# Identifying the Initial Corrosion Fatigue Failure Based on Dropping Electrochemical Potential
## 3 Results and Discussion
### Estimating Early Failure from Electrical Potential
To prevent intermissions, another 10 balances are generated once 10 positive balances are completed to delay the final intermission allowing for more conditions during the 50 s period. Therefore, the second condition can newly be fulfilled after each intermission. Condition 3:  
The third condition comprises the calculation of the gradient (m) and the coefficient of determination (\(R^{2}\)) of each time course after fulfillment of condition 2. The gradient describes the intensity of the direction of the trend. The coefficient of determination allows for data points to be described as a function [28; 29]; here, a polynomial of first degree or a best-fit straight line is appropriate. The gradient is determined via the least square method (Equation (7)). \[m=\frac{\sum_{i=1}^{n}\left(x_{i}-\overline{x}_{j}\right)\cdot\left(y_{i}- \overline{y}_{j}\right)}{\sum_{i=1}^{n}\left(x_{i}-\overline{x}_{j}\right)^{2}} \tag{7}\]  
\(m=gradient\)  
\(x_{i}=number\)\(of\)\(data\)\(points\)\(within\)\(time\)\(course\)\((1\)\(to\)\(10)\)  
\(\overline{x}_{j}=arithmetic\)\(mean\)\(of\)\(numbers\)  
\(y_{i}=data\)\(point\)\(of\)\(time\)\(course\)  
\(\overline{y}_{j}=arithmetic\)\(mean\)\(of\)\(time\)\(course\)  
Because the consecutive numbering of data points as x-values is consistent in all time courses the following simplification is possible (Equation (8)):  
* The arithmetic mean of numbering is \(\overline{x}_{j}=5.5\) for the mean of time courses 1 to 10;
* \(\sum_{i=1}^{n}\left(x_{i}-\overline{x}_{j}\right)^{2}\) is the distance between \(x\)-values and the arithmetic mean of numbering of the particular time course with respect to the Cartesian grid to the second power. Because the numbering remains the same in each time course, it contributes as a constant to simplify Equation (7) to calculate the gradient. Inserting numbers 1 to 10, the gradient is 82.5. \[m=\frac{\sum_{i=1}^{n}\left(x_{i}-5.5\right)\cdot\left(y_{i}-\overline{y}_{j} \right)}{82.5} \tag{8}\]  
Equation (9) gives the calculation of the coefficient of determination (\(R^{2}\)). Figure 7:0  
The simplification for determining the gradient also applies for the coefficient of determination (\(R^{2}\)) (Equation (10)). Figure 7:1  
To fulfill condition 3, condition 1 and 2 have to be fulfilled. The gradient is set for m \(\leq-1\) (subcondition 1) and is sufficiently high so that times courses only apply at the end of the endurance test. The coefficient of determination (\(R^{2}\)) is set for \(R^{2}\geq 0.8\) (subcondition 2). This only occurs for time courses where most values are smaller than their precursor, so that these only scatter slightly from the partial regression line. The necessary conditions may be summarized as follows:  
Condition 1* Each data point is compared to zero;
* For positive values, the time course is marked;
* After completion of 10 positive values, the time course is eliminated and not evaluated;
* Only fully negative data series are evaluated. * Condition 2
* Arithmetic means are calculated for all time courses completed with negative num-bers (non-marked);
* Balances of arithmetic mean values are calculated giving a trend for balance \(\geq\) 0 (zero is interpreted as positive value);
* Intermission for \(\geq\)10 positive balances leads to a delay in data points. * Condition 3
* Conditions 1 and 2 need to be fulfilled;
* Gradient (\(m\)) and coefficient of determination (\(R^{2}\)) immediately after fulfillment of condition 2;
* \(m\leq-1\);
* \(R^{2}\geq\) 0.8;
* Two sub-conditions need to be fulfilled;
* Determination of experiment for three consecutive time courses applying to both sub-conditions. * Sub-condition 1
* If \(m\leq-1\), mark the time course;
* If \(m\geq-1\), reset. * Sub-condition 2
* If \(R^{2}\geq\) 0.8, mark the time course;
* If \(R^{2}\leq\) 0.8 reset. During a first test run, the algorithm was reliable for 14 out of 15 specimens giving a functional probability of 93.3%. Greater reliability may be achieved by adapting the boundary conditions and will be subject to further research.