Issue 78

A. Zakharov et alii, Fracture and Structural Integrity, 78 (2026) 260-274; DOI: 10.3221/IGF-ESIS.78,16

where  is a counterclockwise contour surrounding the crack tip, extending from the lower to the upper crack flank, s is the arc length along the integration path; n i is the outward unit normal to the contour; and i u are the dimensionless displacements.

u E

i



(17)

u

i



w

0

The dimensionless strain energy density (SED) in the formula (16) is described by the following equation:                2 2 1 1 1 2 3 6 1 n f e kk e n W n (18) In Eqs. (16) and (18), the stress tensor and stress invariant are normalised by the yield stress  0 . To illustrate the extent to which the applied stress σ n can be expected to influence the relation between the J-integral and plastic SIF in the extensive plasticity regime under biaxial loading, a wide range of loading biaxiality ratios should be considered. There are several possible ways to use fully plastic strain hardening solutions to determine the behaviour at extensive plasticity regime, based on realistic tensile stress–strain curves involving both the elastic and plastic parts of the strain energy density. The method proposed by Lee and Liebowitz [20] for elastic–plastic solids is one of them. By substituting Eq.(18) into Eq.(16), the J-integral can be expressed in terms of dimensionless angular functions of stresses and displacements and their derivatives with respect to the polar angle:





2





 1 2





 

r

n

1

  

  





FEM

n

2

2

1

0

 e



 e

  d









J

cos

f

kk

E

n

3

6

1





(19)

















u

u

u

u

  

  

  

  





  2





r

r





     d



 r

 r

  d

r

co

s

s

in





rr

rr

0

0













r

r









where:

x r   





sin





cos

 





r

y      r





cos

(20)





s

in





r

dy r 

dS rd 

cos , 



For convenience’s sake, we introduce the following dimensionless quantities for the J-integral:





2

 

  



w

E







FEM

FEM

FEM

  FEM

0



, where

(21)

J

J

 , ij j x i n u dS J





J

Wdy

f

f

f

f

2

E



w

0





Based on the J-integral formulation for extensive plasticity defined by Eqs. (19) and (21), the plastic SIF can be expressed as follows:

 1

f FEM n

   

   

J

 1

1

(22)



 K w K  1 n

K

,

P

P

P

0



I

n

264