PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1237–1248 Author name / Structural Integrity Procedia 00 (2019) 000–000

1242

6

n n f nH q q 1 1 1 1 

'

 E q E q E f f 2 1 2

2

f

0





,

(19)

 1 2

1

n

n

   ,

  



1 1

n



  2

  '

  0 1 2 

''

'

 q q q q 2 2 n

  q E q E E q E q E E 



1 n

(20)

3

1

2

1

1

2

3

f

f

f

f

f

f

f

f

3

1 n

n

E

nH

where the modulus of elasticity and its derivatives are calculated at a R R  . The equilibrium equation for the cross-section of the external crack arm is written as   2 2 A f dA N  ,

(21)

where 2 N and 2 A are, respectively, the axial force and the area of the external crack arm cross-section. By combining of (15) and (21), one derives                 4 1 4 2 3 3 1 3 2 2 2 1 2 2 1 2 4 3 2 2 q R R q R R q R R N  . (22) Equations (18), (19), (20) and (22) should be solved with respect to f  , 1 f q , 2 f q and 3 f q by using the MatLab computer program. In order to calculate the complementary strain energy density in the cross-section of the external crack arm behind the crack front, the stress obtained by (15) is inserted in (5). The complementary strain energy density in the cross-section of the bar ahead of the crack front is derived by formula (5). For this purpose,  is replaced with g  where g  is the stress in the bar cross-section ahead of the crack front. Since in the bar cross-section ahead of the crack front (Fig. 1) R varies in the interval the distribution of the normal stresses is obtained by formula (8). For this purpose,  , 1 q , 2 q and 3 q are replaced, respectively, with g  , 1 g q , 2 g q and 3 g q where g  is the longitudinal strain in the bar cross-section ahead of the crack front. Equations (10), (11), (12) and (14) are used to determine g  , 1 g q , 2 g q and 3 g q . For this purpose, 1 N , 1 R ,  , 1 q , 2 q and 3 q are replaced, respectively, with N , 2 R , g  , 1 g q , 2 g q and 3 g q . After that, g  is inserted in (8) to obtain * 0 u . After obtaining of * 01 u , * 02 u and * 0 u they are inserted in (1) to derive the strain energy release rate. The integration in (1) should be performed by using the MatLab computer program for particular bar configuration, material properties 2 0 R R   , (23)

and loading conditions. 3. Example problem

In the present section, the general solution procedure developed in section 2 of the present paper is applied to analyze the longitudinal fracture behaviour of the cantilever shown schematically in Fig. 2. There is a cylindrical