Issue 48

O. Plekhov et alii, Frattura ed Integrità Strutturale, 48 (2019) 451-458; DOI: 10.3221/IGF-ESIS.48.43

n

e   



GA B

where – elastic limit. The energy of plastic deformation in representative volume near the crack tip can be estimated as follows 0 , , A n  – material constants, 0 0     , , oct e  e      

n                n e  oct e  An  

1

3 2

3











(3)

U

d

.

p

oct

oct

2 1 n

e

0

The energy increment caused by crack advance under monotonic loading can be written as

n

0     e    



d

3 2

n

dU An 

e 



(4)

dl

,

p

dl

where l - crack length.

1 2

1 2

r f

el

 

 



Kf

p e

 (here K – stress intensity factor, r p

– estimation for plastic zone size, r,

oct  

e



 

Using definition

e 

r

e 

r

3

 – polar coordinates, f e

– function of polar coordinate  , we can rewrite Eqn. (4) as

n

0     e    

 

d d

3 2

n

dU An 

e 



(5)

dl

.

p

d dl 

where

r

  

 



df

d

1

  

 

p

e









cos .   

e f C

f

sin

(6)

e



dl

r

d

p e r f r

2

To analyse plastic deformation at the crack tip under cyclic loading we need to divide energy dissipation in cyclic and monotonic plastic zones at the crack tip

tot p U U U   cyc p p

mon

(7)

.

The energy of representative volume at cyclic zone can be estimated as 3 2 cyc p ec pc U    ,

(8)

 



1 1 s G G





 – characteristic size of the yield surface,



, oct c   



  

 – amplitude of plastic deformation under

where ec

pc

ec

an assumption of the validity of Ramberg-Osgood relationship ship, , oct c  – stress change in the representative volume. The full energy of cyclic plastic zone can be calculated as a double integral over the region (S) bounded on the outside of the monotonic plastic deformation zone and inside of the fracture zone







  

  

2 0 3 1 1 2 2 ec s S G G      

, oct c

cyc p







(9)

U

rdrd

1



ec

, p c e r r f  , for cyclic-fracture zone boundary –

The simple approximation of plastic zone boundary can be given by

455