Issue 47

V. Rizov, Frattura ed Integrità Strutturale, (2047) 468-481; DOI: 10.3221/IGF-ESIS.47.37 473 The complementary strain energy cumulated in portion, 1 2 3 1 2 2 l l a x l l      , of the un-cracked part of the beam is written as 2 1 2 2 2 * * 0 1 2 2 1 2 i i i h y l i n R R i h a y U u dx dy dz          (14) where n is the number of layers in the un-cracked beam portion, 2 i y and 2 1 i y  are the coordinates, respectively, of the left-hand and right-hand lateral surfaces of the i -th layer, * 0 i R u is the complementary strain energy density in the same layer, 2 y and 2 z , are the centroidal axes of the cross-section of the un-cracked portion of the beam. The complementary strain energy density in the i -th layer of the un-cracked beam portion can be obtained by (12). For this purpose, i  , 1 y , 1 i y and 1 1 i y  have to be replaced, respectively, with i R  , 2 y , 2 i y and 2 1 i y  where i R  is the distribution of the normal stresses in the cross-section of the i -th layer of the un-cracked beam portion. In order to perform the integration in (5), i  has to be presented as a function of 1 y and 1 z . It is obvious that i  can not be determined explicitly from Eq. (6). Therefore, i  is expanded in series of Taylor by keeping the first six members             2 2 1 1 1 1 1 2 1 1 1 2 2 2 1 1 1 2 1 1 1 , 0 , 0 , 0 ( , ) ( , 0) ( ) 2! , 0 , 0 ( ) 2! i ai i ai i ai i i ai ai ai i ai i ai ai y y y y z y y y z y y y z y y y y y z z y z z                             (15) where   1 1 1 / 2 ai i i y y y    , 1 1 1 1 i i y y y    and 1 / 2 / 2 h z h    (Fig. 2). Formula (15) is re-written as 2 2 1 1 1 2 1 3 1 4 1 5 1 1 6 1 ( , ) ( ) ( ) ( ) i i i ai i i ai i ai i y z y y z y y y y z z                 (16) where the coefficients, 1 i  , 2 i  , 3 i  , 4 i  , 5 i  and 6 i  , are determined in the following manner. First, the distribution of lengthwise strains in the cross-section of the left-hand crack arm is written as 1 1 z    (17) where 1  is the curvature the left-hand crack arm. It should be noted that formula (17) follows from the fact that validity of the Bernoulli’s hypothesis for plane sections is assumed in the present paper since the span to height ratio of the beam under consideration is large. Concerning the application of the Bernoulli’s hypothesis for plane sections, it should also be mentioned that since the beam portion, 2 4 B B , in which the delamination crack is located, is loaded in pure bending (Fig. 1), the only non-zero strains are the lengthwise strains. Thus, according to the small strains compatibility equations, the lengthwise strains are distributed linearly along the cross-section height. By substituting of (7), (16) and (17) in (6), one arrives at   2 2 1 2 1 3 1 4 1 5 1 1 6 1 1 1 1 1 1 1 1 1 2 2 1 2 1 3 1 4 1 5 1 1 6 1 1 ( ) ( ) ( ) cos 2 ( ) ( ) ( ) i i i i i i ai i i ai i ai i i d f i i m i i ai i i ai i ai i m i y y z y y y y z z z y y E E y y y y z y y y y z z H                                            (18)