Issue 46

V. Rizov, Frattura ed Integrità Strutturale, 46 (2018) 158-177; DOI: 10.3221/IGF-ESIS.46.16

s and i p , vary continuously in the radial direction of the i -th layer according to the following hyperbolic

The properties, i

laws:

s

B



s

(6)

i

r r 

i

i



s

1

D



r

r

i



i

i

1

p

B



p

(7)

i

r r 

i

i



p

1

D



r

r

i



i

i

1

s and

i p ,

B s ,

D s ,

B p and

i D p are material properties ( i D

s and

D p control the material gradient of i

where

i

i

i

i

1 i r  are shown in Fig. 2. In (6) and (7), the radius, r , varies in the interval   1 ; i i r r  .

r and

respectively), i

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 



(9)



s

p

B

L B



i

i

i 

i 



s s

2   i i p p i

B D L B D



i

i

i

2   i i

i 



(10)

2

i     L B p 

s

 

B

i

i 

i 



i 

1    i r

r

(11)

i

i D i s r

i 

1  

(12)

i 

i D i p r

i 

1  

(13)

i 

0 D s  and

0 B p  in (8), one obtains

It should be noted that by substituting of

i

i

1  i n 

1





2

2







F

r

r

(14)



L

i

i

1

s



i

1

B

i

161