# Fracture Toughness Determination on an SCB Specim by Meshless Methods
## 4 Analysis, Results and Discussion
In this work, two different versions of NNRPIM were used, named NNRPIMv1 and NNRPIMv2 referring to the first-degree and the second-degree influence-cell, respectively. Referring to the previous works conducted on meshless methods by the authors [16,19,26],  
Figure 2: Half of SCB specimen studied in this work: (**a**) meshless method model discretized by irregular nodal distribution and (**b**) FEM model discretized by triangular mesh, presenting the considered central coordinates, dimensions are in mm. the parameters and the coefficients influencing the NNRPIM analysis were considered the same. This study follows the Radial Basis Function (RBF) formulation and definition as described in the previously published works by the authors, c.f. [19; 25]. As Figure 2 depicts, the whole problem domain contains 4756 S3 elements and 2436 nodes. In all numerical analyses, the discretization density of the problem domain, FE mesh as well as the nodal distribution of meshless methods, may affect the solution accuracy. Denser discretizations defined, more accurate results acquired. Hence, an interest region was defined in the problem domain possessing a denser discretization. This interest region is identified as the cracked area started from (\(P_{B}\)), right after the crack tip of the first crack length \(a_{1}=7.5\) mm. It is dimensioned as \(5\times 15\) mm\({}^{2}\). This interest region was defined based on the first example, \(a_{1}=7.5\) mm and remains the same for the other crack length examples. It is demonstrated as a dash rectangular region highlighted in red as presented in Figure 2. It must be noted that this interest area includes a total number of 1037 nodes and 1920 elements of type S3. Such interest region consideration permits to guarantee that the other crack tip will be located in a region with a denser FE mesh/meshless nodal distribution. Concerning the FE mesh properties and meshless nodal distribution, it must be noted that the Authors have already studied convergence studies [26; 39] on different densities for similar problems and it was concluded that this discretization owns sufficient potential to produce accurate and robust results. In this study, referring to Table 2, three different crack lengths, \((a)\), are considered. Besides, three various span lengths \((S)\) are defined as another parametric value due to the considerable role of span length on the magnitude of bending moment around span coordinates leading to capture different results on the fracture characterization. Owing to the abovementioned description, the numerical analyses have been performed and Table 3 reports the obtained results for the calculation of fracture toughness using all numerical simulations. It must be noted that the theoretical solution was calculated based on Equation (3) in which \(Y_{I}\) was derived from Equation (5). As explained in Section 3, the numerical fracture toughness was calculated in a defined window dimensioned as \(3\times 5\) (mm\({}^{2}\)) considering a gap of \(g\cong 0.5\) (mm) ahead of the crack tip as illustrated in Figure 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(S\) (mm)} & \multirow{2}{*}{\(a\) (mm)} & \multicolumn{5}{c}{\(K_{Ie}\)(MPa\(\sqrt{\)mm)}} \\ \cline{3-7}  & & **Theoretical** & **FEM** & **RPIM** & **NNRPIMv1** & **NNRPIMv2** \\ \hline \multirow{3}{*}{12.5} & 7.5 & 242.30 & 242.03 & 243.32 & 236.83 & 237.13 \\  & 10.0 & 325.65 & 325.43 & 323.40 & 330.11 & 329.69 \\  & 12.5 & 461.17 & 461.89 & 460.92 & 462.45 & 463.03 \\ \hline \multirow{3}{*}{15.0} & 7.5 & 316.81 & 315.10 & 318.74 & 316.40 & 315.89 \\  & 10.0 & 420.24 & 419.15 & 421.71 & 417.18 & 423.17 \\  & 12.5 & 585.10 & 584.57 & 586.38 & 583.31 & 585.31 \\ \hline \multirow{3}{*}{17.5} & 7.5 & 391.32 & 391.12 & 392.73 & 383.03 & 386.58 \\  & 10.0 & 514.82 & 513.49 & 514.71 & 513.33 & 514.63 \\ \cline{1-1}  & 12.5 & 709.02 & 710.71 & 710.23 & 711.92 & 707.85 \\ \hline \end{tabular}
\end{table} 
Table 3: \(K_{Ie}\) obtained from theoretical formulation and numerical analyses.