Issue 47

V. Rizov, Frattura ed Integrità Strutturale, (2047) 468-481; DOI: 10.3221/IGF-ESIS.47.37 476 The J -integral in segment, 1  , is written as 1 1 1 1 0 1 cos i L i i y i n L xi yi i y u v J u p p ds x x                           (34) where 0 i L u is the strain energy density in the i -th layer of the left-hand crack arm, α is the angle between the outwards normal vector to the contour of integration and the crack direction, xi p and yi p are the components of stress vector in the i -th layer of the left-hand crack arm, u and v are the components of displacement vector with respect to the crack tip coordinate system xy ( x is directed along the delamination crack), ds is a differential element along the contour of integration. The strain energy density in the i -th layer of the left-hand crack arm is obtained by applying the following formula [17, 18]:   1 2 0 1 2 1 i i i i m m i i L i m i i u E m H       (35) By substituting of (7) in (35), one obtains   1 2 0 1 1 1 1 1 1 2 cos 1 2 i i i i i i m m i i L i m d f i i i i u y y E E m H y y                        . (36) The other components of the J -integral in segment, 1  , are written as xi i p    (37) 0 yi p  (38) 1 ds dy  (39) cos 1    (40) It should be noted that formula (16) is used to obtain the stress, i  , in (37). The partial derivative, / u x   , that is involved in (34) is expressed as 1 1 u z x       . (41) The J -integral is segment, 2  , is written as 2 1 2 2 0 1 cos i i i y i n R R xRi yRi R i R R y u v J u p p ds x x                           (42) where the strain energy density, 0 i R u , in the i -th layer of the un-cracked beam portion is obtained by formula (36). For this purpose, i  , 1 y , 1 i y and 1 1 i y  are replaced, respectively, with i R  , 2 y , 2 i y and 2 1 i y  . The other components of 2 J  are written as