Issue 46

V. Rizov, Frattura ed Integrità Strutturale, 46 (2018) 158-177; DOI: 10.3221/IGF-ESIS.46.16 161 The properties, i s and i p , vary continuously in the radial direction of the i -th layer according to the following hyperbolic laws: 1 1 i i B i i D i i s s r r s r r      (6) 1 1 i i B i i D i i p p r r p r r      (7) where i B s , i D s , i B p and i D p are material properties ( i D s and i D p control the material gradient of i s and i p , respectively), i r and 1 i r  are shown in Fig. 2. In (6) and (7), the radius, r , varies in the interval   1 ; i i r r  . It should be mentioned that the distribution of the longitudinal strains is analyzed assuming validity of the hypothesis for plane sections, since the length to diameter of the cross-section ratio of the shaft under consideration is large. Thus, L  is distributed uniformly in the cross-section of the internal crack arm. Hence, by substituting of (5), (6) and (7) in (4), one derives     1 2 2 3 3 1 1 1 2 2 3 i n i i L i i i i i F r r r r                   (8) where 1 i i i B L B i i s p       (9) 2 2 2 i i i i i i B D L B D i i i i i B L B i i s s p p s p                   (10) 1 i i i r r     (11) 1 i D i i i s r     (12) 1 i D i i i p r     (13) It should be noted that by substituting of 0 i D s  and 0 i B p  in (8), one obtains   1 2 2 1 1 1 i i n L i i i B F r r s        (14)