Issue 44

V. Reut et alii, Frattura ed Integrità Strutturale, 44 (2018) 82-93; DOI: 10.3221/IGF-ESIS.44.07 86         1 3 1 2 2 4 0 , ( ) a c c y x Y x Y x G x f d c c                       (10) where     1 2 , Y x Y x are the system of fundamental matrix solutions, , 1, 4 i c i  are known constants,   , G x  is the Green’s matrix function [29]. The expression (10) can be rewritten in scalar form                                 11 12 11 12 11 1 1 1 2 2 3 2 4 0 12 1 1 12 11 1 1 0 0 0 11 1 1 12 2 2 0 0 3 ( ) , ' 1 1 1 , ' sin , sin , 1 1 3 1 cos , cos , 1 1 c c c c c c c c u x Y x c Y x c Y x c Y x c G x d G x d B G x d B x d G B x d B G x d                                                                                (11)                                 21 22 21 22 21 1 1 1 2 2 3 2 4 0 22 1 1 22 21 1 1 0 0 0 21 1 1 22 2 2 0 0 3 ( ) , ' 1 1 1 , ' sin , sin , 1 1 3 1 cos , cos , 1 1 c c c c c c c c v x Y x c Y x c Y x c Y x c G x d G x d B G x d B x d G B x d B G x d                                                                                (12) here     , , ij ij x G x d       , and upper limit of the integrals 1 a   in the first case and a   in the second case. The inverse transformations were applied to the formulae (11)-(12), and the substitution of the displacement functions in the boundary conditions         , 0 , , 0 0, , 0 0 y xy y x p x x B x B         reduce to the system of the singular integral equations. S OLVING OF THE SIE SYSTEM FOR THE TWO CASES he changing of the variable * 2       in the integrals with the limits 0 and  , and   * 0 1 1 0 2 c c c c       in the integrals with the limits 0 c and 1 c were done to pass the integration interval   1 1;1 I   . Similar changes were done in the other equations. We first consider in details the second case. SSIE is written in the form                 1 0 1 1 1 1 1 1 1 1 2 2 1 1 1 , , 1 0, 1 0, Z x d K x r x x I x d K x x I x d K x x I x                                                     (13) T