Issue 46

V. Rizov, Frattura ed Integrità Strutturale, 46 (2018) 158-177; DOI: 10.3221/IGF-ESIS.46.16 168 1 2 2 2 M MII b b U M G r a r a                (51) where  is the angle of rotation of the end section of the shaft due to the bending, M U is the strain energy cumulated in half of the shaft as a result of the bending. It should be noted that the bending induces stresses not only in the un-cracked shaft portion and the internal crack arm, but also in the external crack arm. By using methods of Mechanics of materials,  is obtained as   L H a l a       (52) where L  and H  are the curvatures of the crack arms and the un-cracked shaft portion, respectively. Since the bending generates mode II crack loading conditions, the two crack arms deform with the same curvature. Therefore, L  is determined in the following way. First, the equation for equilibrium of the cross-section of the internal crack arm is used 1 1 1 i i n d i i A M z dA       (53) where d M is the bending moment in the internal crack arm. The distribution of the longitudinal normal stress, i  , in the i -th layer, induced by the bending of the shaft, are expressed by (5). The distribution of the longitudinal strains,  , is written as 1 L z    (54) By substituting of (5), (6) and (7) in (23), one derives     1 4 4 5 5 1 1 1 1 1 4 5 i n d L i i i L i i i i M r r r r                     (55) where i i i B s    (56) i i D i B i s s    (57) The radiuses, i r and 1 i r  , in (55) are shown in Fig. 6. It should be noted that at 0 i B p  and 0 i D s  Eqn. (55) transforms in   1 4 4 1 1 1 4 i i n L d i i i B M r r s        (58) which is exact match of the equation for equilibrium of multilayered circular shaft made of homogeneous linear-elastic layers loaded in bending [14] assuming that 1/ i B s is the modulus of elasticity in the i -th layer.