PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1237–1248 Author name / Structural Integrity Procedia 00 (2019) 000–000

1240

4

  2 0

  0

'' i 

' 

( ) R

(0)   

R

R





,

(6)

1!

2!

Where

1 0 R R   .

(7)

By introducing of coefficients, 1 q , 2 q and 3 q , formula (6) takes the form

2 1 ( ) R q q R q R     . 3 2

(8)

The unknown coefficients, 1 q , 2 q and 3 q , are determined by using the Ramberg-Osgood equation (2). For this purpose, by substituting of (8) in (2), one obtains

n 1





2

  E q q R q R 2 3 2 1

n H q q R q R 1 3 1 2  







.

(9)

There are four unknowns,  , 1 q , 2 q and 3 q , in equation (9) where  is the longitudinal strain in the internal crack arm. These unknowns are determined in the following way. First, one equation is derived by substituting of 0  R in (9)

n 1 1 .

H q

q 1 

  E

(10)

n 1

0  R in the first and second derivatives of (9) with respect

Further, two other equations are obtained by substituting of

to R

1 1



'

n nH q q 1 1 n

 E q E q E 2 1 2

0





2

,

(11)

 1 2

1

n

n

  

   ,



1 1

n



  2

  '

  0 1 2 

''

'

 q q q q n n 2 2

  q E q E E q E q E E 



(12)

3

1

2

1

1

2

1

3

3

1 n

n

E

nH

where ' E and '' E are, respectively, the first and second derivatives of the modulus of elasticity with respect to R (it should be noted that these derivatives are obtained at 0  R ). Finally, one equation is written by considering the equilibrium of the elementary forces in the cross-section of the internal crack arm behind the crack front