PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 97–105 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

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5

0 1 = N ,

(15)

Fl M = − 1 .

(16)

The present analysis is performed assuming validity of the Bernoulli’s hypothesis for plane sections since the beam under consideration has a high length to height ratio.

Fig. 2. Cross-section of the lower crack arm (the position of the neutral axis is marked by 1 1 n n − ). Concerning the application of Bernoulli’s hypothesis in the present paper, it ca n be noticed also that the beam portion between the supports is loaded in pure bending (Fig. 1). Therefore, the only non-zero strain in the beam portion between the supports is the longitudinal strain,  . Thus, according to the small strain compatibility equations,  is distributed linearly along the beam height ) ( 1 1 1 1 n z z = −   , (17) where 1 1 n z is the neutral axis coordinate (the neutral axis shifts from the centroid, since the modulus of elasticity varies transversally to the beam). The cross-section of the lower crack arm is shown schematically in Fig. 2. After substituting of  in (13) and (14), the two equations for equilibrium are solved with respect to 1  and 1 1 n z by using the MatLab computer program. The components of J -integral in segments 1 A are written as  = − x p , (18) 0 = y p , (19) 1 cos = −  , (20) 1 ds dz = , (21) where the coordinate, 1 z , varies in the interval / 2, / 2] [ 1 1 h h − .