Contact with break-off under bending of an elastic strip with a rigid disk

We consider the problem of the theory of elasticity of the contact interaction of a rigid circular disk and an elastic strip, which rests upon two supports with disturbance of contact in the middle part of the contact region. On the basis of the Wiener–Hopf method, an integral equation of the problem is reduced to an infinite system of algebraic equations. The size of the zone of break-off of the boundary of the strip from the disk and the distribution of contact stresses are determined.     

The impact of a plane punch on an elastic half-plane

The transient dynamic contact problem of the impact of a plane absolutely rigid punch on an elastic half-plane is considered. The solution of the integral equation of this problem in terms of the unknown Laplace transform of the contact stresses at the punch base is constructed by a special method of successive approximations. The solution of the transient dynamic contact problem is obtained after applying an inverse Laplace transformation to the solution of the integral equation over the whole time range of the impact process, and the law of the penetration of the punch into the elastic medium is determined from a Volterra-type integrodifferential equation. The conditions for the punch to begin to separate from the elastic half-plane are formulated from the solution obtained, and all the stages of the separation process are investigated in detail. The law of the punch motion on the elastic half-plane and the width of the contact area, which varies during the separation, are then determined from the solution of the Volterra-type integrodifferential equation when an additional condition is satisfied.     

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.     

Problem of an elastic half-plane with a crack emerging orthogonally at the boundary

The problem of the half-plane, in which a finite crack emerges orthogonally at the boundary, is studied. On the edges of the crack a self-balancing load is applied. A detailed investigation is carried out for an integral equation with respect to the unknown derivative of the displacement jump, to which the problem can be reduced. The exact solution of the integral equation is constructed with the aid of the Mellin transform and the Riemann boundary value problem for the halfplane. The asymptotic behavior of the solution at both ends of the crack is elucidated. First the asymptotic behavior of the solution at the point of emergence of the crack is obtained and the dependence of this asymptotic behavior on the type of the load is established. For a special form of the load one obtains a simple expression of the stress intensity coefficient. In the case of a general load, the asymptotic behavior is used for the construction of an effective approximate solution on the basis of the method of orthogonal polynomials. As a result, the problem reduces to an infinite algebraic system, solvable by the reduction method.Translated from Dinamicheskie Sistemy, No. 4, pp. 45–51, 1985.     

Optimization of the stress tensor in an elastic anisotropic half-plane

We consider the problem of optimizing the components of the stress tensor and their integral characteristics. The normal and tangential forces prescribed on the boundary of the elastic anisotropic half-plane y≥0 are chosen from certain function classes of curvilinear strip type. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 110–116.     

Circular inclusion in an infinite elastic strip

This paper is concerned with the problem of a circular inclusion undergoing spontaneous dimensional changes in an infinite elastic strip with the straight edges free from displacements. Consequent elastic fields both in the inclusion and the surrounding strip are determined with the aid of complex variable technique. Closed form expressions for two sets of complex potentials and for various stress components are provided. Boundary stresses have been computed and their behaviour examined for varying situations.

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Zusammenfassung Die Arbeit betrifft das Problem eines kreisrunden Fremdkörpers, der sich innerhalb eines unendlichen elastischen Streifens mit verschiebungslosen Rändern deformiert. Die elastischen Felder im Fremdkörper und im Streifen werden mit der komplexen Methode bestimmt, und es werden für zwei Sätze komplexer Potentiale und für verschiedene Spannungskomponenten geschlossene Lösungen angegeben. Die Randspannungen werden berechnet und für verschiedene Fälle untersucht.

     

Interaction of elastic waves with a curvilinear crack in a half-plane

We consider the problem of the interaction of monochromatic displacement waves with a curvilinear crack-cut in a half-plane. We find integral representations of the solution. The boundary-value problem is reduced to a system of singular integral equations. A parametric investigation is carried out for the effect of the form of the load, the fastening conditions on the boundary of the half-plane, and the curvature of the crack on the dynamic coefficients of the stress intensity.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 77–82, 1988.     

A note on composition operators in a half-plane

We provide conditions for composition operators on the Hardy space of the upper half-plane to have a closed range. We also show that, unlike the situation of the unit disk, the operator of composition with an analytic self-map Φ of the upper half-plane can be similar to an isometry even when Φ is far from being an inner function.     

The dynamics of the motion of a flat punch on the boundary of an elastic half-plane

The dynamic contact problem of the motion of a flat punch on the boundary of an elastic half-plane is considered. During motion, the punch deforms the elastic half-plane, penetrating it in such a manner that its base remains parallel to the boundary of the half-plane at each instant of time. In movable coordinates connected to the moving punch, the contact problem reduces to solving a two-dimensional integral equation, whose two-dimensional kernel depends on the difference between the arguments for each of the variables. An approximate solution of the integral equation of the problem is constructed in the form of a Neumann series, whose zeroth term is represented in the form of the superposition of the solutions of two-dimensional integral equations on the coordinate semiaxis minus the solution of the integral equation on the entire axis. This approach provides a way to construct the solution of the two-dimensional integral equation of the problem in four velocity ranges of motion of the punch, which cover the entire spectrum of its velocities, as well as to perform a detailed analysis of the special features of the contact stresses and vertical displacements of the free surface on the boundary of the contract area. An approximate method for solving the integral equation, which is based on a special approximation of the integrand of the kernel of the integral equation in the complex plane, is proposed for obtaining effective solutions of the problem that do not contain singular quadratures.     

On an integral equation of the problem for an elastic strip with a slit

A new singular integral equation is obtained that describes the elastic equilibrium of a strip with both an inner and an edge slit (crack) and has a considerable advantage over existing equations /1–9/, etc.) from the viewpoint of a numerical realization and clarification of the analytical relationship with an analogous equation for a half-plane. Numerical results are given of a computation of the stress intensity coefficients at the tips of the inner and edge cracks that refine data in the literature.     

Asymptotic analysis of the interaction of a finite number of holes in an elastic plane or half-plane

The problem of the interaction of a finite number of holes in an elastic plane or half-plane is considered. The analysis is based on the complex potential method of Muskhelishvili as well as on the theory of compound asymptotic expansions by Maz'ya. An asymptotic expansion of the complex potentials in terms of relative hole radii is constructed. This expansion is uniformly valid in the whole domain. The method leads to a simple procedure which does not involve any coupled system of linear equations. The successive closed-form approximations can be obtained in an iterative manner to an arbitrary order without any need for numerical approximation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dynamic saint-venant principle for an elastic semi-infinite strip

We consider the problem of longitudinal deformation of an elastic semi-infinite strip under the action of various kinds of self-balanced harmonic loads. Using numerical analysis we show that in a region of relatively low frequencies, when only one propagating wave exists in the wave conductor, one can formulate an analog of Saint-Venant's principle. When this is done, the given frequency range decomposes into two subranges. In the first of these the main role in the Fourier series expansion of the load is played by the non-self-balanced component, while in the second it is necessary to take account of the first harmonic as well. One figure. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 40–46, 1991.     

Integral representations for some classes of functions holomorphic in a strip or in a half-plane

В данной работе рассм атриваются классы фу нкцийf(z), голоморфные в област иa (?∞<a<b≦+∞) приp≧1 иs≧0, и у довлетворяющие одному из следующих условий:

1. Еслиb≦+∞, то $$\int\limits_a^b {(\int\limits_{ - \infty }^{ + \infty } {\left| {f\left( {x + iy} \right)} \right|^p } dy)^s dx< + \infty .} $$
2. Еслиb=+∞, иa=0, то $$\int\limits_0^u {(\int\limits_{ - \infty }^{ + \infty } {\left| {f\left( {x + iy} \right)} \right|^p } dy)^s dx \leqq \varrho \left( u \right), u > 0,} $$ где?(u) — функция опред еленного роста.

Результаты работы су щественно обобщают т еорему Пэли—Винера о параме трическом представлений класс аH ₂ на полуплоскости.     

Impression of a semiinfinite stamp in an elastic strip with regard for friction and adhesion

We consider the problem of contact interaction between a semiinfinite stamp with rectilinear base and an elastic strip with one rigid side. Friction forces in the contact region are taken into account. These forces lead to the division of the contact region into slipping and adhesion zones. With the use of the Wiener–Hopf method, a system of integral equations is reduced to an infinite system of algebraic equations. The computational results of stresses and strains at the boundary and at inner points of the elastic strip are presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 138–149, January–March, 2008.     

On generalized formulation of the equilibrium problem of an elastic strip

A “nonenergetic” formulation of the boundary value problems of statics of an elastic strip based on the principle of admissible displacements, is studied. The formulation makes possible, in particular, the study of problems concerning the strips of infinite energy, while retaining the external form of the “energetic” formulation /1–3/, and produces unique solvability of the problem under weaker restrictions imposed on the external loads. Such a formulation is also possible for other problems of the theory of elasticity.