Issue 46

V. Rizov, Frattura ed Integrità Strutturale, 46 (2018) 158-177; DOI: 10.3221/IGF-ESIS.46.16 162 which is exact match of the equation for equilibrium of multilayered circular shaft made by homogeneous linear-elastic layers loaded in centric tension [14]. This fact is an indication for consistency of Eqn. (8) since at 0 i D s  and 0 i B p  the non-linear stress-strain relation (5) transforms in the Hooke’s law assuming that 1/ i B s is the modulus of elasticity in the i -th layer. Eqn. (8) should be solved with respect to L  by using the MatLab computer program. Eqn. (8) is applied also to determine H  . For this purpose, 1 n and L  are replaced, respectively, with n and H  in (8), (9) and (10). Here, n is the number of layers in the un-cracked shaft portion. Figure 3 : Non-linear    diagram. Since the external crack arm is free of stresses (Fig. 1), the strain energy cumulated in half of the shaft as a result of the centric tension is written as F FL FH U U U   (15) where FL U and FH U are the strain energies in the internal crack arm and the un-cracked shaft portion, respectively. The strain energy in the internal crack arm is obtained by addition of strain energies cumulated in the layers 1 0 1 i i i n FL FL i A U a u dA      (16) where 0 i FL u is the strain energy density in the i -th layer. The strain energy density is equal to the area, OPQ , enclosed by the stress-strain curve (Fig. 3). Thus, 0 i FL u is written as 0 0 L i FL i u d      (17) By substituting of (5) in (17), one derives 0 1 ln ln i i i i i FL L L i i i i i s s s s u p p p p p                     (18) The strain energy cumulated in the un-cracked shaft portion is expressed as