PSI - Issue 16

Ivan Shtoyko et al. / Procedia Structural Integrity 16 (2019) 148–152 Ivan Shtoyko, Jesús Toribio, Viktor Kharin, Myroslava Hredil / Structural Integrity Procedia 00 (2019) 000 – 000

150

3

Table 1. The chemical composition of the Х52 steel. Element С Si Mn Cr

Fe

Ni

Mo

S

Cu

Al

wt. %

0.206

0.293

1.257

0.014

0.017

0.006

0.009

0.011

0.034

balance

Table 2. Mechanical properties of the Х52 steel. Yield strength  ys , MPa

Ultimate tensile strength  UTS , MPa

Elongation  , %

410 30.2 According to the experimental research of Gabetta et al. (2008), a crack in the pipe under long-term static loading and soil corrosion propagates mainly with constant rate . For the non-operated pipe the rate is (0) = 1,03 ⋅ 10 −3 m/year and for the steel after 30 years of operation (30) = 8,03 ⋅ 10 −3 m/year. Based on these experimental data, the equation for approximate determination of the crack growth rate dependent on the operation time t , i.e., ≈ ( ) , for the pipe made of the X52 steel may be written as follows: ( ) = 10 −3 [1,03 + 0,23( 0 + )] (m/year), (1) where 0 – is the time of pipe operation before gas pipeline residual lifetime evaluation. Following Andreikiv et al. (2007, 2017), the energy approach is implemented to solve the problem. As a result, it is reduced to the mathematical model that renders the following governing equation: = ( )√1 + −2 2 2 , (2) = 0, = 0, (0, ) = 0 ( ); = ∗ , ( ∗ , /2) = ℎ, where , – coordinates of polar system Oαρ which follows the shape of the corrosion-mechanical crack. The equation (2) is nonlinear partial differential equation. It may be solved approximately using the next assumption: the initial crack has semi-elliptical configuration and its propagation rate is constant. Then, it can be supposed that the crack only slightly differs from the elliptical shape during its propagation. Under these assumptions, the equation (2) reduces to the next differential equation system: = 10 −3 [1,03 + 0,23( 0 + )], = 10 −3 [1,03 + 0,23( 0 + )], (3) = 0, (0) = 0 , (0) = 0 ; = ∗ , ( ∗ ) = ∗ , ( ∗ ) = ℎ. (4) Solving the system of differential equations (3) subjected to the initial and final conditions (4), of the changes of the semi-axes of the elliptical crack front are obtained as follows: ( ) = 0 + 10 −3 (1,03 + 0,23 0 + 0,01 2 ), (5) ( ) = 0 + 10 −3 (1,03 + 0,23 0 + 0,01 2 ). Applying the final condition (4) to the solution for b ( t ) in (5), the next equation for the residual lifetime t * is obtained: ∗ 2 + (103 + 23 0 ) ∗ − 10 5 (ℎ − 0 ) = 0 . (6) 528