PSI - Issue 39

Dmitry O. Reznikov / Procedia Structural Integrity 39 (2022) 256–265 ReznikovD.O./ Structural Integrity Procedia 00 (2019) 000–000

260

5

ratios of overload cycles) are unknown random variables, the mean of random values of maximum nominal hoop stresses E {ΔΣ} = E {Σ max }– S min and the stress ratios E { R Σ }= S min / E {Σ max } are used in the equation describing the kinetics of the crack growth under underloads. Then one may write down the following equation:

2

    

2 m m  

 



Y E 

2

 

m

.



m

   

  



(5)

1

a

2 a C N 



    

  

N N 

N

k

2 1 { } E R  

r

k

r





Then, taking into account the equation (3), the crack depth after N r cycles of constant amplitude loading and N k overloads will be determined by the expression:

2

    

      

    

m

2 m m 

   

   

   

   

2

m



{ } 

m

S

E



(6)

  Y 

 

  

m

1   

a

a

N C

N C  





2

0

N N 

r

k

2

1

1 { } E R 

R



r

k

r



The criterion of brittle fracture is applied:

K Y

a K  

 

.

(7)

max

I

C

Ic

where Σ max is the maximum nominal hoop stress in the loading cycle that occurs during the most severe overload in the series of N k overloads (Σ max = max {Σ 1 , Σ 2 ,…, Σ Nk }), K Ic is the fracture toughness. In this case, one can make a conservative assumption that the maximum stress Σ max occurs in the last overload, when the function ξ of the load bearing capacity of the component with a cyclically growing crack reaches its minimum value. The condition for brittle fracture of the component under consideration is:

I Ic K K 

(8)

The expression for the critical crack depth a C can be obtained from equation (7):

2





K

.

(9)

a

 

Ic

max     Y 

C

In this case, the fracture condition (8) for a given level of maximum stresses (Σ max ) can be expressed in terms of the crack depth: r k N N C a a   . (10) Taking into account equations (6) and (9), the condition of the pipeline component fracture after the application of N r cycles of regular loading and N k overloads takes the form:

2

    

     

    

m

2 m m 

2

.

   

   

  

   



  

2

m



{ } 

1     m

S

E

K

(11)



  Y 

m

a

N C

k N C   



  

Ic

2

 

 

0

r

2

1

1 { } E R 

R

Y









max

r

