PSI - Issue 18

A.P. Zakharov et al. / Procedia Structural Integrity 18 (2019) 749–756 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

751

3

The governing parameter of the crack-tip elastic – plastic stress – strain field in the form of the I n -integral in Eq.1 and Eq.2 is a function of the material strain hardening exponent n and angular stress/displacement distributions. Shlyannikov and Tumanov (2014) suggested the numerical procedure for calculating I n - integral for the different cracked bodies by means of the elastic – plastic FE-analysis of the near crack-tip stress-strain fields.   1 cos FEM FEM n n e n I     

    

n    

1

  

  

  

  

  

(4)

FEM

FEM

du

du

FEM FEM

FEM FEM

u

u

sin













r 



r



 

rr

r





d

d





1

 





FEM FEM FEM FEM rr r r u u     

d   

cos



n

1



ij i u  ~ , ~ – dimensionless functions of stresses and displacements.

where θ – polar angle,

It is well known that small-scale analysis of fracture is usually considered applicable for investigating the plastic zone size at the crack tip; for example, for an infinite cracked plate, less than one order of the crack length. However, at higher values of applied nominal stress, in what is referred to as the large-scale yielding range, the plastic zone is no longer small compared to the crack length and the plastic SIF required some modification. For large scale yielding, J -integral cannot be calculated simply, in general, since it depends in a complicated way on the geometry, load level and nonlinear stress-strain behavior. To avoid increasing the differences in the values of the plastic SIF in a full range of biaxial nonlinear deformation, one can use Lee and Liebowitz’s algorithm (197 7) for numerical determination of the J -integral under large-scale yielding conditions:   2 0 , f f ij j i x J w W dy n u ds E       (5) where where  is a curve that surrounds the crack tip, starting from the lower crack flank, traversing counterclockwise, and ending on the upper crack flank; s is the arc length; n i is the outward unit vector normal to the curve; and the dimensionless displacement. The dimensionless strain energy density containing in the expression (7) is described by the following equation:

n

1

1 2 









  



n   e

2

2

1

W







e 



(6)

f

kk

n

3

6

1   

It should be noted that the J f -integral components are determined through the corresponding nodal stresses and displacements of the specified cracked FE problem. Substituting Eq.(6) into Eq.(5) gives expression for the J integral as the dimensionless stress and displacements angular functions accounting for their derivatives with respect to the polar angle:





2

r

n

0 

1

1 2 









  

  



FEM

n

2

2

1



J

d  

cos

e 



e 









f

kk

E

n

3

6

1



(7)













u

u

u

u









  

  

  

  





r

d     

d  

cos

sin







 

r 

r 

r

r





rr

rr

0

0





r

r





















Given the J -integral formulation for large-scale yielding in the form of Eq. (7) the expression for the plastic