Issue 49

A. Baryakh et alii, Frattura ed Integrità Strutturale, 49 (2019) 257-266; DOI: 10.3221/IGF-ESIS.49.25 262 where m l    . A general solution of this linear differential equation with constant coefficients takes the following form:   1 2 3 4 ( ) ( ) ( ) ( ) f y C ch y C sh y C ych y C ysh y         (13) Then a stress function is determined according to the following expression:   1 2 3 4 sin ( ) ( ) ( ) ( ) x C ch y C sh y C ych y C ysh y           (14) and the relevant stress components are calculated according to the following formulas:     2 2 2 1 2 3 4 2 sin ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) x x C ch y C sh y C sh y ych y C ch y ysh y y                               2 2 1 2 3 4 2 sin ( ) ( ) ( ) ( ) y x C ch y C sh y C ych y C ysh y x                 (15)     2 1 2 3 4 cos ( ) ( ) ( ) ( ) ( ) ( ) xy x C sh y C ch y C ch y ysh y C sh y ych y x y                                where constants 1 2 3 4 , , , C C C C are determined by relevant stress boundary conditions. For the test problem (Fig. 2), the boundary conditions can be represented in the form of symmetric expansion into a Fourier series: 0 1 cos m m m x q A A l       (16) where the expansion coefficients are defined by the following expressions: 0 , qa A l  2 sin 1 cos a m a m a q m x l A q dx l l m        (17) With account of the boundary conditions, the constants 1 2 3 4 , , , C C C C in expressions for stresses (15): 1 2 4 2 2 3 2 ( ) ( ) (2 ) 2 cos 2 ( ) (2 ) 2 cos 0 q sh h hch h C sh h h a q sh h C sh h c a C C                        (18) where q- is determined by expression (16). The results of an analytical solution for middle line (y=0) on which only a normal stress takes place: