Issue 48

T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 509 Figure 7 : The energy release rate G e as a function of the crack inclination q and the crack initiation angle a . Figure 8 : The energy release rate G e as a function of the crack inclination q , the crack initiation angle a and elastic modulus of the interfacial zone 2 E . Finally, the boundary conditions (8) and (9) in the inner problem formulation (7), lead to the solution of the following system of the singular integral equations following from the above mentioned transformations of (10) ( ) ( ) ( ) ( ) ( ) ( ) 1 (0) 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ 1 0 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; ; d ˆ ˆ 1 yy X XYY Y YYY T X B H X B H X X m e p k é ù æ ö÷ ç ê ú = X X + X + X X÷ ç ÷ ç ê ú è ø + -X ë û ò , (24) ( ) ( ) ( ) ( ) ( ) ( ) 1 (0) 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ 1 0 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; ; d ˆ ˆ 1 xy X XXY Y YXY T X B H X B H X X m e p k é ù æ ö÷ ç ê ú = X + X + X X X ÷ ç ÷ ç ê ú è ø + -X ë û ò (25) and integral equations ensuring the closed crack tips ( ) ( ) 1 1 ˆ ˆ 0 0 ˆ ˆ ˆ ˆ d 0, d 0. X Y B B X X= X X= ò ò (26) The outer stresses ( ) (0) ˆˆ ˆ yy T x , ( ) (0) ˆˆ ˆ xy T x are the solution of (4), (5) and (6) at the points ˆ ˆ x X e = of the crack e g in the case of the loaded but uncracked matrix 1 W and also linearly transformed with respect to the local coordinate system ˆˆ xy . The unknown Burgers vector densities ( ) ˆ ˆ X B X and ( ) ˆ ˆ Y B X are found in the form ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1 , 1 Y Y X X B B l l c c - - X = X -X X X = X -X X , (27) where the value of the exponent l is different from the value -0.5 due to the mismatch of the materials of the matrix 1 W and interfacial zone 2 W . The functions ( ) ˆ ˆ Y c X and ( ) ˆ ˆ X c X are continuous in the interval ( ) ˆ 0,1 XÎ and approximated by Jacobi polynomials. The stress intensity factors I K and II K at the crack tip lying in the matrix 1 W can be evaluated from the dislocation densities ( ) ˆ ˆ Y B X and ( ) ˆ ˆ X B X as follows ( ) 1 1 2 ˆ ˆ , , 1 2 1 1 I II Y X K l m p c k + = + , (28)