PSI - Issue 51

Victor Rizov et al. / Procedia Structural Integrity 51 (2023) 44–50 V. Rizov / Structural Integrity Procedia 00 (2022) 000–000

46

3

the vertical notch, crack arms 1, 2 and 3 are stress free. First, crack 1 is analyzed. For this purpose, the strain energy release rate (SERR), G , is obtained by applying the following formula (Rizov (2017)):

*

bda dU

2

G 

,

(1)

1

* U is the complementary strain energy (CSE),

1 da is an elementary increase of crack 1.

where

The CSE is written as (Fig. 1)

* 1 * U U U U U     , 4 * 3 * 2 *

(2)

* 3 U and

* 1 U ,

* 2 U ,

* 4 U are the CSE in crack arm 4 (the boundaries of this crack arm are

l x  1 ,

where

3 1 x l a   ,

(the  1 z axis is shown in Fig. 1)), in part, 2 3 S S , of the beam (the

4 1 / 2 z h h   and

/2

1 z h 

4 1 /2 z h h h    and 3

3 1 x l a   ,

/2

2 1 x l a   ,

1 z h 

boundaries of this part of the beam are

), in part,

3 4 S S , of the beam (the boundaries of this part of the beam are

/ 2 h h    and

2 1 x l a   ,

1 1 x l a   ,

z

1

1

/2

2

1 z h 

l a x

l

1 1   

), and in the un-cracked portion,

, of the beam, respectively.

The CSE in crack arm 4 is expressed as

U a u dA A * 01 ( ) 3 * 1 1   ,

(3)

* 01 u is the complementary strain energy density (CSED).

where 1 A is the cross-section,

The following stress-strain relation is used (Ignatiev et al. (2014)):

n

   

   



      D 1 1

  



,

(4)

L

where  is the stress,  is the strain, D , L and n are material parameters. D is distributed as

m

m h D D  1

h 2

  

  

D D

z

 



0

,

(5)

0

1

where

z h    .

2 h

(6)

1

2

In (5), 0 D and 1 D are the values of D in the upper and lower surface, m is a material parameter that controls the inhomogeneity along the height. The CSED is (Rizov (2017))