PSI - Issue 13

Ivan Shatskyi et al. / Procedia Structural Integrity 13 (2018) 1476–1481 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

1478

3

[ ]( ) | [ ] ( ) | 0 n y n y n n u x h x    ,

n n x y n N x

M x

hN x

( , 0) 0  ,

( , 0)sgn[ ]  ,

( , 0)



y n

y n

y

n

n

n



( , 0) n n x y n (2) and also conditions of the free edge for slits (Williams (1961), Isida (1977), Berezhnitskii et al. (1979), Savruk (1981)): ( , 0) 0 n y n N x  , ( , 0) 0 n n x y n N x  , ( , 0) 0 n y n M x  , ( , 0) n n x y n n M x C   , 2 1 , 1, n n x L L n N N     . (3) Bending and twisting moments and membrane forces are set up on infiniteness: , , , 0, ( , ) y y x x xy xy y x xy M m M m M m N N N x y           . (4) Here and hereafter N ij are membrane forces, M ij are moments, M * ij is generalized twisting moment, [ ], [ ] n n y x u u and [ ] [ / ], [ ] [ / ] n n y n x n w y w x         are respectively the jumps of the displacements in the middle surface and the rotation angles of the normal on the cuts L n , C n are arbitrary constants. 2.2. Integral equations To construct the solution of the problem (1) – (4) we used the method of singular integral equations. Integral expressions of the forces and moments on the cutting line irrespective of the type look like this:   11 12 1 ( , 0) ( , )[ ] ( ) ( , )[ ] ( ) 4 k n k k k N l y n nk n y nk n x k l B N x K x u K x u d               ,   21 22 1 ( , 0) ( , )[ ] ( ) ( , )[ ] ( ) 4 k n n k k k N l x y n nk n y nk n x k l B N x K x u K x u d               ,   0 33 34 1 ( , 0) ( , )[ ] ( ) ( , )[ ] ( ) 4 k n k k k N l y n n nk n y nk n x k l Da M x m K x K x d                  ,   0 43 44 1 ( , 0) ( , )[ ] ( ) ( , )[ ] ( ) 4 k n n k k k N l x y n n nk n y nk n x k l Da M x p K x K x d                   , (5) where 0 0 , n n m p are functions of the main stress state, 2 B Eh  , 3 2 2 (3(1 )) D Eh    , (3 )(1 ) a      , E and  are Young's modulus and Poisson's ratio for the plate material. Dash indicates the derivative with respect to a coordinate. Kernels of expressions (5) (Savruk (1981)) contain singular additive of Cauchy type at n k  . Having satisfied the boundary conditions (2) for cracks and (3) for slits, we received the system of singular integral equations: [ ] ( ) [ ] ( )sgn[ ]( ) 0 n n y n yn n y n u x h x x       ,   21 22 1 ( , )[ ] ( ) ( , )[ ] ( ) 0 4 k k k k N l nk n y nk n x k l B K x u K x u d               ,   11 12 1 sgn[ ]( ) ( , )[ ] ( ) ( , )[ ] ( ) 4 k k k k N l y n nk n y nk n x k l Bh x K x u K x u d                  33 34 0 1 ( , )[ ] ( ) ( , )[ ] ( ) 4 k k k k N l nk n y nk n x n k l Da K x K x d m                  , n M x C  , 1 1 ( , 0) 0,  ; x L L n N    1, n y n n n N x ;