Issue 45

O. Reut et alii, Frattura ed Integrità Strutturale, 45 (2018) 183-190; DOI: 10.3221/IGF-ESIS.45.16 183 The discontinuous solutions of Lame’s equations for a conical defect O. Reut, N. Vaysfeld Odessa Mechnikov University, Institute of Mathematics, Economics and Mechanics, Ukraine reut@onu.edu.ua , vaysfeld@onu.edu.ua A BSTRACT . 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. K EYWORDS . Conical defect; Helmholtz’s equation; Wave potential; Integral Transformation; Lame’s equations. Citation : Reut, O., Vaysfeld, N., The discontinuous solutions of Lame’s equations for a conical defect, Frattura ed Integrità Strutturale, 45 (2018) 183-190. Received : 15.05.2018 Accepted : 24.06.2018 Published: 01.07.2018 Copyright: © 2018 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. I NTRODUCTION he urgency of the problem of elastic waves diffraction is due to the need to take into account the presence of heterogeneities during the development of new composite materials, geophysical and seismological studies. These processes take place under dynamic loads of different nature. In order to facilitate engineering design, preliminary calculations are required on the basis of appropriate mathematical models that provide an opportunity to analyze the effects of such dynamic stress concentrators as inclusions, cavities, cracks, holes, etc. On the other hand, the problems of elastic wave diffraction are one of the classical problems of the mechanics of deformable bodies. 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 [1-14]. One of the powerful methods for solving problems of wave diffraction on defects of various forms is the method of discontinuous solutions. This method was created by G. Ya. Popov [15]. He gave a definition of a discontinuous solution of the differential equation in partial derivatives, namely: a discontinuous solution of a certain differential equation in partial derivatives is such a solution that satisfies this equation throughout the region of determining an unknown function, with the exception of the defect points, in the transition through which an unknown function has discontinuities with jumps of the unknown function itself and its normal derivative. T