## 3 Results
### Cracked Pipes Without Patch Repair  
To verify the proposed method used for calculating fracture parameters, unpatched cylindrical pipes with an axial crack length of \(2a\) and axial length \(L\), as shown in Figure 6, are analyzed. In these analyses, the nondimensional SIF, \(F\), is considered, and its value for the opening mode is calculated as follows [34]  
\[F=\frac{\sqrt{GE}}{q\frac{\mathbb{R}}{I}\sqrt{\pi a}} \tag{21}\]  
where \(G\) represents the energy release rate calculated using the VCCT, \(E\) denotes the elastic modulus of the pipe, \(q\) is the internal pressure, and \(R\) and \(t\) indicate the pipe's radius and thickness, respectively. Regarding the material properties of the steel pipe, \(E\) is 200 GPa and [PERSON]'s ratio is 0.3. The geometrical parameters \(R\), \(L\), and \(t\) are specified as 60 mm, 150 mm, and 3 mm, respectively. The internal pressure, \(q\), is set to 1 MPa. Due to symmetry, only one-quarter of the computational domain is considered. Figure 7 illustrates the finite element mesh of the computational domain, modeled using the proposed approach. A 7 \(\times\) 6 element mesh is constructed in both the \(x\)- and \(\theta\)-directions, with a single layer of elements applied in the thickness direction. In Figure 7, the meshing strategy follows a functionally refined model.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Laminate** & **Present Analysis** & **LW Element [25]** & **Reference [32]** & **Reference [33]** \\ \hline One layer (0) & 9.52 & 9.52 & 8.85 & 9.31 \\ Two layers (0/90) & 4.56 & 4.56 & - & - \\ \hline \end{tabular}
\end{table}
Table 1: Maximum displacements of the clamped cylindrical shell.  
Figure 5: Convergence of maximum deflections in a cylindrical shell with two layers.  
Generally, when a VCCT is applied to compute fracture parameters, very fine elements of size \(\Delta a\) are used near the crack tip. This often results in large aspect ratios of elements, which can lead to the performance degradation of elements in conventional finite element methods. According to reference [35], elements with a \(p\)-level of 5 or higher, formulated using Lobatto shape functions, can tolerate large aspect ratios: up to 1000 with less than 3% of relative error. In the present model, all elements have aspect ratios of less than 500, ensuring numerical stability. Stress oscillations typically occur near crack tips in the VCCT when the virtual crack closure length \(\Delta a\) approaches zero [36]. Therefore, it is essential to determine an appropriate range for \(\Delta a\) before estimating fracture parameters. Reference [37] recommend \(\Delta a/a>0.05\) when using the VCCT with conventional finite elements with first- or second-order shape functions based on Lagrangian polynomials. Figure 8 illustrates the sensitivity of \(\Delta a\) in the VCCT across different \(p\)-levels in the present model when the crack length \(a\) is 9 mm. It is observed that the computed nondimensional SIFs remain consistent within the range \(0.015\leq\Delta a/a\leq 0.05\). Table 2 presents the computed nondimensional SIFs for cracked pipes with increasing crack sizes ranging from 3 mm to 18 mm, and these values are compared with the results of other solutions. The discrepancies remain within a relative error of 10%, and the results obtained using the present model fall within this acceptable range.  
Figure 6: Geometric configuration of an axially cracked pipe.  
Figure 7: Finite element mesh of the present analysis.