PSI - Issue 60

D. Sen et al. / Procedia Structural Integrity 60 (2024) 44–59 Deeprodyuti Sen/ Structural Integrity Procedia 00 (2024) 000 – 000

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5

The coefficients ‘ g j ’ were obtained by Scarth (2002) using a series of finite element analysis and are

tabulated in the CSA Standard.

c. Closed form solutions for the crack mouth opening displacement for the virtual crack of size ‘ s ’ ahead of blunt notch under the application of stress field represented by cubic polynomial was developed by Scarth (2002). The opening displacement for the stress term is given by the following equation:-

j

A

o x       s

j

' 5.1869

v

h f

s



(5)

Tj

j j

( 1) 

E j

d. Similar to the case of a strip-yield model, an estimate of fictitious crack length ‘ s ’ is obtained from the condition that the stress singularity vanishes at the trailing edge of the process zone       , , 0 I I H K s x K s p    (6) e. The resultant crack mouth opening displacement considering the superposition of stresses can be expressed as, ( , ( )) ( , ) T T T H v v s x v s p    (7)

f. The final expression of Threshold Peak Stress is as follows

g f p

TH  

o o C

(8)

2

3

o   s     x

o   s     x

o   s     x

1 1 1 g f q g f q  o o o

g f q

g f q





2 2 2

3 3 3

g. The non- linear algebraic equation in ‘ s ’ to be solved is as follows , 2

0     s   x

v x

p

5.1869

g f

Z



c

C

0 0

1

'

E

o

   

2

1 1   f q  

  

  

o      

  

o   s    

h g

h g

h g s

(9)

h

f q

h

f q

h











3

3

1

1

2

2

2 2

3 3

o

o

o

2

3

4

g

g x

g x





0

0

0

Z



1

2

3

o   s     x

o   s     x

o   s     x

1 1 1 g f q g f q  o o o

g f q

g f q





2 2 2

3 3 3

Eq. 9 is solved numerically to obtain a converged value of ‘s’ and subsequently the peak threshold stress σ TH is estimated from eq. 8. In the solution procedure described above, the sampling length ‘x o ’ for fitting the stress profile is taken as ‘s’ i.e. the process zone length. By doing so, eq. 9 simplifies, but it should be noted that parameters such as ‘h i ’ and ‘g i ’ are implicit functions of ‘s’ . Once a converged value of ‘s’ is obtained, the parameters ‘h i ’ and ‘g i ’ are updated to satisfy eq. 9.