## 3 Results and Discussion
### Application of ANN Model for the Estimation of Critical Crack Size of a Pipe  
In fracture mechanics, the critical crack size (\(a_{\rm C}\)) is the maximum size of a crack that an engineering component can tolerate without failing. The \(a_{\rm C}\) is a vital consideration in engineering design and structural integrity assessments. Engineers need to ensure that critical crack sizes are adequately controlled and do not reach unsafe levels that could lead to catastrophic failures. To estimate the critical crack size (\(a_{\rm C}\)), the \(K_{\rm I}\) at the crack tip is determined. If the \(K_{\rm I}\) is higher than the fracture toughness of the material (\(K_{\rm C}\)), the crack propagates rapidly, and a sudden fracture of an engineering component is possible [4]. As a case study, the \(a_{\rm C}\) of a 304 stainless steel cracked pipe with a \(t/R_{\rm i}\) of 0.012, a thickness of 9.5 mm, and subjected to an internal pressure of 1.5 MPa [25] was estimated using FEA, API 579-1/ASME FFS-1, and ANN. The relationships between \(K_{\rm I}\) and \(a\) obtained from FEA, API 579-1/ASME FFS-1, and ANN are shown in Figure 9. It is observed that \(K_{\rm I}\) increases with \(a\). To identify the \(a_{\rm C}\), the fracture toughness (\(K_{\rm C}\)) of 304 stainless steel (i.e., 120 MPa.m\({}^{1/2}\)[26]) is indicated in the figure. At the point where the calculated \(K_{\rm I}\) equaled the \(K_{\rm C}\) value, the corresponding crack size was determined as the \(a_{\rm C}\). The \(a_{\rm C}\) values obtained from FEA, API 579-1/ASME FFS-1, and ANN are 6.78, 6.50, and 6.82 mm, respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Zone**} & \multicolumn{2}{c}{**Testing Data**} & \multicolumn{2}{c}{**Normalized \(K_{\rm I}\)**} & \multicolumn{2}{c}{**Difference (\%)**} \\ \cline{2-9}  & \(t/R_{\rm i}\) & \(a/t\) & **FEA** & **ANN** & **BiLerp** & **ANN vs. FEA** & **BiLerp vs. FEA** \\ \hline \multirow{3}{*}{linearity zone} & 0.002 & 0.1 & 2.3657 & 2.4114 & 2.4676 & 1.93 & 4.31 \\  & 0.02 & 0.1 & 2.3717 & 2.4074 & 2.4578 & 1.51 & 3.63 \\  & 0.25 & 0.1 & 2.2993 & 2.3358 & 2.3715 & 1.59 & 3.14 \\ \hline \multirow{3}{*}{nonlinearity zone} & 0.002 & 0.5 & 5.6197 & 5.5627 & 6.1076 & 0.59 & 8.68 \\  & 0.002 & 0.7 & 12.5583 & 12.6429 & 15.5746 & 0.67 & 24.02 \\ \cline{1-1}  & 0.012 & 0.5 & 5.5399 & 5.5885 & 5.9715 & 0.88 & 7.79 \\ \cline{1-1}  & 0.012 & 0.7 & 11.9121 & 11.7481 & 14.1883 & \(-\)1.38 & 19.11 \\ \cline{1-1}  & 0.02 & 0.5 & 5.4693 & 5.5277 & 5.8699 & 1.07 & 7.33 \\ \cline{1-1}  & 0.04 & 0.7 & 10.3687 & 10.1446 & 11.5745 & \(-\)2.16 & 11.63 \\ \cline{1-1}  & 0.075 & 0.5 & 5.0328 & 5.0686 & 5.2976 & 0.71 & 5.26 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison between the predictions of normalized \(K_{\rm I}\) from the bilinear interpolation of API 579-1/ASME FFS-1 and the ANN model. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Zone**} & \multicolumn{2}{c}{**Testing Data**} & \multicolumn{2}{c}{**Normalized \(K_{\rm I}\)**} & \multicolumn{2}{c}{**Difference (\%)**} \\ \cline{2-9}  & \(t/R_{\rm i}\) & \(a/t\) & **FEA** & **ANN** & **Lerp** & **ANN vs. FEA** & **Lerp vs. FEA** \\ \hline \multirow{3}{*}{linearity zone} & 0.002 & 0.4 & 4.2120 & 4.2022 & 4.2052 & \(-\)0.23 & \(-\)0.16 \\  & 0.15 & 0.4 & 3.6844 & 3.7103 & 3.6821 & 0.70 & \(-\)0.06 \\  & 0.25 & 0.4 & 3.3961 & 3.3419 & 3.3976 & \(-\)1.60 & 0.04 \\ \hline \multirow{3}{*}{nonlinearity zone} & 0.002 & 0.6 & 8.0180 & 8.0103 & 8.0101 & \(-\)0.10 & \(-\)0.10 \\  & 0.006 & 0.6 & 7.9288 & 7.9085 & 7.9087 & \(-\)0.26 & \(-\)0.25 \\ \cline{1-1}  & 0.006 & 0.8 & 22.1026 & 22.0536 & 22.1218 & \(-\)0.22 & 0.09 \\ \cline{1-1}  & 0.012 & 0.6 & 7.7972 & 7.7747 & 7.7838 & \(-\)0.29 & \(-\)0.17 \\ \cline{1-1}  & 0.02 & 0.6 & 7.6293 & 7.6160 & 7.6130 & \(-\)0.17 & \(-\)0.21 \\ \cline{1-1}  & 0.02 & 0.8 & 18.9014 & 18.8772 & 18.9039 & \(-\)0.13 & 0.01 \\ \cline{1-1}  & 0.04 & 0.8 & 15.7327 & 15.7047 & 15.9116 & \(-\)0.18 & 1.14 \\ \hline \end{tabular}
\end{table}