Issue 46

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

1 3 / B B s s ratio: curve 1 – for the shaft configuration

Figure 10 : The strain energy release rate in non-dimensional form plotted against

shown in Fig. 7a (linear-elastic solution), curve 2 – for the shaft configuration shown in Fig. 7a (non-linear solution), curve 3 – for the shaft configuration shown in Fig. 8a (linear-elastic solution), curve 4 – for the shaft configuration shown in Fig. 8a (non-linear solution).

* MTL

* L U and

* 0

* MTL

* 0

U

u

U

u

Formula (46) is applied to determine

. For this purpose,

are replaced with

and

,

L

MTL

i

i

respectively. Formula (48) is used to obtain the complementary strain energy density, * 0 i MTL u

, in the i- th layer of the

internal crack arm. For this purpose, * 0 i L u

 are replaced, respectively, with * 0 u

and  , where  is expressed

and L

MTL

i

by (54). * MQ U

* L U ,

* 0

* 0

* 0

* MQ

1 n and

2 n and

u

u

u

is obtained by replacing of

, respectively, with

,

in (46).

is

U

L

MQ

MQ

i

i

i

* 0

* 0

and  in (48) and taking into account the fact that

 , respectively, with

u

u

and L

determined by replacing of

L

MQ

i

i

the external crack arm is loaded in bending only. Thus, * 0 i MQ u

is written as

  

   

2

  

  



s

s

s

s

1

* 0

i

i  

i

i

   





u

ln

ln

(64)

MQ





s

p

p

p

p

p p

i

i

i

i

i

i

i

i

The complementary strain energy cumulated in the un-cracked shaft portion as a result of bending and torsion is calculated by (49). For this purpose, * H U and * 0 i H u are replaced with * MTH U and * 0 i MTH u , respectively. Formula (48) is used to obtain the complementary strain energy density, * 0 i MTH u . For this purpose, * 0 i L u , L  and  are replaced, respectively, with * 0 i MTH u , b  and H  , where H  is found by (38). Finally, * MTL U , * MQ U and * MTH U are added-up and substituted in (44). The result is

   

   

2  i n 

n

i n  

1

1 







* 0

* 0

* 0



u dA u dA u dA  

(65)

G

MTL

MQ

MTH

r 

i

i

i







i

i

i

1

1

1

b

A

A

A

i

i

i

The integration in (65) should be performed by the MatLab computer program. It should be noted that the strain energy release rate obtained by (65) is exact match of the strain energy release rate calculated by (62), which is a verification of the

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