PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 63–74 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

71

9

By substituting of (25) in (24), one derives

R

R

R

2



2

2





  

  

1

3

 

 

 

* u R dR d IN A A 0

* u R dR d A A EX 0

* u R dR d UC A A

G

=

+

−







.

(26)

R

2



p

1

1

+

3

R

0

0

0

0

0

3

The complementary strain energy density, *

0 IN u , that is involved in (26) is derived by applying the formula (Rizov,

2017):

*

u

u

= − 

.

(27)

IN

IN

0

0

By substituting of (19) and (20) in (22) and replacing of  with

int x  , one obtains

2

x 

x 

L

Q L

L

Q L

  

 −  

  

  

*

u

ln

ln

x 

=

− + int

+

.

(28)

int

IN

0

2

2

+ L Q Q 

Q

Q

int

int x  is replaced with

Formula (28) is applied also to determine * 0

1 + p EX u

1 + p x  . The strain energy

. For this purpose,

1 r x  in (22). It should be noted that R , * 0 IN u , * 0 1 + p EX u and

density, * 0

int x  with

1 UC u , is derived by replacing of

* 0 1 UC u which are involved in (26) are obtained by (17) and (22) at x a = . The integration in (26) is performed by the MatLab computer program. The fact that the strain energy release rate obtained by (26) matches exactly that found by (21) is a verification of the general approach for analyzing the strain energy release rate developed in the present paper. The solution to the strain energy release rate (21) is applied to evaluate the influence of the continuously varying radius of the cross-section along the beam length, the crack location in radial direction, the material inhomogeneity, the crack length and the material non-linearity on the longitudinal fracture behaviour of the cantilever beam shown in Fig. 2.

2 1 / R R ratio (curve 1 – at non-linear mechanical behaviour of

Fig. 3. The strain energy release rate in non-dimensional form plotted against

the material, curve 2 – at linear-elastic behaviour).

0 1 / G GL R N = . It is

The strain energy release rate is expressed in non-dimensional form by using the formula

10 = b F N and

0.150 = l m,

1 = R

3 1 = F N.

0.005

assumed that

m,