Equilibrium of an elastic layer weakened by plane cracks

The spatial problem of the elastic equilibrium of a layer in whose middle plane there is a system of cracks is considered. The cracks are maintained open under the action of a normal load applied to their edges. The layer faces are compressed between two rigid smooth foundations. The problem is reduced to solving an integral equation of the first kind. The asymptotic methods of “large and small λ” /1/ as well as the method of successive approximations and a variational method are used to construct the solutions of this equation for elliptically and rectangularly shaped cracks in different ranges of variation of the geomtrical parameters.     

Geometrically nonlinear dilation of an orthotropic plane with an elliptical hole

In a geometrically nonlinear formulation we obtain expressions for the stresses and displacements on the edge of an elliptical hole in an orthotropic plane dilated by a strain at infinity in the direction of the principal axes of anisotropy. We exhibit the dependence of the concentration of stresses on the magnitude of the load acting on the plane. We prove that the stresses are finite at the ends of the crack and we compute their values. We confirm the need to take account of the geometric nonlinearity of the problem for low-modulus orthotropic materials.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 90–96.     

Stressed state of an elastic plane weakened by a narrow slot with partly closed edges

An elastic plane weakened by a narrow rectilinear slot with rounded ends is considered. The plane is compressed by force P at angle a to the slot axis and with force P in the perpendicular direction. Central areas of the slot edges close under the action of compression. Their reaction in relation to the ratio of parameters of the problem has the nature of sticking together or Coulomb friction. The stress-strain state of the system described is studied.Translated from Dinamicheskie Sistemy, No. 4, pp. 25–33, 1985.     

The equilibrium of an elastic space weakened by two spherical cavities and an external circular crack

Using the relationship between the basic solutions of Laplace's equation in toroidal and spherical coordinates, the Fourier method is employed to solve the problem of the equilibrium of an elastic space weakened by two spherical cavities and an external circular crack. The proposed approach leads to an infinite system of linear algebraic equations of the second kind with exponentially decaying matrix coefficients. A small-parameter expansion is used to obtain an asymptotic formula for the normal stress intensity factor.     

Deformation of a composite elastic plane weakened by a periodic system of the arbitrarily loaded slits

The paper deals with the problems of periodic system of cuts distributed along the boundary of a bond connecting two elastic half-planes and acted upon by nonperiodic loads. In one problem it is assumed that the cuts are open, with normal and tangential stresses applied to their edges, while in another problem the edges touch each other and are loaded by tangential stresses. The method of solution is based on the simultaneous use of the discrete Fourier transformation and the theory of boundary value problems for automorphous analytic functions. The solutions are otained in quadratures. Other classes of problems to which the proposed methods can be applied, are described.

Generally speaking, in the case of irregular loads, the solution is usually based on the theory of representation of the symmetry groups /1,2/, and in the case of certain types of symmetry, particularly the translational, on the discrete Fourier transforms /3– 6/. However the objects of transformation may be different in one and the same problem, and their choice affects significantly the solvability of the boundary value problem for the transformed quantities in the cell of periods. Below two problems of the theory of cracks are solved in quadratures to illustrate the effective simultaneous use of the discrete Fourier transformation and the Muskhelishvili method.     

A consistent solution of the problem of one-sided tension of an elastic plane with an astroidal hole

We consider the stress-strain state of an elastic plate with a hole in the form of an astroid under one-sided tension. The contact of edges of the hole near two opposite tips of the astroid is taken into account, which eliminates the contradiction with the classical solution concerning the overlapping of edges of the hole. We determined both the length of the regions of contact pressure and the distribution of contact stresses. For two contact-free tips of the hole, the value of the stress intensity factor is calculated.     

The decay of an arbitrary initial discontinuity in an elastic medium

The solution of the problem of the decay of an arbitrary discontinuity in elastic theory is studied. It is assumed that a plane boundary separates an elastic homogeneous, non-heat-conducting medium into two half-spaces with different elastic properties and densities. Each of the media possesses an arbitrary kind of homogneous initial strain (stress) and velocity. In the sequel the stresses and velocities of the media are assumed to be continuous at the boundary. This results in the formation of a system of plane selfsimilar waves (simple and shock), which propagate in each of the half-spaces. The problem is solved under the assumption of weak non-linearity and anisotropy of the materials. This permits an approximate evaluation of the stress and strain at the contact discontinuity. After this the problem on the decay of an arbitrary initial discontinuity is reduced to two problems on the sudden change of load on a half-space boundary, which are solved independently for each of the media.     

Realizations for Schur upper triangular operators centered at an arbitrary point

Reproducing kernel spaces introduced by L. de Branges and J. Rovnyak provide isometric, coisometric and unitary realizations for Schur functions, i.e. for matrix-valued functions analytic and contractive in the open unit disk. In our previous paper [12] we showed that similar realizations exist in the nonstationary setting, i.e. when one considers upper triangular contractions (which appear in time-variant system theory as transfer functions of dissipative systems) rather than Schur functions and diagonal operators rather than complex numbers. We considered in [12] realizations centered at the origin. In the present paper we study realizations of a more general kind, centered at an arbitrary diagonal operator. Analogous realizations (centered at a point of the open unit disk) for Schur functions were introduced and studied in [3] and [4].     

Electromagnetic fields induced by electric current in an infinite plate with an elliptical hole

Analytical solutions to the electromagnetic field in a thinconductive plate with an elliptical hole are derived by meansof complex potentials and conformal mapping techniques. Thesteady-state current field in a thin conductive plate is twodimensional (2D) and is explored by a standard complex variabletechnique. The current is disturbed around the elliptical hole,and produces a three dimensional magnetic field. In this case,using the complex variable method to solve the real magneticfield can be challenging. The magnetic boundary conditions takedifferent forms for the soft ferromagnetic and the para- ordiamagnetic materials under consideration. A simplified analysistaking account of the magnitude of the magnetic permeabilityof the magnetic material and air surrounding the material isproposed to reduce the magnetic field in a thin plate to 2Dcalculations. The magnetic field distributions are derived foreach material and the equations of the magnetic components atthe tip of elliptical hole are presented.     

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.     

The diffraction of a plane compressional wave by a spherical cavity in an elastic medium

Zusammenfassung Ein sphärischer Hohlraum in einem unendlichen homogenen und isotropen elastischen Medium wird von einer ebenen Druckstosswelle überstrichen, deren Front sich gleichförmig fortbewegt. Das durch Ablenkung am Hohlraum erzeugte Verschiebungs- und spannungsfeld wird mittels einer Integraltransformation bestimmt. Insbesondere wird die starre Teilbewegung der Begrenzung des Hohlraums ermittelt. Das Problem wird für eine einstufige Druckwelle gelöst. Die Ergebnisse lassen sich als Einflussgrössen betrachten, mit denen sich die Duhamel-Integrale für allgemeinere Druckwellen berechnen lassen.

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This work was performed under Contract No. AF 29(601)-5132 (Project No. 1080, Task No. 108007) for Air Force Special Weapons Center, Kirtland Air Force Base, New Mexico, USA.     

Approximation of convolutions of probability distributions by infinitely divisible laws under weakened moment restrictions

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 194, pp. 79–90, 1992.     

A case of plane rotations of an elastic pendulum

The motion of a point mass, suspended on a spring in a uniform gravity field, is investigated. The spring is assumed to be weightless and to possess linear elasticity. Motion occurs in a specified fixed vertical plane. It is shown that a pendulum motion exists in which the angle, made by the axis of the spring and the vertical, varies uniformly with time. The problem of the orbital stability of this motion is solved.     

On the one-dimensional scattering of plane waves on an arbitrary potential

Theoretical and Mathematical Physics -