Issue 45

O. Reut et alii, Frattura ed Integrità Strutturale, 45 (2018) 183-190; DOI: 10.3221/IGF-ESIS.45.16 189 1 2 2 1 2 0 ( , ) ( , ) 2 ( , ) 2 (sin ) ( , ) ( , ) , n n n n k s r s s ctg w r in v r J rs s d s c G                                    (17) The jump of wave function 2 ( , ) n r   is found by the analogical procedure 2 0 ( , ) ( ( , ) ( , ) ( , ) n n n k r s r u r J rs s d               (18) The jumps of the wave functions derivatives are constructed using this scheme. For example, the expression for the derivative of wave potential jump 2 ( , ) n r   has the form   1 2 1 0 ( , ) ( ( , ) sin ( , ) ) r n n n r r v in d                 (19) As a result, all transformations of the wave functions and their derivatives jumps are expressed through the transformations of stress and displacements jumps. There are substituted to the corresponded formulae of the Helmholtz’s equation discontinuous solution and inversion of the integrals transformations is done. The following substitution of these formulae to the expressions (13) finalizes the deriving of the discontinuous solutions of Lame’s equations in a case of steady state oscillations for the defect (1). C ONCLUSIONS 1. The discontinuous solution of Helmholtz’s equation is derived for a conical defect. 2. To construct the discontinuous solutions of Lame’s equations for a conical defect one must express the jumps of wave functions and their derivatives through the displacements and stress jumps. The substitution of these formulas to the discontinuous solution of Helmholtz’s equation leads to the discontinuous solutions of the Lame’s equations. 3. In the case when a defect is a crack, the jumps of the stress are equal to zero, when a defect is a thin shell adherent to a medium the jumps of the displacements are equal to zero. So, the derived formulae can be used for the different types of the conical defects situated in a medium. 4. The derived formulae will be significantly simplified when the axisymmetrical problem is solved. In this case parameter n should be equal to zero in all final expressions. A CKNOWLEDGMENTS he authors are grateful to Simon Peter Dyke for his attention and great help in the editing of the manuscript’s text. R EFERENCES [1] Babeshko, V. A., Babeshko, O. M. and Evdokimova, O. V. (2010). On the method of block element, Mechanics of Solids, 45, pp. 437-444. DOI: 10.3103/S0025654410030143. [2] Grinchenko, V. T. and Meleshko, V. V. (1981). Harmonical oscillations and waves in elastic bodies (in Russian), Naukova dumka, Kyiv. T