Issue 78

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

where ε denotes the strain, σ the stress, σ 0 the yield stress, E the elastic modulus, n the strain hardening exponent and α the Ramberg-Osgood model hardening coefficient. When the nonlinearity is introduced through this constitutive equation for a strain-hardening material, the singular solution developed by Hutchinson [1,2] takes the following form:             1 1 , n ij P ij r K r (2)



1

    , r

    



P K r

(3)



n

1

e

e

             1 , n P n n P ij P ij r K r

(4)

 1

  ,

  

 K r u 1 n n P

  



u r

i

,

1,2

(5)

i

i

1 2

3 2

  

  

(6)

  e





s s

1

ij ij

,max

max

  , ij r    is the dimensionless angular stress function,

here r and  are polar coordinates with origin at the crack tip,

P K is

e  is the equivalent (von Mises) stress normalized by yield stress σ 0 , and r is the crack tip distance

the plastic SIF,

normalized by crack length r r a  . The corresponding strain components are expressed as:

3 4







n

1

         





rr

rr

e

rr

3 4





(7)



n

1

         







rr

e

rr

3 2

 1  





n

1

r 

    







r

e

r

Small scale yielding As shown by Shih [6], the plastic SIF in pure Mode I (or pure Mode II) can be related directly to the elastic SIF via the J integral [5]. For the general case of mixed mode biaxial loading, the components of the J - integral in the local Cartesian coordinates system are expressed through the elastic SIF K 1 and K 2 as follows:            2 2 1 2 1 1 4 x J K K E (8)         1 2 2 1 1 4 y J K K E (9) where E is the Young’s modulus, ν is the Poisson’s ratio, k = 3-4 ν . For an infinite-size CCP subjected to biaxial loading, the plastic SIF is related to the J-integral by Shih’s relation in SSY regime, which is represented as [6]:

 0 2



n

2

2 y

1

   J J

(10)

J

I K

x

n P



E

262