Issue 47

V. Rizov, Frattura ed Integrità Strutturale, (2047) 468-481; DOI: 10.3221/IGF-ESIS.47.37 475   1 1 2 2 6 1 3 1 6 2 1 2 2 cos 0 i i i i m m m m i i i i ai i i i i i i i i m y m m                              (28) Eqns. (19), (24) – (28) can be written for each layer of the left-hand crack arm. In this way, 6 L n equations with 6 1 L n  unknowns, 1 i  , 2 i  , 3 i  , 4 i  , 5 i  , 6 i  and 1  , where 1, 2, ..., L i n  , can be constructed. Another equation can be written by considering the equilibrium of the elementary forces in the left-hand crack arm cross-section 1 1 1 2 1 1 1 1 2 i L i h y i n i i h y M z dy dz          (29) where M is the bending moment in the left-hand crack arm. It is obvious that (Fig. 1) 1 M Fl  (30) By substituting of (16) in (29), one derives       3 3 2 2 3 1 1 1 5 1 1 1 1 12 24 L i n i i i i i ai i ai i h h M y y y y y y                        (31) Eqns. (19), (24) – (28) and (31) should be solved with respect to 1 i  , 2 i  , 3 i  , 4 i  , 5 i  , 6 i  and 1  by using the MatLab computer program. Formula (16) is applied also to present i R  as a function of 2 y and 2 z . For this purpose, 1 i  , 2 i  , 3 i  , 4 i  , 5 i  , 6 i  , 1 y and 1 z are replaced with 1 Ri  , 2 Ri  , 3 Ri  , 4 Ri  , 5 Ri  , 6 Ri  , 2 y and 2 z , respectively. It should be noted that Eqs. (19), (24) – (28) and (31) can be used also to determine 1 Ri  , 2 Ri  , 3 Ri  , 4 Ri  , 5 Ri  , 6 Ri  and 2  where 2  is the curvature of the portion, 1 2 3 1 2 2 l l a x l l      , of the un-cracked part of the beam. For this purpose, L n , 1 i y , 1 1 i y  , 1 i  , 2 i  , 3 i  , 4 i  , 5 i  , 6 i  and 1  are replaced, respectively, with n , 2 i y , 2 1 i y  , 1 Ri  , 2 Ri  , 3 Ri  , 4 Ri  , 5 Ri  , 6 Ri  and 2  in (19), (22) – (28) and (31). By substituting of (4), (5) and (14) in (3), one arrives at 1 1 2 1 1 2 2 2 * * 0 1 1 0 2 2 1 1 2 2 1 1 2 i i L i i i i h h y y i n i n L R i i h h y y G u dy dx u dy dz h h                           (32) where * 0 i L u and * 0 i R u are obtained by (12), (13), (16), (19), (20), (23), (25) – (28) and (31) at 1 x a  . It should be noted that the term in the brackets in (32) is doubled in view of the symmetry (Fig. 1). The integration in (32) should be carried-out by using the MatLab computer program. The delamination fracture behavior is analyzed also by applying the J -integral approach [19] in order to verify the solution to the strain energy release rate (32). The integration of the J -integral is performed along the integration contour,  , showed by a dashed line in Fig. 1. Since the right-hand crack arm is free of stresses, the solution of the J -integral is written as   1 2 2 J J J     (33) where 1 J  and 2 J  are the values of the J -integral, respectively, in segments, 1  and 2  , of the integration contour (segments, 1  and 2  , coincide with the cross-section of the left-hand crack arm and un-cracked beam portion, respectively). The term in brackets in (33) is doubled because of the symmetry.