Issue 77

M. Rehaman et alii, Fracture and Structural Integrity, 77 (2026) 45-55; DOI: 10.3221/IGF-ESIS.77.04

initiation angle estimated from FE analyses was compared with the proposed GMPZR fracture criterion, other theoretical fracture criteria and available experimental results, iv) from the FE analyses an attempt is been made to estimate approximate formulation for GMPZR and crack initiation angles based on stress intensity factors under mixed-mode (I/II) loading, and v) finally, the effects of T-stress on crack initiation angle estimated from GMPZR criterion were also studied

G ENERALIZED MINIMUM PLASTIC ZONE RADIUS CRITERION

A

generalized minimum plastic zone radius (GMPZR) criterion is an extension of the MPZR [9, 19] criterion. It states that the direction of crack initiation coincides with the direction of minimum plastic zone radius evaluated from the von Mises yield criterion. The radius of PZ ahead at the crack tip contains stress intensity factors and T stress. The crack initiation can be determined by minimizing r [9, 19]:

2        

2 r    

r        





0

0

(1)

   

o

o

The stresses near the crack tip expressed in terms of Cartesian co-ordinates (x, y) are given as:

          

3                             2 2 3 2 2            



     

         

 

2        

3             cos 2 2  

  

  

cos 1 sin sin 

2 cos 

sin

2

0 0                 T

xx         yy xy     



2 2  

3             2 

K

K

I

II



cos 1 sin sin 



sin sin

(2)

cos

2





r

r

2

2

    

   

cos sin cos 2 2  

3             2 



3                2  

 

cos 1 sin sin 2 2  

zz xz yz      

0

Plane stress

(3)

where K I and K I I are the stress intensity factors in Mode I and Mode II, respectively. The Von Mises yield criteria for a 3D object can be written in the following form       2 2 2 2 2 xx yy yy zz zz xx y             

(4)

One can substitute the singular stress field of Eq. (2) in the above yield criterion Eq. (4) and solve for the PZ radius r. It can be expressed as follows.



  2 II K 







1

3 4

15

9

  





2

2

2



1 cos2 2cos   





, , ) 



( , r K K T

K

2sin

cos2

I

II

I

2



2

4

2 4

4

y





2

K T r 





T             

13

3 8

2 2

1

3 5 cos

  

  

I

sin 2 sin 3 2sin    









K K

cos

(5)

I II



4

2 2 2 2

2

y

y

K T r 





2 2 13  

3 5

  

II



sin sin 2 2 4 2 



2



y

By substituting Eq. (5) in Eq. (1), we get,

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