Numerical and Asymptotic Solution of the Problem of Oscillations of an Inhomogeneous Waveguide with an Annular Crack of
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SICAL PROBLEMS OF LINEAR ACOUSTICS AND WAVE THEORY
Numerical and Asymptotic Solution of the Problem of Oscillations of an Inhomogeneous Waveguide With an Annular Crack of Finite Width A. O. Vatulyana, b, * and V. O. Yurova, ** a
Southern Federal University, Rostov-on-Don, 344006 Russia Southern Mathematical Institute, Vladikavkaz Scientific Center, Russian Academy of Sciences, Vladikavkaz, 362027 Russia *е-mail: [email protected] **e-mail: [email protected]
b
Received January 27, 2020; revised April 23, 2020; accepted April 28, 2020
Abstract—The problem of waves in an inhomogeneous cylindrical waveguide with an annular crack is considered. A system of integral equations with hypersingular kernels is obtained for finding jumps in radial and longitudinal displacements on the sides of a crack. For the solution, a scheme based on the boundary element method was used. An asymptotic solution to the system of integral equations is constructed for when the defect width tends to zero. The results of computational experiments on comparing solutions obtained by two methods are presented. Keywords: cylindrical waveguide, heterogeneity, crack, system of integral equations, asymptotic analysis DOI: 10.1134/S1063771020050140
INTRODUCTION The use of new materials entails the active development of nondestructive testing methods for structural elements made from them. In this process, research into the wave propagation in inhomogeneous waveguides occupies a certain niche. Despite the fact that homogeneous waveguides have been studied in sufficient detail [1] by analytical methods, the study of wave processes in functionally gradient and piecewise inhomogeneous waveguides is mainly done numerically. To successfully solve the inverse problem of identifying hidden (internal) defects, it is necessary to develop methods for solving direct problems on wave propagation in inhomogeneous waveguides with defects. To study wave processes in homogeneous structures with defects, methods based on the regularities of Lamb wave propagation are mainly used [2]. In [3], delamination dimensions and depths are identified by varying the resonance frequencies. Further progress in studying wave processes in inhomogeneous structures is usually made by division into piecewise-homogeneous zones or by constructing solutions for simple inhomogeneities. An exponential law was used in [4] to describe the deformation of an inhomogeneous plane with a cut. In [5], a method was found for constructing an approximate solution to the problem of a disk-shaped crack in a functionally gradient space by considering a problem in which an arbitrary law of heterogeneity is replaced with an approximation by a certain system of functions.
The mathematical apparatus for solving problems on waves in extended objects with defects is usually based on application of the boundary integral equations (BIE) method [6, 7] or the finite element method (FEM) [8, 9], supplemented by nonreflecting boundary conditions. BIE arise when solving problems with open cracks, and the inte
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