Issue 44

V. Reut et alii, Frattura ed Integrità Strutturale, 44 (2018) 82-93; DOI: 10.3221/IGF-ESIS.44.07 85             0 1 0 1 , 0 , 0 , 0, , 0 , 0 , 0, xy xy xy y y y x B x B x B c x c x B x B x B c x c                     (7) One needs to solve the corresponding boundary value problems to estimate the stress state of the semi-strip and the concentration of the stresses at the crack’s tips. G ENERAL SOLVING SCHEME FOR THE SEMI - STRIP STRESS STATE ESTIMATION ccording to the approach [29], the Fourier’s transformation was applied to the system of Lame’s equilibrium Eqs. (2) and to the boundary conditions (1), (3)-(4), (1), (5) by the generalized scheme [30]. The initial problem was reduced to a vector boundary problem [31]         2 0 0, 0 L y x f x y y a           (8) here         2 2 " 2 ' L y x Iy x Qy x Py x              is the differential operator of the second order, I is an identity matrix,       u x y x v x              1 0 1 1 0 1 P                      1 0 1 1 0 1 Q                           1 2 1 2 3 1 3 '( ) sin cos ' 1 1 1 1 1 ( ) sin ' cos 1 1 x b x b x f x x b x b x                                                         0 , y x v x y    is an unknown function. So     0 ' , ' y v x y x    ,   0 ' y u x y      , and the second boundary condition in (3) is satisfied automatically. The components of the vector   y x   are the Fourier transformation of the displacements     0 ( ) cos , ( ) sin , u x y u x y dy v x y v x y                          (9) The solution of the vector boundary problem was obtained in the form [27, 28] A