Issue 46

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

g

B



g

(26)

i

r r 

i

i



g

1

D



r

r

i



i

i

1

where

i i r r r   

(27)

1

r and

1 i r  , are shown in Fig. 2. In (25) and (26), i B f , i D f , i B g and i D

g are material properties (

D f and

The radiuses, i

i

i g , respectively). The distribution of shear strains in radial direction is written as D f govern the gradient of i f and i

r







(28)

m

r

b

By substituting of (24), (25), (26) and (28) in (23), one derives

1  i n 

 

2

1 4

1 5

  

  









4

4   r

5







i 

T

r

r

r

5

(29)





m i

i

i

i

i

1

1

r



i

1

b

where

1

i 



(30)

i 





f

g

B i

B m





i 

(31)

i

i



 2

D i g r

 

  

2

i 





i i r

1

2





r

1

i

i b

i 

f

D  i



i 

(32)

i

f

B



i 

(33)

i





i i r

1

0 B g  and

0 D f  Eqn. (29) transforms in

It should be mentioned that at

i

i

1  i n 

 

1





4

4





T

r

r

(34)



m

i

i

1

r

f

2



i

1

b

B

i

The fact that (34) is exact match of the equation for equilibrium of a multilayered circular shaft made by linear-elastic homogeneous layers loaded in torsion [14] is an indication for consistency of (34) since at 0 i B g  and 0 i D f  the non linear stress-strain relation (24) transforms in the Hooke’s law assuming that 1/ i B f is the shear modulus in the i -th layer. Eqn. (29) should be solved with respect to m  by using the MatLab computer program. Eqn. (29) is used also to determine the shear strain at the periphery of the cross-section of the un-cracked shaft portion. For this purpose, 1 n , b r and m  are replaced, respectively, with n , R and q  in (29) and (31).

164