PSI - Issue 28

Alla V. Balueva et al. / Procedia Structural Integrity 28 (2020) 873–885 Author name / Structural Integrity Procedia 00 (2019) 000–000

883

11

2.3. Dimensionless analysis and comparison of real and ideal gas approximations

* / a a a   , on the

From (1.18), in real gas approximation, the dependence of the normalized crack growth,

dimensionless growth time, i t  = t i / t

* , where a * = α 2 , t * = α 2 / β can be written as follows:

   

  

3   2

3 2

a



2 a   

(2.1)

ln

1   

t a

 



2

2



2(

)

a





whereas from the previous work (Balueva and Dashevskiy, 1999), for the ideal gas approximation, this expression was

t a  

(2.2)

2

When the crack radius a  , then (2.1) becomes:

a        

2

3 2





2 a   

(2.3)

ln

t a

 

2

2

After differentiating both sides with respect to t , we come up with the following expression for the velocity of the crack growth a  in the approximation for real gas:

3 1

1



2

(2.4)

a   

a 

a 







2

2

2

a

2

a

whereas for ideal gas, from (2.2), the crack velocity:

a   

(2.5)

2

which means that in ideal gas approximation the crack spreads at the constant speed. From (2.4) one can see, that in real gas approximation, the crack even at infinity, does not grow at constant velocity, however, asymptotically still approaches a constant, besides the same as for the ideal gas approximation (expression (2.5)):

a as a    

(2.6)

2

It is also in good agreement with experiments, where slow crack growth with a constant speed is typical for the Hydrogen Induced Cracking (e.g., Balueva and Germanovich, 2012). 3. Conclusions In this paper we obtained an analytical, closed form solution for the internal crack growth under extreme pressures. This result will fill gaps in estimations of longevity in pipelines with cracks because now these calculations can be done not only limited to ideal gas approximation, but with real gas under pressures as they are in industry.