Issue 70

S.K. Shandiz et alii, Frattura ed Integrità Strutturale, 70 (2024) 24-54; DOI: 10.3221/IGF-ESIS.70.02

  

  

2 K K 

2

2

(1 )  

K

a



(1) W b 



da

I

II

III

(45)



E

E

0

where E E   in case of plane stress, for plane strain 2 /(1 ) E E     , and a is the crack depth. The stress intensity factors of modes I, II, and III are represented by K I , K II , and K III , respectively. Writing the excess energy due to the presence of a crack and neglecting axial force, will read as [71]:

 

  

2 IM IP IIP K K K   2 )

(

a



(1) W b

 

da

(46)

E 

0

M F 

P F  l

P F 

a s

a s

II a s

6

( )

3

( )

( )

I

I







K

K

K

,

,

(47)

IM

IP

IIP

2

2

bh

bh

bh

4

2 cos / 1.122 0.561 0.085 0.18 s s s s     0.923 0.199[1 sin / 2] s    2     s   

2



( ) s

tan

,

F I



s

(48)

3

s

2 ( ) (3 2 ) s s s  

F

II



s

1

In the above equations, s is the ratio of crack depth to beam depth. The flexibility matrix of an intact element is expressed as:

2 (0) W



u

(49)

(0)









, i j

,

1,2,

,

c

P P P M

ij

1

2

 

P P

i

j

The coefficient of additional flexibility due to the presence of cracks is defined as follows:

2 (1) W



u

(50)

(1)









, i j

,

1,2,

,

c 

P P P M

ij

1

2

 

P P

i

j

Finally, the total flexibility will read as,

(0) ij ij   c c c    (1) ij

(51)

The equilibrium conditions are described as follows:









T

 T

[ ]

(52)

P M P M

P M  1 i i







i

i

i

i

1

1

1

T

1 L   

1 0

  

(53)

 

[ ] T

0 1 0 1 



According to the principle of virtual work, the stiffness matrix of the damaged element can be reached as [37,71]:

[ ] [ ] [ ][ ] T c  K T c T 

(54)

36