Issue 45

O. Reut et alii, Frattura ed Integrità Strutturale, 45 (2018) 183-190; DOI: 10.3221/IGF-ESIS.45.16 187         1 2 2 1 2 0 ( ) ( ), 2 ( , ) , 1/ 2 2 ( ) ( ), 1/ 2 i i k J q H xq x sh K x K i J x d k x J xq H q x k                                       The discontinuous solution can be simplified 0 ( , ; , ) , sin ( , ( , ; , ) , ) , , n n n n n G x x G x d x rq q q q q                                               (12) here     ( , ; , ) cos cos ( , ) n n n kn k k k k n G x P P J x            . It was stated that the limit values of the wave potential near the branches of the defect (1) have the form 0 1 , 0 , 2 ( , ; , ) sin ( , , ( , ; , ) ) , , 0 0 n n n n n n x x q q G x G x d x rq q q q                                                                   These formulas are derived by the use of the known facts of potential theory such as a discontinuity of a double layer’s potential and normal derivative of plane layer potential. The application of inverse integral Fourier’s transformation to formula (12) completes the construction of discontinuous solution of Helmholtz’s equation. C ONSTRUCTION OF THE DISCONTINUOUS SOLUTIONS OF L AME ’ S EQUATIONS ccordingly to the discontinuous solution method [15] to derive the discontinuous solutions of Lame’s equations, one must find the formulae expressing the jumps of the wave potentials and their derivatives through the jumps of the displacements and stress. The wave potential functions     , , , , , , 1, 2 j r r j        satisfy the Helmholtz’s Eqns. (2) with the velocities 1 2 , c c correspondently. Well known formulas in the Fourier’s transformation domain (3) are written in the form [11]: 1 2 ( , ) ( , ) ( , ) n n n n u r r r r           , 1 1 2 1 ( , ) ( ( , ) ( ( , )) ) sin ( ) ( , ) n n n n v r r r r r in r                 , (13) 1 2 1 ( , ) ( sin ) ( ( , ) ( ( , )) ) ( , ) n n n n w r in r r r r r               , here ( , ) ( , ), ( , ) ( , ), ( , ) ( , ) n n n r n n n U r u r U r v r U r w r            . The stress transformations are expressed through the displacements and hence through the wave potentials as well. To use the discontinuous solution of the Helmholtz’s Eqn. (2) derived earlier, the jumps’ transformations of the wave functions     , , , , 1, 2 n jn r r j      should be expressed through the jumps of the stress and displacements. This procedure is enough complicated, so here its scheme is shown. One must use such equalities 2 2 2 ( , ) ( , ), n n c n r L r       1 ( , ) ( , ), n n c n r L r       2 2 2 2 ( ) c L y r y r c       . With regard of these formulas, it is possible to express the jumps of the mechanical characteristic through the jumps of the wave potentials: A