Issue 45

O. Reut et alii, Frattura ed Integrità Strutturale, 45 (2018) 183-190; DOI: 10.3221/IGF-ESIS.45.16 184 The jumps of the function and its normal derivative are given by the conditions of the original problem, both are established under certain conditions of the problem solving. Under a defect one must understand 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. G.Ya. Popov proposed a method for constructing such solutions for defects and bodies, described in the orthogonal curvilinear coordinate systems. It is essential, corresponding to this method, to construct a discontinuous solution of a Helmholtz’s equation (or the Laplace equation in a static statement of a problem) and to further construct discontinuous solutions of the equations of motion (or equilibrium equations respectively). It is possible to realize this thanks to formulas connecting the wave potentials with the displacements and stress. In [15] the solutions of elasticity static problems were constructed for linear, circular, spherical and cylindrical defects. The method of discontinuous solutions was extended to the problem of wave diffraction in papers [16-18]. In [19] the method of discontinuous solutions is extended to the defect of an arbitrary form. The novelty of the proposed paper is in the construction of the discontinuous solutions of Lame’s equations for a conical defect in the case of a steady state loading. As a first stage of the solution deriving, a discontinuous solution of the Helmholtz equation for a case of a conical shape defect is constructed. The special scheme is proposed to find the unknown jumps of the displacements and stress. The derived formulae of Lame’s equations discontinuous solutions can be applied for the solving of the boundary problems of elasticity for the different types of the conical defects, such as a crack, a thin inclusion adherent with a medium, partially adherent inclusion etc. For the case of an axisymmetric problem for a conical defect all obtained formulae are substantially simplified. S TATEMENT OF THE PROBLEM et’s consider an acoustic medium containing a conical defect whose surface is described in a spherical coordinate system by the correspondences: , 0 , a r b             (1) The steady-state oscillations of the media are described by the Helmholtz equation         2 2 2 ' , , ' , , , , 0 i r r r r r c                      (2) here     , , , , , i t r t r e          , c is the wave’s speed in the acoustic medium. Here it was agreed to disregard the designation "tilda" over a letter and to introduce the following new notations 2 2 2 2 i q c c            , where  is the frequency of the incident wave,     2 2 sin , , , , sin sin r r                    , (here the point over the letter denotes a derivative with regard to a second variable). The aim is the deriving of the discontinuous solution of the Eqn. (2) for the defect (1) located in acoustic medium. D ERIVING OF THE DISCONTINUOUS SOLUTION IN THE TRANSFORMATION DOMAIN he integral Fourier transformation is applied to the equation with regard of variable      , , , in n r e r d             (3) In the transformation’s domain (3) the Eqn. (2) takes a form L T