PSI - Issue 51

4

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

47

* 01 u   

u

,

(7)

01

where 01 u is the strain energy density (SED). 01 u is obtained by integrating (4). The result is

1

n 



DL

DL

  

   

1

01 u D

  



.

(8)

1

1

n

L

n





By combining (4), (7) and (8), one obtains

1

n

n



DL

DL





1      

   

  

   

* 01 u D

1



1

1

L

n

L

n





.

(9)

The distribution of  along the height of crack arm 4 is   n z z 2 1 2     ,

(10)

n z 2 are the curvature and the coordinate of the neutral axis. 1  and

n z 2 are obtained from

1  and

where

following equations:

A    ( )

N

dA

,

(11)

1

1

A ( ) 1   

M

z dA 2

,

(12)

1

1 N is the axial force,

1 M is the bending moment.

where

2 3 S S and 3 4 S S of the beam and in the un-cracked beam portion are determined

The CSE cumulated in parts

analogically. By substituting CSE in (1), one arrives at

b ( ) 3 2     A

   ,

  u dA A * 04 ( ) 4

u dA * 03

G



(13)

where the integration is carried-out by the MatLab. The SERR for cracks 2 and 3 are analyzed in analogical manner. The solutions of SERR are verified by the J -integral (Broek (1986)). For crack 1 the integration contour, B , is used (Fig. 1). The J -integral is written as   2 1 2 B B B J J J   , (14)

1 B J

2 B J

1 B and

2 B , of the integration contour, respectively.

and

are the J -integrals in segments,

where