Issue 78

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

Substitution of Eqs. (8) and (9) into Eq. (10) results in the following equation for plastic SIF:

 1



    

 



n

1

2





2

2 K K

  2

K K

4



    I w

   1 1 4

1

2

1 2

(11)

 

K

P

2

 0

 

n

The elastic SIF for a finite-size CCP subjected to biaxial loading can be calculated as:

  

(12)



 

 a F

K

n

K

1

1

where a is the crack length,  n is the nominal applied stress,  is the loading biaxiality ratio, w is plate width, F Ki (  ) is specimen geometry correction factor. In the numerical calculations of this work using the finite element method, the value of the parameter w was chosen to be several times larger than the crack size a in order to simulate a state close to an infinite size cracked body. For convenience in subsequent comparisons, the elastic SIF is introduced in the following dimensionless form:          1 1 1 n K K K a F (13) For Mode I loading, the expression for the plastic SIF given by Eq. (11) can be expressed as:

 1

2

   

   



                     0 1 1 4 n n a I w

n

1

SSY

2

(14)



K

K

P

1

The I n -integral in Eq.10, Eq.11 and Eq.14 governs the elastic–plastic crack-tip stress–strain field. This parameter is a function of the material's strain hardening exponent n and the angular distributions of the stress and displacement components. Shlyannikov and Tumanov [7] proposed a numerical procedure for the computation of the I n -integral for a wide range of cracked bodies configurations, based on elastic–plastic FE analysis of the near-tip fields:

         

          

      

      

 

  

FEM

du 

FEM FEM r rr u      



d



n

  1   

cos

sin

n FEM e













1

n







  

  

FEM

du 

(15)







, 

, , M n a w

. 

FEM

I

d



FEM FEM

u 

r  







n

p

r



d







1





cos

FEM FEM FEM FEM rr r r u u         





1

n  

Extensive plasticity The applicability of SSY fracture analysis is commonly limited to cases where the plastic zone at the crack tip is small relative to the crack size. For an infinite cracked plate, this condition is satisfied when the plastic zone size is less than about one order of the crack length and the nominal stress (normalized by the yield stress) does not exceed ~0.2. Within this regime, a direct relationship exists between the plastic SIF and the elastic SIF. However, as the applied nominal stresses increase to the level at which extensive plasticity develops, the plastic zone becomes comparable to the crack length, and the plastic SIF formulation requires appropriate modification. To avoid significant variations in the plastic SIF over the entire range of biaxial loading, the numerical procedure proposed by Lee and Liebowitz [20] can be adopted to determine the J-integral in extensive plasticity regime:         2 0 , f f ij j i x w J W dy n u ds E (16)

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