Issue 48

T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 506 and 1 2 3 W= W ÈW ÈW with outward normal n . The boundary problem (4), (5) and (6) can only be solved using some numerical methods - the boundary element method in this case. This method ensures that the boundary conditions in (6) are fulfilled at discrete points along the boundary of the domain W . On the other hand, the conditions of the stress and displacement compatibility at the interfaces of 1 W , 2 W and 3 W are satisfied automatically via the fundamental solution described in previous section. The fundamental solution also makes it possible to express the boundary conditions for the inner problem represented in (3) by the stress tensor     0 / T x   , which must be independent of the crack length parameter e . The inner problem can be formulated in such a way that a crack of the unit length 1 1 e e g g = º is initiated in arbitrary direction along the scaled ˆ ˆ / X x e = -axis from the interface between the unloaded matrix 1 ¥ W and the interfacial zone 2 W . The domain 1 ¥ W is a domain 1 W with an outer boundary extended to infinity. The crack faces are loaded with radial and tangential stresses ( ) (0) T qq x and ( ) (0) r T q x from the outer problem (4), (5) and (6), which are generated in the uncracked matrix 1 W at points of the crack e g . Thus, ( ) (0) 0 ⋅ = T X  for e ¥ ÎW X (7) ( ) (0) ˆ e = ⋅ t T X n  for 1 g Î X (8) ( ) (0) 0  T X  for ¥ X (9) where e ¥ W is a cracked domain 1 2 3 ¥ ¥ W =W ÈW ÈW . The inner problem is due to the scaled coordinate / X x e = independent of the parameter e . Similarly to [9], the crack in (7), (8) and (9) is modelled using the continuously distributed dislocation technique [3], [5]. If the dislocation with Burgers vector i x y b b b = + is situated at the point 1 / z e ¥ Z= ÎW , then the stress tensor components YY T and XY T appearing at the point 1 / z e ¥ = ÎW Z can be written as follows using the potentials (1) and (2), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 0 2 1 i 1 1 ; i ; 2 1 , ; , ; 1 , ; , ; YY XY n n n n n n n n n T T b b b n bh k bh k b n h k bp k m k g g g g ¥ - - - - - - = - - - - - - é -Z ê Z + Z = - + + + ê+ -Z -Z -Z êë é + - Z + Z + êë ùù + + Z - Z ú úû û å Z Z Z Z Z Z Z Z ZZ Z (10) The dislocations will be distributed along the ˆ ˆ / X x e = -axis, i.e. for ˆ ˆ / 0 Y y e = = , see Fig. 1. Hence the standard transformations of vectors and tensors with respect to the rotations through angles a and q in XY and ˆ ˆ XY planes, respectively, must be applied to the Eqn. (10), see [5]. The singular part represented by the first three expressions in (10) is simply transformed to the Cauchy type singular kernel. The transformed regular part of (10) represented by the infinite sum can be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 ˆ ˆ ˆ 3 2 ˆ ˆ ; ; cos sin ; cos ; cos sin 2 ; sin ; cos sin ; sin sin 2 , XXX XYY XXY XYY YXX YYY YXY H X H H H H H H q q q q q q q q q q X = + - + + + - X Ξ X Ξ X Ξ X Ξ X Ξ X Ξ (11) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 ˆ ˆ ˆ 2 3 ˆ ˆ ; ; sin ; cos sin ; sin sin 2 ; cos sin ; cos ; cos sin 2 , XXX XYY XXY YYY YXX YYY YXY H X H H H H H H q q q q q q q q q q X =- - + + + + - X Ξ X Ξ X Ξ X Ξ X Ξ X Ξ (12)