Issue 48

T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 505 applied in the method of the boundary integral equations as the outer solution of the problem. The basis for the expression of the fundamental solution is the complex analysis in the plane elasticity represented by Muskhelishvili’s complex potentials, e.g. [10], which for the matrix 1 W can be written in the form (the series of complex potential for domains 2 W and 3 W can be found in [3], [4]) ( ) ( ) ( ) ( ) ( ) 1 0 0 ; ; , ; ln , ; s n n n n n n z z h k z z h k z f z f z g z g z g g z ¥ ¥ - - - - = = = + =- - + å å , (1) ( ) ( ) ( ) ( ) ( ) 1 0 0 ; ; , ; ln , ; s n n n n n n z z p k z k z p k z z z y z y z g z g z g g g z z ¥ ¥ - - - - = = = + =- - + + - å å , (2) where ( ) 1 1 1 1 , for point force, 2 1 i 1, for dislocation. 1 f k b k k g p k m g k ìïï =- = ïï + ïíïï = = ïï + ïî and 1 m is shear modulus and 1 k is Kolosov’s constant of the material of the domain 1 W . The functions ( ) ; s z f z and ( ) ; s z y z are the well-known complex potentials giving at the point i z x y   the response of the point force i x y f f f   or edge dislocation i x y b b b   situated at point i z x h = + in the infinite complex plane. The coefficients   , ; n h k    and   , ; n p k    are determined from the conditions of the continuity of displacements and the resultants of the tractions along the interfaces among the matrix 1 W , interfacial zone 2 W and inclusion, see [3], [4]. A SYMPTOTIC ANALYSIS AND STRESS INTENSITY FACTOR EVALUATION he purpose of using the methods analyzing the path of the crack initiated from interface between the interfacial zone 2 W and the matrix 1 W follows from the stress composite expansion applied at the point of the crack initiation represented by the parameter a , see Fig. 1. The composite expansion, i.e. an asymptotic expansion with respect to the small parameter e , is introduced under the assumption that the crack e g is small with respect to the dimensions of the inclusion 3 W and can be written as follows, see [3], ( ) ( ) ( ) ( ) ( ) 0 (0) (0) ˆ , / , e e = + + T x T x T x T x  (3) where [ ] , x y º x ,   (0) T x is the outer stress,   (0) / T x   is the inner stress and   (0) ˆ T x is an error when the members associated with n e , where 0 n  , are neglected in the asymptotic expansion of the stress   , T x  . The outer stress   (0) T x is defined in the crack-free domain 1 W as the solution of the following boundary problem ( ) (0) 0 ⋅ = T x for Î W x (4) ( ) (0) P ⋅ = T x n t for Î ¶W x (5) where ( ) ( ) ( ) ( ) 1 2 , 0 for 1, 0 , 0, for 0, 1 P T T ì =  ïï= íï =  ïî n t n (6) T