Issue 46

V. Rizov, Frattura ed Integrità Strutturale, 46 (2018) 158-177; DOI: 10.3221/IGF-ESIS.46.16 163   0 1 i i i n FH FH i A U l a u dA      (19) where the strain energy density in the i -th layer, 0 i FH u , is obtained by formula (18). For this purpose, L  is replaced with H  . By substituting of (3), (15), (16) and (19) in (2), one obtains   1 0 0 1 1 1 i i i i i n i n II L H FL FH i i b b A A F G u dA u dA r r                         (20) Apparently, the torsion moment, T , induces mode III crack loading conditions (Fig. 1). By analyzing the balance of the energy, the mode III component of the strain energy release rate, III G , is written as 1 2 2 2 T III b b U T G r a r a                (21) where  is the angle of twist of the end section of the shaft, T U is the strain energy cumulated in half of the shaft as a result of the torsion. In (21), the expression in the brackets is doubled in view of the symmetry (Fig. 1). By applying methods of Mechanics of materials, one obtains   q m b a l a r R       (22) where m  and q  are the shear strains at the periphery of the cross-sections of the internal crack arm and the un-cracked shaft portion, respectively. The shear strain at the periphery of the cross-section of the internal crack arm is determined by using the following equation for equilibrium of the cross-section of the internal crack arm: 1 1 i i n i i A T rdA       (23) where i  is the distribution of the shear stresses in the i -th layer induced by the torsion. In the present paper, the mechanical behavior of the functionally graded material in torsion is described by the following non-linear stress-strain relation [13]: i i i f g      (24) where  is the shear strain, i f and i g are the distributions of the material properties in the i -th layer. The continuous variation of i f and i g in the radial direction of the i -th layer is described by the following hyperbolic laws: 1 1 i i B i i D i i f f r r f r r      (25)