Diffraction of elastic waves

Guz, A. N.; Kubenko, V. D.; Cherevko, M. A.

doi:10.1007/bf00883678

Cited by 107 publications

(55 citation statements)

References 31 publications

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“…The construction of their analytical solutions, analysis of wave fields in the vicinity of t defects constitute a broad class of problems whose decompositions require the involvement of complex mathematical apparatus. The development of this mathematical apparatus has been carried out by many scientists [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. One of the powerful methods for solving problems of wave diffraction on defects of various forms is the method of discontinuous solutions.…”

Section: Introductionmentioning

confidence: 99%

The discontinuous solutions of Lame’s equations for a conical defect

Vaysfeld

Reut

2018

Frattura Ed Integrità Strutturale

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ABSTRACT. In this article the discontinuous solutions of Lame's equations are constructed for the case of a conical defect. Under a defect one considers a part of a surface (mathematical cut on the surface) when passing through which function and its normal derivative have discontinuities of continuity of the first kind. A discontinuous solution of a certain differential equation in the partial derivatives is a solution that satisfies this equation throughout the region of determining an unknown function, with the exception of the defect points. To construct such a solution the method of integral transformations is used with a generalized scheme. Here this approach is applied to construct the discontinuous solution of Helmholtz's equation for a conical defect. On the base of it the discontinuous solutions of Lame's equations are derived for a case of steady state loading of a medium.

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Section: Introductionmentioning

confidence: 99%

The discontinuous solutions of Lame’s equations for a conical defect

Vaysfeld

Reut

2018

Frattura Ed Integrità Strutturale

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“…In this connection, efficient parallel algorithms based on substantiated analytical methods [3] take on special significance. In solving plane and antiplane problems of diffraction theory [4,5], the method of integral equations [6][7][8] is highly efficient. Additional advantages of this approach lie in reducing the number of spatial variables, a rather high convergence rate, and the possibility of applying various efficient numerical solution methods [6].…”

Section: Introductionmentioning

confidence: 99%

Framework analysis of the object domain of stationary dynamic problems of the elasticity theory for isotropic media with arbitrary discontinuities

Панченко

Назаренко

2013

Cybern Syst Anal

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A general scheme of framework analysis of an object domain is considered. The domain involves a wide class of problems of the theory of dynamic elasticity, which allows one to develop a software tool for the synthesis of applications (a CASE method) of parallel high-precision numerical solution of this class of problems. This article proposes a scheme of parallel high-precision solution to systems of singular integral equations that occur in the analysis of certain problems of diffraction of elastic harmonic waves on systems of discontinuities of an arbitrary cross-section. New computational results are presented.Keywords: relational framework, relational database schema, object domain, dynamic problem of elasticity theory, isotropic medium with arbitrary discontinuities, CASE tool.

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“…We sequentially apply the Laplace transform with respect to time and an integral transform with respect to q to the wave equations (1.3) [5]: The transformations reduce the system of equations (1.3) to systems of ordinary differential equations of the second order: which automatically satisfies the boundary condition (1.2). This can be demonstrated with the formulas relating the displacements and stresses with the wave potentials [4]:…”

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confidence: 99%

Dynamics of a truncated elastic cone

2011

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The problem of determining the nonstationary wave field of an elastic truncated cone with nonzero dead weight is formulated in terms of wave functions. The Laplace transform with respect to time and an integral transform with respect to time polar angle are used to reduce the problem to a one-dimensional vector problem in the transform domain. The transforms of the wave functions are expanded into series in inverse powers of the Laplace transform parameter, which makes it possible to study the wave process at the initial instants of interaction. A method is proposed to solve the problem for an elastic cone doubly truncated by spherical surfaces Keywords: truncated elastic cone, nonstationary wave field, Laplace transform, transform with respect to polar angle, cone doubly truncated by spherical surfaces, initial instants of interaction Introduction. The stress state of conical bodies under static loads is studied in many publications. For example, the axisymmetric equilibrium of a circular cone was first addressed in [10]; the equilibrium of a cone subjected to a concentrated force at the vertex was studied in [14]; the equilibrium of a cone subjected to a bending moment at the vertex was studied in [11]; a mixed problem for an infinite cone was solved in [13]; the stress state of a truncated semi-infinite cone was analyzed in [15]; the exact solution of an axisymmetric problem for a cone with a center of rotation at the vertex was first obtained in [6]; the exact solution of an axisymmetric problem for a circular hollow cone was found in [7].Far fewer publications address dynamic elastic problems for conical bodies. Experimental results on the propagation of elastic waves in a truncated elastic cone are presented in [12]; the analytic solution of an axisymmetric dynamic problem for a hemispherical dome, which is a special case of an elastic cone twice truncated by spherical surfaces, was found in [9]; a dynamic problem for an elastic cone under a dynamic load was solved in [2] disregarding the dead weight.1. Problem Formulation. Consider a truncated elastic cone at rest. Its position in a spherical coordinate system is described by 0 0 < < < < -£ £ r a, , q w p j p (Fig. 1, section zOx). A nonstationary axisymmetric load f t ( , ) 0 is applied at time t = 0 to the spherical surface r a = of the cone:

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Diffraction of elastic waves

Cited by 107 publications

References 31 publications

The discontinuous solutions of Lame’s equations for a conical defect

The discontinuous solutions of Lame’s equations for a conical defect

Framework analysis of the object domain of stationary dynamic problems of the elasticity theory for isotropic media with arbitrary discontinuities

Dynamics of a truncated elastic cone

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