PSI - Issue 33

I.A. Evstafeva et al. / Procedia Structural Integrity 33 (2021) 933–941 Author name / Structural Integrity Procedia 00 (2019) 000–000

936

4

3. Solution of the problem 3.1. Equivalent stress

According to the known solution of the thermoelasticity for a pipe subjected to internal and external pressures of media with different temperatures, the hoop stresses on the inner and outer surfaces can be express as follow



    

( E T T R    ) 2 R r

2

2

2

p r p R p p  

1

2

( ) r      ( ) r E T

( ) r

;

R









r

R

r

2 R



R

2 R r 

2

2 R r 

2 R r

2

2(1 )  

 

ln

r





    

( E T T r    ) 2 R r

2 p r p R p p   2

2

1

2

( ) R R R      ( ) ( ) E T

.

r









r

R

r

2 R



R

2 R r 

2

2 R r 

2 R r

2

2(1 )  

 

ln

r



Rewriting these expressions in terms of the pipe thickness h R r   and midsurface radius c R , then expanding them into a Taylor series in powers of / c h R , and neglecting the terms / c h R of a degree greater than first, gives

pR



E T  

( ) r

;

c    p



(3) (4)

R

2(1 )  

h pR



E T  

( ) R

.

c    p



r

2(1 )  

h

These expressions differ from the classical thin-shell theory formulas only by constant terms (that do not change during the corrosion process). 3.2. Refined solution Since thin-walled shells are considered, the change in the midsurface radius c R during the corrosion process can be neglected. Substituting Eqs. (3) and (4) into the combination of Eqs. (1) and (2) yields the following equation for the evolution of the pipe thickness with time

pR

pR

( dh d R r  )





  

  

  

  









R R R T T  

 

 

 

exp

exp

,

th   

th

A m

A m

T T 

r 



  

c

c

(5)



R R

r

r

r

r

dt

dt

h

h

where

2(1 ) . r R r E T A a m p m E T A a m p m               2(1 ) R R R r R r r

;

0 0 0 (0) h h R r    ) results in the

Separating the variables in Eq. (5) and integrating (with initial condition

following solution

0 ln , h h M Ah M t A A Ah M      2

(6)

0

where