# Fracture Toughness Determination on an SCB Specim by Meshless Methods
## 5 Conclusions  
This study deals with the fracture characterization on a marble SCB specimen using advanced discretization techniques. The numerical model was prepared through the FEM in FEMAP software and the essential data of FE mesh was exported to feed the numerical algorithms developed in MATLAB(r). The numerical elastostatic algorithm was implemented based on the FEM fundamentals and RBF formulation in which the meshless nodal discretization was generated based on the FE mesh background, which was built by the standard 2D constant strain triangle elements. Then, respecting the FE mesh property, maintaining the nodal points, the RPIM and NNRPIM meshless models were generated in which the latter one employed on both first- and second-influence cell versions. As an output, the fracture toughness, \(K_{lc}\), and the strain energy release rate, \(G_{c}\), were computed. The obtained numerical results (from FEM and meshless models) on \(K_{lc}\) and \(G_{c}\) were compared to a reference theoretical solution proposed for the SCB specimen available in the literature. It aimed at describing how the fracture toughness of the studied specimen can be affected by the crack and span lengths defined in the geometry. However, the results infer that \(K_{lc}\), \(G_{c}\) and stress gradient around the crack tip accounted for a greater value if the crack length grows or the span length increases. The acquired results imply that  
Figure 7: The stress profile in y-direction, \(\sigma_{yy}\), on the cracked area for normalized span equal to \(S/R=0.7\) 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. although the FEM solutions were closer to the reference solution, meshless methods results were sufficiently accurate and promising. A general discussion on the meshless method privileges can be drawn as:  
The meshless methods are capable to offer numerically advantageous models to evaluate the fracture characterization of the engineering structures since their formulations are based on the FE mesh background (nodes) and some difficulties that FEM models may face are already overcome. These issues can be identified as the dependency on the element type and shape, FE distortions, convergence study on the nonlinearity, high degrees of freedom and FE remeshing in terms of crack propagation problems. Besides, in terms of computational cost, it must be mentioned that the meshless methods would have the potential to reduce the computational costs since they are solving the problems only using the nodal distribution in the problem domain. Moreover, it will be feasible to allocate denser discretization in the interest region/influence domain of the corresponding model. This aspect is more significant in solving large structures with complex material behaviors, for instance, elastoplastic analyses, damage models and porous materials. However, a simple model was studied in this work but it can be pointed out that the meshless methods formulations extended to the LEFM are practically capable to simulate the fracture problems providing sufficiently accurate results to validate the supporting methodology. Conceptualization, J.B.; methodology, F.M.S., B.V.F. and J.B.; software, J.B.; validation, F.M.S. and B.V.F. ; investigation, F.M.S., B.V.F. and J.B.; writing--original draft preparation, F.M.S. ; writing--review and editing, B.V.F. ; supervision, J.B. All authors have read and agreed to the published version of the manuscript. This research received no external funding. [PERSON] gratefully acknowledges the funding received from Ministerio da Educacao e Ciencia, Fundacao para a Ciencia e a Tecnologia (Portugal), under grant PTDC/EME-EME/29339/2017.