Observation and Modeling of Stress Corrosion Cracking in High Pressure Gas Pipe Steel

Abstract

Stress corrosion cracking (SCC) is commonly observed to form a colony of closely spaced multiple cracks. Four stages of SCC colony evolution are discussed. The first is the colony initiation stage (CIS), which is associated with formation of corrosion pits randomly distributed over a certain domain of the surface exposed to an aggressive environment. Electrochemical processes play a leading role in CIS. The individual crack growth (ICG) driven by a combination of mechanical stresses and electrochemical processes constitutes the second stage. At the end of the second stage, the individual cracks reach certain proximity of one another resulting in much crack interaction. This becomes a transition to the third, strong crack interaction and clusters formation, stage. Cluster growth and individual crack or a cluster instability leading to the ultimate failure constitute the final, fourth stage of the SCC evolution process. In this article, we present observations and a general approach to modeling the first two stages of SCC, i.e., CIS and ICG, that together constitute the major part of the total lifetime of an engineering structure serving under SCC conditions. A computer simulation of individual SC crack growth is developed and compared with a large set of SCC observation data.

Appendix A

1.1 Representation of the concentration of hydrogen at the crack tip

The distribution of the concentration of hydrogen, C(x, t), satisfies the diffusion equation

$$ {\frac{\partial C(x,t)}{\partial t}} = D{\frac{{\partial^{2} C(x,t)}}{{\partial x^{2} }}} $$

(A1)

The vector flux of hydrogen is $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q} (x,t)_{|x = 0} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q} (t) = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q} \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{e}_{1} ,$ and the boundary and initial conditions are defined as

$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q} (x,t)_{|x = 0} = \left\{ {\begin{array}{*{20}c} { - k(R,t) \cdot \left[ {C(x,t)_{|x = 0} - C_{\max } } \right],\begin{array}{*{20}c} {\text{if}} & {C(x,t)_{{\left| {x = 0} \right.}} > 0} \\ \end{array} } \\ {0,\quad \begin{array}{*{20}c} {\text{if}} & {C(x,t)_{{\left| {x = 0} \right.}} = 0} \\ \end{array} } \\ \end{array} \quad {\text{and}}\;C(x,t)_{|t = 0} = 0} \right. $$

(A2)

where C _max stands for the saturated hydrogen density and the coefficient k depends on the stress ratio and time. In addition, the solution of Eq. [A1] can be expressed as a standard form:

$$ C(x,t) = {\frac{C(0,t)}{{\sqrt {\pi 4Dt} }}} \cdot \exp \left[ { - {\frac{{x^{2} }}{4Dt}}} \right] $$

(A3)

We can define C(0, t) = C ₀(t); thus, Eq. [A3] represents the shape of the distribution in time. Green’s function for density C(x, t) due to unit mass concentrated at (ξ, τ) can be represented as

$$ G(x,t;\xi ,\tau ) = {\frac{1}{{\sqrt {4\pi D(t - \tau )} }}}\exp \left[ { - {\frac{{(x - \xi )^{2} }}{4D(t - \tau )}}} \right] $$

(A4)

For τ domain and x = 0, i.e., hydrogen diffused only from the crack tip,

$$ G(x,t;\xi ,\tau ) = {\frac{\Updelta M(\tau )}{{\sqrt {4\pi D(t - \tau )} }}}\exp \left[ { - {\frac{{(x - \xi )^{2} }}{4D(t - \tau )}}} \right] $$

(A5)

where

$$ \Updelta M(t) = \dot{C}_{0} \Updelta \tau = k\left[ {C_{\max } - \left. {C(x,\tau )} \right|_{x = 0} } \right] \cdot \Updelta \tau $$

(A6)

Plugging Eq. [A6] into Eq. [A5] leads to

$$ G(x,t;0,\tau ) = \int_{0}^{t} {{\frac{{k\left \lfloor {C_{\max } - \left. {C(x,\tau )} \right|_{x = 0} } \right \rfloor}}{{\sqrt {4\pi D(t - \tau )} }}}\exp \left[ { - {\frac{{x^{2} }}{4D(t - \tau )}}} \right]} d\tau $$

(A7)

Plugging Eq. [A7] into Eq. [A6] leads to

$$ C(x,t) = \int_{0}^{t} {{\frac{{k\left \lfloor {C_{\max } - \left. {C(x,\tau )} \right|_{x = 0} } \right \rfloor}}{{\sqrt {4\pi D(t - \tau )} }}}\exp \left[ { - {\frac{{x^{2} }}{4D(t - \tau )}}} \right]} d\tau $$

(A8)

For $\left. {C(x,t)} \right|_{x = 0} = C_{0} (t),$ Eq. [A8] can be rewritten as

$$ C_{0} (t) + \int_{0}^{t} {{\frac{k}{{\sqrt {4\pi D(t - \tau )} }}}C_{0} (\tau )} d\tau = {\frac{{kC_{\max } }}{{\sqrt {4\pi D} }}} \cdot \int_{0}^{t} {{\frac{d\tau }{{\sqrt {t - \tau } }}}} $$

(A9)

So, Eq. [A9] can be expressed as an integral equation as

$$ C_{0} (t) + \int_{0}^{t} {{\frac{k}{{\sqrt {4\pi D(t - \tau )} }}}C_{0} (\tau )} d\tau = {\frac{{kC_{\max } }}{{\sqrt {4\pi D} }}} \cdot \sqrt t $$

(A10)

Equation [A10] is a known special kind of integral equation, i.e., Volterra type of the second kind, so Eq. [A10] can be solved by using Laplace transforms and inverse Laplace transforms. The solution of the integral equation is as follows:

$$ C_{0} (t) = - {\frac{{2\sqrt D C_{\max } }}{{k\sqrt {\pi t} }}} + C_{\max } + {\frac{{2C_{\max } \sqrt D }}{k}}\left( {{\frac{1}{{\sqrt {\pi t} }}} - {\frac{{\exp \left[ {{\frac{{k^{2} t}}{4D}}} \right] \cdot k\left( {1 - {\text{erf}}\left[ {{\frac{k\sqrt t }{2\sqrt D }}} \right]} \right)}}{2\sqrt D }}} \right) $$

(A11)

Finally, Eq. [A11] can be expressed simply as

$$ C_{0} (t) = C_{\max } \left( {1 - \exp \left[ {{\frac{{k^{2} t}}{4D}}} \right]\left\{ {1 - {\text{erf}}\left[ {{\frac{k\sqrt t }{2\sqrt D }}} \right]} \right\}} \right) $$

(A12)

The general solution of C(x, t) can be obtained by plugging Eq. [A12] into Eq. [A8] as follows, and can be solved by numerical approaches:

$$ C(x,t) = {\frac{{kC_{\max } }}{{2\sqrt {\pi D} }}}\int_{0}^{t} {{\frac{{\exp \left[ {{\frac{{k^{2} \tau }}{4D}}} \right]\left\{ {1 - {\text{erf}}\left[ {{\frac{k\sqrt \tau }{2\sqrt D }}} \right]} \right\}}}{{\sqrt {t - \tau } }}}\exp \left[ { - {\frac{{x^{2} }}{4D(t - \tau )}}} \right]} d\tau $$

(A13)