圆弧形裂纹问题中的应力对数奇异性

研究了无限大板上的一条圆孤形裂纹, 又在裂纹表面作用有反对称载荷. 换言之, 裂纹两侧表面的载荷是大小相等方向相同的. 上述问题可用复变函数方法来解决. 应力和位移分量通过两个复位函数来表示. 经过一系列推导, 此问题可归结为复变函数的黎曼-希尔巴德(Riemann-Hilbert) 问题, 并且可用闭合形式得出解答. 裂纹端的应力强度因子用通常方法定出. 在裂纹端邻域, 得到的复位函数中有对数函数部分. 由这个对数函数部分, 可以定义和得出裂纹端的对数奇异性, 此对数奇异性系数用闭合型式得出.     

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.     

Buckling of an elastic plate subjected to unilateral constraints

Summary In this paper we show how it is possible to extend the well-known variational properties of the critical loads for elastic structures to the case of unilateral constraints. In particular we examine the buckling of an elastic plate unitalerally constrained between two elastic half-spaces and loaded by axial forces. For this problem we discuss the existence of the lowest positive and greatest negative critical load multiplier and provide a characterization of these multipliers by means of the minimal properties of a suitable functional.

---

Sommario In questo lavoro viene mostrato come sia possibile estendere le ben note proprietà variazionali dei carichi critici di una struttura elastica dal caso dei vincoli bilaterali a quello dei vincoli unilaterali. In particolare viene esaminato il problema di svergolamento di una piastra rettangolare elastica, sollecitata da forze assiali agenti nelle due direzioni e vincolata unilateralmente tra due semispazi elastici. Per tale problema si discute l'esistenza del più piccolo moltiplicatore positivo dei carichi e del più grande negativo per cui si ha instabilità. Inoltre viene fornita una caratterizzazione variazionale di tali moltiplicatori mediante le proprietà di minimo di un opportuno funzionale.

     

Stress singularity due to traction discontinuity on an anisotropic elastic half-plane

In a half-plane problem with x₁ paralleling with the straight boundary and x₂ pointing into the medium, the stress components on the boundary whose acting plane is perpendicular to x₁ direction may be denoted by t₁ = [σ₁₁, σ₁₂, σ₁₃]^T. Stress components σ₁₁ and σ₁₃ are of more interests since σ₁₂ is completely determined by the boundary conditions. For isotropic materials, it is known that under uniform normal loading σ₁₁ is constant in the loaded region and vanishes in the unloaded part. Under uniform shear loading, σ₁₁ will have a logarithmic singularity at the end points of shear loading. In this paper, the behavior of the stress components σ₁₁ and σ₁₃ induced by traction-discontinuity on general anisotropic elastic surfaces is studied. By analyzing the problem of uniform tractions applied on the half-plane boundary over a finite loaded region, exact expressions of the stress components σ₁₁ and σ₁₃ are obtained which reveal that these components consist of in general a constant term and a logarithmic term in the loaded region, while only a logarithmic term exists in unloaded region. Whether the constant term or the logarithmic term will appear or not completely depends on what values of the elements of matrices Ω and Γ will take for a material under consideration. Elements for both matrices are expressed explicitly in terms of elastic stiffness. Results for monoclinic and orthotropic materials are all deduced. The isotropic material is a special case of the present results.     

The strain energy function of a metal subjected to finite elastic volume change

The form of the strain energy function for a metal subjected to finite elastic volume change and infinitesimal elastic shearing strain is considered.     

Plane steady vibration of an orthogonal elastic wedge

The problem is reduced to a single Fredholm integral equation by using a superposition of two half-plane solutions. The treatment is exact within the theory of linear elastodynamics. Dissipation is introduced in the form of linear viscous damping at the surface in order to overcome the mathematical difficulties which are associated with the nondecaying surface waves that occur when no dissipation is present.

---

Zusammenfassung Das Problem ist durch Superposition zweier Lösungen in der Halbebene auf eine einzige Fredholmsche Integralgleichung Zurückgeführt. Die Formulierung ist exakt im Sinne linearer elasto-dynamischer Theorie. Energieverlust ist durch lineare, viskose Dämpfung an der Oberfläche eingeführt um mathematische Schwierigkeiten zu vermeiden die durch nicht abklingende Oberflächenwellen entstehen wenn keine Dämpfung vorhanden ist.

     

Scattering of a Rayleigh wave by an elastic wedge

We study the two-dimensional, steady-state problem of the scattering of waves in a homogeneous, isotropic, linear-elastic wedge where the wedge angel is less than 180°. For incidence of a Rayleigh surface wave, we plot the amplitudes and phases of the surface waves reflected and transmitted by the corner.     

Stress and strain analysis in elastic laminated composite beams

A detailed theoretical analysis of stress and strain in fiber-reinforced, laminated composite beams is presented within the framework of three-dimensional theory of thermo-elastodynamics. Each of reinforcing and matrix layers is made of different anisotropic material and is of different constant thickness. By the use of a variational theorem, the fully non-linear governing equations are consistently obtained for the case when the displacement components of each layer are taken to vary linearly over the cross-section of the layer. Then the linear version of the resulting equations, including the uniqueness of its solution is given, and the general results are briefly discussed.

---

Zusammenfassung In der vorliegenden Abhandlung wird eine detaillierte Spannungs-und Verzerrungsuntersuchung von faserverstärkten laminierten Balken innerhalb der dreidimensionalen thermo-elastodynamischen Theorie durchgeführt. Für jede Verstärkungs-und Bindeschicht wird unterschiedliches anisotropes Verhalten und unterschiedliche Schichtstärke vorausgesetzt. Unter Annahme eines linearen Verlaufs der im Querschnitt liegenden Verformungskomponenten über jede einzelne Schicht werden mit Hilfe eines Satzes der Variationsrechnung die vollständigen nichtlinearen Grundgleichungen abgeleitet. Damit ist auch die linearisierte Form der Gleichungen mit eindeutiger Lösung bestimmt. Eine kurze Untersuchung der Ergebnisse in ihrer allgemeinen Form ist angeschlossen.

     

Vibrations of neo-Hookean elastic wedge

This paper considers small amplitude vibrations superimposed upon large planar deformations of an infinite wedge composed of a neo-Hookean elastic material. It is shown herein that even though the static deformation of the entire wedge and the vibrations of the wedge faces are planar, out-of-plane vibrational modes must necessarily be excited in the wedge interior even to first order in an asymptotic expansion of the motion with small parameter being the amplitude of the vibration applied to the wedge faces. In addition, it is demonstrated that this result is fundamentally due to the non-linearity of the problem by demonstrating that the corresponding problem for an incompressible, isotropic, homogeneous linear elastic wedge does not exhibit the same behavior.     

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.     

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.     

Thermal post-buckling of an elastic beams subjected to a transversely non-uniform temperature rising

IntroductionAxiallycompressedstresseswilloccurinaconstrainedelasticbeamsubjectedtoatemperaturerising .Ifthemagnitudeofthecompressedstressesexceedacertainlimit,thermalbucklingoftheheatedbeam ,whichisoutofitsinitialconfiguration ,willtakeplace .So ,investigationsonthermalbucklingofrodsandbeamsareverynecessaryandimportantforthedesignofstructuresworkinginhightemperatureenvironmentsandofsomethermalsensitiveelasticelements.Becausethermalelasticpost_bucklingofbeamsandrodsareinducedbythethermallyaxial…     

Action of a strip-shaped stamp on an elastic incompressible three-dimensional wedge

Moscow University. Translated from Prikladnaya Mekhanika, Vol. 25, No. 8, pp. 19–26, August, 1989.     

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°.     

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.