved using the Lur'e-Vorovich homogeneous solutions. The most typical results from numerical experiments for a wide range of physical and geometrical parameters are presented __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 58–65, August 2005.     

A variational model for stress analysis in cracked laminates with arbitrary symmetric lay-up under general in-plane loading

The present research work presents a variational approach for stress analysis in a general symmetric laminate, having a uniform distribution of ply cracks in a single orientation, subject to general in-plane loading. Using the principle of minimum complementary energy, an optimal admissible stress field is derived that satisfies equilibrium, boundary and traction continuity conditions. Natural boundary conditions have been derived from the variational principle to overcome the limitations of the existing methodology on the analysis of general symmetric laminates. Thus, a systematic way to formulate boundary value problem for general symmetric laminates containing many cracked and un-cracked plies has been derived, and appropriate mathematical tools can then be employed to solve them. The obtained results are in excellent agreement with the available results in the literature. In the field of matrix cracks analysis for symmetric laminates, the present formulation is the most complete variational model developed so far.     

Dynamic response of a rigid plastic simply supported imperfect beam subjected to a uniform intense dynamic loading

This paper proposes a theoretical analysis for the dynamic response of a rigid perfectly plastic simply supported beam with an imperfection in the midspan cross-section under uniform step, pulse, and impulsive loading when support is assumed to be free to move inward. The complete solutions for an entire dynamic response process are given and the relationship between the distribution of energy dissipated at plastic hinges and the parameter of imperfection is also discussed.     

Molecular model and flow calculation: I. The numerical solutions to multibead-rod models in homogeneous flows

The Galerkin method is used to solve the diffusion equation of the distribution function in configurational space for a multibead-rod model, and the dimensionless components of the extra stress tensor are then calculated by means of the expression of ensemble average. The material functions for steady-state shear flow and uniaxial flow and the mechanical properties of rigid-rodlike molecule suspensions in superposed flows are obtained numerically. The results indicate that it is promising to employ the multibead-rod models without the constitutive equation in numerical simulations of flows of suspensions. The project supported by the National Natural Science Fundation of China.     

桥梁受移动荷载动力响应的一种精细积分法

迄今为止,精细积分法都是应用于荷载作用位置固定不变的问题。本文将精细积分法推广到荷载作用点位置随时间而变化的情形,按有限元方法计算了桥梁受移动荷载作用时的动力响应,并与常规的Newmark方法、解析解做了比较。数值结果表明,精细积分法按本文的策略推广后,计算精度和效率均比通常的数值积分方法得到显著的提高。