Issue 46

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

 , H

 and 2

z are replaced with b

z , respectively ( 2

z is the vertical centroidal axis of the cross-section of the un

and 1

cracked portion of the shaft). By substituting of (52), ML U ,

U and (60) in (51), one derives the following expression for the mode II

U

,

MH

MQ

component of the strain energy release rate as a result of the shaft bending:

   

   

1  i n 

2  i n 

i n  

M G

1













  



u dA u dA u dA  

(61)

MII

L H

ML

MQ

MH

0

0

0





r

r

i

i

i







i

i

i

1

1

1

b

b

A

A

A

i

i

i

The mode III component of the strain energy release rate induced by the shaft torsion is obtained by formula (39). The total strain energy release rate is found by addition of (39) and (61). The result is

   

   

1  i n 

2  i n 

i n  

MG

1 













  



u dA u dA u dA  



L H

ML

MQ

MH

0

0

0



r

r

i

i

i







i

i

i

1

1

1

b

b

A

A

A

i

i

i

   

   

1 R r      q

i n 

 

i n 



T

1

  



m



 

u dA u dA 

(62)

TL

TH

0

0



b r r 

i

i





i

i

1

1

b

b

A

A

i

i

Integration in (62) should be carried-out by the MatLab computer program.

Figure 9 : The strain energy release rate in non-dimensional form plotted against 1 D s property (curve 1 – for the shaft configuration shown in Fig. 7a, curve 2 – for the shaft configuration shown in Fig. 8a, curve 3 – for the shaft configuration shown in Fig. 7b, curve 4 – for the shaft configuration shown in Fig. 8b). Formula (62) is verified by obtaining of G with the help of (44). The complementary strain energy cumulated in half of the shat as a result of bending and torsion is found as

*

* * MTL MQ MTH *

(63)

U U U U   

* MTL

* MTH

* MQ

U

U are the complementary strain energies in the internal crack arm, the external crack arm

where

,

and

U

and the un-cracked shaft portion, respectively. It should be specified that * MQ U

is due to the bending only, since the

external crack arm is not loaded in torsion.

171