# Assessment of AC Corrosion Probability in Buried Pipelines with a FEM-Assisted Stochastic Approach
## Appendix A  
The purpose of this Appendix is to show that the shape of the holiday inside the pipeline insulating coating has a minimal influence on the probability of exceeding or not a certain limit for the current density and, for such a reason, it is reasonable to consider, in the calculation model, only the area \(A\) of the holiday itself. Often, in literature, the holiday in the insulating coating of a pipeline is modelled by supposing that the small portion of the pipeline, in correspondence with the coating defect, which is in contact with the soil, (that is, the exposed area) can be represented by a metallic circular plate placed at the air-soil interface. The circular shape is, of course, an extreme idealization, so one could further proceed by introducing a more realistic shape, that is the ellipse. Starting point is the paper [40]; the authors consider an electrostatic problem involving an elliptic metallic plate of area S\({}_{\text{e}}\) with semi-major axis a and semi-minor axis b having potential V and unknown surface charge density \(\eta=\eta(x,y)\). In practice, they found the solution of the following integral equation:  
\[V=\frac{1}{4\pi\epsilon_{0}}\iint\limits_{S_{e}}\frac{\eta(x^{\prime},y^{ \prime})}{\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}}} dx^{\prime}dy^{\prime}, \tag{1}\]which is given by:  
\[\eta(x,y)=\frac{2e_{0}V}{bK(k)}\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^{2}- \left(\frac{y}{b}\right)^{2}}}, \tag{10}\]  
where \(K(k)\) is the complete elliptic integral of the first kind that is:  
\[K(k)=\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}(\theta)}} \tag{11}\]  
\(k\) is the ellipse eccentricity (\(0\leq k<1\)) expressed by:  
\[k=\sqrt{1-\left(\frac{b}{a}\right)^{2}}, \tag{12}\]  
and \(e_{0}\) is the absolute vacuum permittivity. If the plate is in contact with a conductive medium having resistivity \(\rho\) (in the case of the present paper the soil) and by remembering the relationship between current density \(J(x,y)\) and surface charge density \(\eta(x,y)\) i.e.,:  
\[J(x,y)=\frac{\eta(x,y)}{2\rho e_{0}}, \tag{13}\]  
one finally has the expression of the current density on the elliptic plate, at potential \(V\), and placed at the surface of the soil:  
\[J(x,y)=\frac{V}{\rho bK(k)}\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^{2}- \left(\frac{y}{b}\right)^{2}}}. \tag{14}\]  
By looking at (14), one can notice that the current density is minimum at the center of the plate and tends to infinity by approaching the borders; nevertheless, one is interested in calculating an average (or equivalent) current density that is constant over all the area \(S_{e}\) of the plate so that its flux through it is equal to current leaked to soil \(I_{e}\). By integrating (14) over \(S_{e}\) one gets:  
\[I_{e}=\iint\limits_{S_{e}}\frac{V}{\rho bK(k)}\frac{dxdy}{\sqrt{1-\left(\frac {x}{a}\right)^{2}-\left(\frac{y}{b}\right)^{2}}}=\frac{4 Va}{K(k)\rho}\frac{\pi }{2}. \tag{15}\]  
Finally, by remembering that the area of the ellipse is given by \(S_{e}=\pi ab\), one can obtain the average current density \(J_{e}\) by means of:  
\[J_{e}=\frac{I_{e}}{\pi ab}=\frac{2V}{b\rho K(k)}. \tag{16}\]  
In the particular case when the plate has a circular shape of radius \(r\), one has \(a=b=r,\ k=0\) and \(K(0)=\pi/2\); thus, the average current density \(J_{c}\) is given by:  
\[J_{c}=\frac{4V}{\rho\tau\pi}=\frac{4V}{\rho\sqrt{A\pi}}. \tag{17}\]  
To assess the influence of the eccentricity on the current density leaked to the soil, it is useful to study the ratio \(\lambda\) between \(J_{e}\) and \(J_{c}\) for a given value of the plate area \(A\) in the function of the eccentricity \(k\); to do that, it also necessary to add that the semi-minor axis b can be expressed as:  
\[b=\left(1-k^{2}\right)^{1/4}\sqrt{\frac{A}{\pi}}. \tag{18}\]By considering (A8)-(A10) one finally has:  
\[\eta(x,y)=\frac{2e_{0}V}{bK(k)}\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^{2}- \left(\frac{y}{b}\right)^{2}}}, \tag{10}\]0 (A11)  
Notice that \(\lambda\) is a function of the only eccentricity.