PSI - Issue 28

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

877

5

z

w ( r )

p

c = 0

r = a 0

r = a

c = c 0

Fig. 2. The internal crack growth with applied gas pressure.

Therefore, for each moment of time, two separate mathematical problems need to be solved, namely (1) the diffusion problem, about hydrogen diffusing into crack cavity and (2) the elasticity problem, about finding the crack aperture under the loading created by gas pressure. The two problems are related by an equation of the gas state. In previous work, the Ideal Gas Equation (IGE) was used to relate problems (1) and (2). The disadvantage of using the IGE to describe this relationship is that it only provides an approximation. In this paper, we make a substantial improvement

on the previous model by using the Real Gas Equation (RGE) to relate problems (1) and (2). The expression for the aperture is given as follows [e.g., Timoshenko and Goodier, 1970]:   2 2 2 4 1 ( ) ( ) w r a t r E     

(1.1)

where p is pressure, E is the Young’s Modulus for a given material, and v is the Poisson Ratio. The connection between the fracture toughness K c , crack radius, a , and the gas pressure, p , can be written as:

2

(1.2)

c K

p a





Then, the expression for the crack volume reads

3

4

pa

(1.3)

V





3

E

E

 

where

is the Effective Young’s Modulus .

E



 2

1





From the expression (1.2) of fracture toughness, the critical gas pressure within the crack is given thus: