Optical theorem for the scattering of waves in an elastic plate

The scattering of waves in an elastic plate is investigated. A generalization of the optical theorem is found for the case of a noncompact scatterer. The formulas obtained can be used for indirect control of the correctness of computing diffraction fields.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 20–23, 1981.The author is grateful to D. P. Kouzov for his constant attention to the work.     

Differentiation of the energy functional in the equilibrium problem for a Timoshenko plate with a crack on the boundary of an elastic inclusion

Under consideration is the equilibrium of a composite plate containing a through vertical crack of variable length at the interface between thematrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion, by the Kirchhoff–Love model. Some formula is obtained for the derivative of the energy functional with respect to the crack length.     

Stability of a square plate with an elastic border

We study the problem of stability of a square plate subject to uniform compression in both directions. The elastic border of the plate is characterized by the four stiffness coefficients (corresponding to the number of sides) in relation to the angle of rotation. We obtain an approximate formula for computing the critical load. For the cases when the plate has hinge support along the entire border or is rigidly clamped the results determined by this formula practically coincide with the exact solutions. One table. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 46–50, 1991.     

Modeling a swimming fish with an initial boundary value problem: Unsteady maneuvers of an elastic plate with internal force generation

In order to model unsteady maneuvers in swimming fish, we develop an initial-boundary value problem for a fourth-order hyperbolic partial differential equation in which the fish's body is treated as an inhomogeneous elastic plate. The model is derived from the three-dimensional equations of elastic dynamics, and is essentially a simple variant of the classical Kirchhoff model for a dynamic plate. The model incorporates body forces generating moment to simulate muscle force generation in fish. The initial-boundary value problem is reduced to a beam model in one spatial dimension and formulated computationally using finite differences. Interaction with the surrounding water is represented by nonlinear viscous damping. Two example applications using simple but physically reasonable physiological parameters are presented and interpreted. One models the acceleration from rest to steady swimming, the other a rapid turn from rest.     

The stressed state of an elastic space with a rectangular crack

By using the method of mechanical quadratures we reduce the resolvent integral equation for the problem of an infinite elastic space with a rectangular crack to a system of linear algebraic equations. We give the results of numerical experiments in varying the stress intensity factor on one side of the crack in the case of tension in the direction perpendicular to the plane of the crack. One figure. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 24–28.     

Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensional zero-range processes

We prove a non-equilibrium functional central limit theorem for the position of a tagged particle in mean-zero one-dimensional zero-range process. The asymptotic behavior of the tagged particle is described by a stochastic differential equation governed by the solution of the hydrodynamic equation.     

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.     

On the numerical solution of a hypersingular integral equation for elastic scattering from a planar crack

In this paper we describe a fully discrete quadrature method for the numerical solution of a hypersingular integral equationof the first kind for the scattering of time-harmonic elasticwaves by a cavity crack. We establish convergence of the methodand prove error estimates in a Hölder space setting. Numerical examples illustrate the convergence results. Received 30 November, 1998. Revised 22 November, 1999.     

Thermal stresses in an elastic sphere containing a penny-shaped crack

Zusammenfassung Es wird der Spannungszustand in einer elastischen Kugel untersucht, die im Zentrum einen münzenförmigen Riss mit konstanter Oberflächentemperatur besitzt, während die sphärische Oberfläche auf der Temperatur null gehalten wird. Dabei wird angenommen, dass an der Oberfläche a) die Schubspannung und die radiale Verschiebung, b) die Normalspannung und die Schubspannung null sind.

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This work was supported by a C.S.I.R. grant No. 3(277)/69-GAU-II.     

A normal crack in an elastic half-space with stress-free surface

The problem of an elastic half-space with stress-free surface and a crack of arbitrary shape with prescribed displacements or tractions is reduced to an equivalent system of integral equations on the crack. For a pressurized crack in a plane perpendicular to the free surface, a scalar integral equation is derived. In properly chosen function spaces, unique solvability of the integral equation and regularity of solutions for regular data are proven.     

On bending an elastic plate with a delaminated thin rigid inclusion

Under study is the problem of bending an elastic plate with a thin rigid inclusion which may delaminate and form a crack. We find a system of boundary conditions valid on the faces of the crack and prove the existence of a solution. The problem of bending a plate with a volume rigid inclusion is also considered. We establish the convergence of solutions of this problem to a solution to the original problem as the size of the volume rigid inclusion tends to zero.     

Vibrations of an elastic plate partially immersed in a liquid

The free vibrations of an elastic plate, placed orthogonally at the bottom of an unbounded open reservoir of constant depth, are considered. The reservoir is filled with a compressible liquid, the plate protrudes above the free surface and its rigidly fastened to the bottom. Gravitational effects are ignored. An analytical representation of the acoustic field in the liquid and the vibrational field in the plate is obtained. The natural frequencies and natural forms of vibration are calculated as a function of the height of the liquid level. An approximate approach to calculating the vibrations of the plate is considered.     

Resonances of an elastic plate in a compressible confined fluid

We present a theoretical study of the resonances of a fluid–structureproblem, an elastic plate placed in a duct in the presence ofa compressible fluid. The case of a rigid plate has been largelystudied. Acoustic resonances are then associated to resonantmodes trapped by the plate. Due to the elasticity of the plate,we need to solve a quadratic eigenvalue problem in which theresonance frequencies k solve the equations (k) = k², where are the eigenvalues of a self-adjoint operator of the formA + kB. First, we show how to study the eigenvalues locatedbelow the essential spectrum by using the min–max principle.Then, we study the fixed-point equations. We establish sufficientconditions on the characteristics of the plate and of the fluidto ensure the existence of resonances. Such conditions are validatednumerically.     

The scattering of a plane flexural wave by a sector of a thin elastic plate with a supported edge

A solution is constructed of the problem of the diffraction of a plane flexural wave at the vertex of a thin elastic plate cut out in the form of a sector, on the edge of which the condition of a supported edge (hinged support) is specified.