Issue 44

V. Reut et alii, Frattura ed Integrità Strutturale, 44 (2018) 82-93; DOI: 10.3221/IGF-ESIS.44.07 91 was constructed with the help of the matrix differential calculation, , 1, 2 i c i   are known constants,   , G x   is the Green’s matrix function which was constructed by the use of the matrix semi-infinite Fourier transformation. The components of the Green’s matrix function have the following form               11 1 1 , 2 2 1 x x x x x x e e G x e e x e x e                                                             12 1 , 2 1 x x G x x e x e                               21 1 , 2 1 x x G x x e x e                                 22 1 1 , 2 2 1 x x x x x x e e G x e e x e x e                                                  After inverting the expression (18), and the summation of the weak-convergent integrals, the formulae for the displacements in the quarter plane have the following form                                      2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 1 1 , ' ln ln 4 1 1 2 1 1 x x u x y x y x y x y x y x y x x d x y x y                                                                                                                       2 2 2 2 0 2 2 2 2 2 2 2 2 1 1 , ' 2 2 2 4 1 2 2 2 1 y x y x y y v x y arctg sign x arctg x x x y x y xy x xy y y arctg d x sign y x x y x y x y                                                                                                         These expressions will describe the displacements in the quarter plane if the function   '   is found. To get it, the formulae for the displacements were put in the boundary condition   ( , 0) , 0 y x p x x a     . After changing the variables, the singular integral equation was derived         1 3 1 2 2 3 0 1 h x h h x d q x x x x x                           (19) here     1 ' 2 a              , 2 1 2 3 3 2 4 , , 2 h h h           ,   q x is the known function.