Boundary resonance in a semi-infinite elastic rigidly fixed waveguide

On the basis of a numerical analysis, a boundary resonance in a semi-infinite elastic waveguide with mixed boundary conditions, when a part of the surface is rigidly fixed while a force which is harmonically dependent on time is imposed on the other part, is found and analysed. Here, the frequency found for the boundary resonance lies below the first stopping frequency of the waveguide so that the boundary resonance in the situation being considered is of the same nature as a resonance in a system without damping: an unlimited increase is observed in the amplitudes of the characteristics of the wave field as the frequency of the vibrations tends to a value Ω_e, unlike the conventional form of a resonance with finite amplitudes [1–3]. The specific behaviour of the amplitude of excitation of a normal wave with a purely imaginary propagation constant in the neighbourhood of the frequency of the boundary resonance Ω_e is found.     

The wave field generated by a centre of rotation in an unbounded elastic medium with a semi-infinite conical crack

The dynamic stress intensity factor at the edge of a semi-infinite conical crack when the medium is loaded by a non-stationary centre of rotation is determined. A centre of rotation is understood to be a set of four forces of equal magnitude that act in the same plane and form pairs having the same direction of rotation.¹ If the magnitude of these forces is time dependent, i.e., their application is non-stationary, they form a non-stationary centre of rotation. The solution of the problem required the use of methods of integral transformations and discontinuous solutions, which reduced the problem to an integral differential equation in Laplace transform space. The combined use of the orthogonal polynomial method and time discretization to solve the equation enabled a formula for the stress intensity factor to be obtained.     

Hertz problem for a rigid punch moving across the surface of a semi-infinite elastic solid

The elastodynamic problem of a rigid punch moving at a constant sub-Rayleigh speed across the surface of an elastic half-space is investigated in the present paper. The unknown contact region is determined as part of solution from the unilateral or Signorini conditions. Numerical results are plotted showing how the eccentricity of the contact ellipse changes with the punch speed. Some asymptotic properties of the solution for the case where the punch speed is comparable with the Rayleigh wave speed are explored in details.     

Reflection and refraction of attenuated waves in semi-infinite elastic solid media

In earthquakes it is known that waves of one type starting from the origin inside the earth produce waves of other types after impact at the free surface of the earth. A theoretical discussion is given here dealing with reflection and refraction of waves which have their amplitudes increasing with depth similar in nature to the initial diverging earthquake waves which after reflection produce the apparent surface waves in which the displacement decreases with depth. In one case it has been found that at a free surface a plane wave of SV type, travelling parallel to the surface, is capable of generating surface waves after internal reflection which under a certain state of attenuation are identical with the Rayleigh Waves.     

On the axisymmetric interaction of a rigid disc with a semi-infinite solid

An analytical treatment is presented for the determination of the response of a vertically-loaded disc embedded in a semi-infinite elastic medium. By means of Love's method of potential and a set of relaxed boundary conditions, the mixed boundary value problem is formulated as dual integral equations with the aid of Hankel transforms. On the reduction of the dual integral equations to a Fredholm integral equation which features a closed-form kernel, solutions to the inclusion problem are computed. In addition to including existing solutions for zero and infinite embedment as degenerate cases, the present analysis reveals a severe boundary-layer phenomenon which is apt to be of significance to this class of problems in general. As illustrations, numerical results on the load-displacement relation, the response of the embedding medium, as well as the contact load distribution are included.

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Résumé Une solution analytique est présentée pour déterminer la réponse d'un disque rigide horizontal chargé verticalement et enfoui dans un milieu semi infini élastique. En utilisant la méthode des potentiels de Love et en imposant un ensemble de conditions aux frontières de type mixte, le problème est formulé à l'aide d'une paire d'équations intégrales au moyen de la transformée d'Hankel. On réduit la paire d'équations intégrales à une simple équation intégrale de type Fredholm pour laquelle on obtient une expression analytique du kernel. En plus d'inclure les solutions classiques d'un disque enfoui à une profondeur nulle ou infinie, le présent travail révèle certains aspects de ce problème général d'interaction, aspects qui n'apparaissent pas nécessairement dans les études précédentes. Une méthode est exposée pour le traitement numérique des différentes intégrales rencontrées. Sont inclus, à titre d'exemples, la distribution de la contrainte verticale de contact, le ratio force-déplacement, et quelques caractéristiques concernant la réponse du milieu.

     

Propagation of shear waves in a semi-infinite elastic waveguide with a local constriction

A procedure is proposed for computing longitudinal-shear stresses and displacements in a semi-infinite waveguide with two lateral cutouts. Numerical results are reported for the stresses on the cutout contours and for the transmission coefficient of the elastic wave.Sumy Branch, Khar'kov Polytechnical Institute. Translated from Dinamicheskie Sistemy, No. 10, pp. 14–20, 1992.     

Nonlinear resonances of a semi-infinite cable on a nonlinear elastic foundation

The Multiple Time Scale (MTS) method is applied to the study of nonlinear resonances of a semi-infinite cable resting on a nonlinear elastic foundation, subject to a constant uniformly distributed load and to a linear viscous damping force. The zero order solution provides the static displacement, which is governed by a nonlinear equation which has been solved in closed form. The first order solution provides the linear resonances, which are seen to be functions of the nonlinearity parameter and of the static displacement at the finite boundary only. Although the first-order governing equation is linear, it has non constant coefficients and cannot be solved in closed form, so that a numerical solution is considered; the eigenfrequencies obtained in this way are also compared with the approximate eigenvalues obtained by the WKB method. At the second order of the MTS expansion, we see that the solution is independent of the intermediate time scale; some additional terms are present, including a time-independent shift of the average position of the oscillations. Finally, the nonlinear frequency–amplitude response curves, which are investigated in detail and which represent the main result of this work, are obtained from the solvability condition at the third order.     

Ovals in a plane coordinatised by a regular nearfield of dimension 2 over its centre

We give a characterisation of the Rosati oval in the regular nearfield plane of dimension 2 over its centre.     

Two-parameter asymptotic approximations in the analysis of a thin solid fixed on a small part of its boundary

Planar elasticity problems are considered for thin domains fixedalong a small part of the end region boundary. The analysisinvolves two small parameters: the normalized thickness of thebody and the normalized length of the fixed part of the boundary.The aim of the paper is to derive an asymptotic approximationof the solution to a boundary-value problem in such a domainand, in particular, analyze the ‘effective boundary conditions’,which occur for the leading-order terms of the asymptotics.We include applications for problems of both anti-plane shearand plane strain elasticity.     

Note on the reconstruction of infinite graphs with a fixed finite number of components

For every positive integer c, we construct a pair G_c, H_c of infinite, nonisomorphic graphs both having exactly c components such that G_c and H_c are hypomorphic, i.e., G_c and H_c have the same families of vertex-deleted subgraphs. This solves a problem of Bondy and Hemminger. Furthermore, the pair G₁, H₁ is an example for a pair of non-isomorphic, hypomorphic, connected graphs also having connected complements—a property not shared by any of the previously known counterexamples to the reconstruction conjecture for infinite graphs.     

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.