# Fracture Toughness Determination on an SCB Specim by Meshless Methods
## 4 Analysis, Results and Discussion
Moreover, Table 4 shows the deviation computed amongst the numerical results compared to the theoretical solution. For all numerical calculations on fracture toughness, it must be stated that to avoid the plastic effect that occurred at the crack tip, a gap of \(g\cong 0.5\)\((\mathrm{mm})\) was considered after the crack tip. This gap was defined following the plastic radius, \(r_{c}\), computation derived from Equation (8) as calculated by [PERSON] et al. [39] for the similar SCB specimen. Besides, Table 5 presents the critical strain energy calculation derived from the numerical analyses calculated through Equation (7). In conclusion, \(G_{\mathrm{C}}\) raised if the crack and span lengths increased. Straightforwardly, the higher fracture toughness and \(G_{\mathrm{C}}\), then the greater the material's fracture resistance, which then possibly translates into higher damage tolerance. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Deviation (\%)**} \\ \cline{3-6} S (mm) & \(a\) (mm) &
\begin{tabular}{c} **Theoretical** \\ **and FEM** [FOOTNOTE:]Footnote : 100 \(\times\left|\left(K_{lc}^{TH}-K_{lc}^{FEM}\right)/K_{lc}^{TH}\right|\); \(  
Figure 4 depicts the fracture toughness variation in terms of various \(\alpha\) for different normalized span \((S/R)\) obtained from theoretical and numerical solutions. In these graphs, a reasonable increment in fracture toughness is noticeable as the crack length grows. All numerical methods provided similar results to the ones obtained from the theoretical formulation. The acquired curves experienced a parabolic behavior as anticipated by the former studies [32; 33]. Figure 4: Fracture toughness for a range of normalized crack length (\(\alpha\)) and normalized span \((S/R)\) obtained by different methods: (**a**) theoretical formulation, (**b**) FEM, (**c**) RPIM, (**d**) NNRPIMv1 and (**e**) NNRPIMv2. The stress distribution in \(y\)-direction, \(\sigma_{yy}\), was also obtained from all numerical studies for different crack lengths and various normalized spans \((S/R)\), as demonstrated in Figures 5-7. It must be noted that the stress contours were extracted in a 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.  
Figure 5: The stress distribution in y-direction, \(\sigma_{yy}\), on the cracked area for normalized span equal to \(S/R=0.5\) obtained by different numerical methods; top row: \(a=7.5\) (mm), middle row: \(a=10.0\) (mm) and bottom row: \(a=12.5\) (mm), stress value in MPa. Figure 6: The stress profile in y-direction, \(\sigma_{yy}\), on the cracked area for normalized span equal to \(S/R=0.6\) obtained by different numerical methods; top row: \(a=7.5\) (mm), middle row: \(a=10.0\) (mm) and bottom row: \(a=12.5\) (mm), stress value in MPa. The smooth contours on the integration nodes of meshless methods and the FEM infer that the density and the area of the stress distribution enhance by increasing the crack length and span. The cracked area accounted for the maximum stress conforming to the LEFM theory. As the crack length raised, the stress singularities increased around the crack tip. In most cases, NNRPIMv1 and NNRPIMv2 depict a much similar stress distribution in comparison to the RPIM and FEM even though there is no significant difference amongst the profiles. The verified results acquired from different methods in terms of accuracy would prove the potential capability of meshless methods to provide accurate solutions. Therefore, it can be inferred that the obtained results have shown promising accuracy leading to validate the numerical methodologies.