# Fracture Toughness Determination on an SCB Specim by Meshless Methods
## 4 Analysis, Results and Discussion  
A pre-defined cracked SCB specimen made of marble was considered in this study, as schematically depicted in Figure 1. Regarding geometrical characteristics, the specimen radius was considered as \(R=25\) (mm) and thickness was defined as \(B=1\) (mm). The mechanical parameters for the studied marble specimen are considered according to a reference article [36], reported in Table 1.  
Besides, the crack and the span lengths were accordingly defined as \(a\) and \(S\). In this study, different crack lengths and spans were taken into account led to obtaining the numerical results using meshless method formulations, see Table 2.  
As mentioned before, the critical mode I SIF, \(K_{IC}\), would be the most important parameter in fracture mechanics determining the resistance of the material to the impending urge to propagate the initial crack. This analysis is implemented on half of SCB specimen \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Young’s modulus & \(E=75,500.0\) (MPa) \\ Poisson’s ratio & \(\
u=0.3\) \\ Tensile strength & \(\sigma_{t}=16.1\) (MPa) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of the studied SCB specimen, marble. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Span Length, \(S(\mathbf{mm})\)** & **Crack Length, \(a(\mathbf{mm})\)** & **S/R** & \(\alpha=a\)/\(R\) \\ \hline \(S_{1}=12.5\) & \(a_{1}=7.5\) & 0.5 & \(\alpha_{1}=0.3\) \\ \(S_{2}=15.0\) & \(a_{2}=10.0\) & 0.6 & \(\alpha_{2}=0.4\) \\ \(S_{3}=17.5\) & \(a_{3}=12.5\) & 0.7 & \(\alpha_{3}=0.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specification on the span and crack lengths studied in this work. 2D domain under 3-point-bending condition due to symmetric geometry, as shown in Figure 2. The specimen contains an initial crack located in the middle of the SCB specimen for which the calculation of fracture attributes is taken into account. Regarding the natural boundary condition, a compressive concentrated load was considered with a magnitude of \(P=1.0\) (kN) applied to the model vertically aligned with the initial crack. The essential boundary conditions were applied to the model constraining the nodes in the horizontal direction right above the crack tip to the end where the concentrated load is applied. On the other hand, depending on the various span lengths, \(S\), a nodal restriction was imposed to restrict the model in the vertical direction to satisfy the requirements of the SCB specimen. Therefore, (as shown in Figure 2a), to simulate the boundary conditions regarding the span, an individual node (\(P_{A}\)) was singularly selected to restrict \(v(x,y)\) displacement field in the corresponding point. Furthermore, a vertical line was assumed from the crack tip, (\(P_{B}\)), to the point, (\(P_{C}\)), under force to be bounded to \(u(x,y)\). The geometry of the model and the FE mesh was built through FEMAP(r) software. Then, the coordinates of the nodes/elements with their connectivity were imported as an input to the developed numerical algorithms implemented in MATLAB(r). The numerical algorithms were based on a FEM code and used the FE mesh background to generate the nodal discretization. It was then extended to the LEFM theory to calculate the fracture toughness and strain energy release rate. All stress and displacement types of boundary conditions were fully defined in the numerical models as explained. Regarding the FEM study, the problem domain was discretized by the standard 2D constant strain triangle elements, type S3. Then, the same discretization was used remaining nodal points to be analyzed by meshless methods. For meshless method analyses, although in previous works [37,38], the number of nodes within an influence domain was suggested to be between \(n=[9,16]\) nodes, [PERSON] et al. [26] recommended that each influence-domain should possess 20 nodes in order to get reliable results.