# Fracture Toughness Determination on an SCB Specim by Meshless Methods
## 1 Introduction
The necessity of mesh refinement in the problems that the geometry changes or contains a large distortion, FEM shows some shortcomings that leads researchers to emerge new techniques. One of the earliest methods proposed by [PERSON] et al. [14] so-called Element-Free Galerkin (EFG) is capable to simulate cracks propagating statically and dynamically. Furthermore, the EFG is able to give valid values for irregular nodal distribution, which is one of the advantages of this technique. [PERSON] et al. [15] proposed a variant of the EFG method integrating the application of the level set and diffraction methods to simulate multiple interacting cracks using a coarse meshless nodal discretization. This methodology was deployed to model kinked cracks that have knee singularity. On the other hand, one of the main factors affecting the acceptance of a numerical technique is computational cost. Generally, the cost is measured for a prescribed accuracy for a problem. Therefore, by taking into account that denser discretizations would increase the computational cost, it is necessary to optimize the cost meticulously without a negative effect on accuracy. In this regard, meshless methods are flexible to be optimized and may be adjusted to produce reliable results. Besides, it is worth mentioning that meshless methods provide the opportunity to impose the displacement constraints on the stiffness matrix directly, but the EFG method employs the Lagrange-multipliers for this purpose which is not as straightforward as it is for meshless methods. Meshless methods have been successfully implemented to address fracture and damage characterizations for a variety of material behaviors, c.f. [16; 17; 18; 19; 20]. So far, the FEM formulation with quadratic elements has been employed to calculate mode I and mixed-mode SIFs for SCB specimens [9; 13; 21; 22]. Nevertheless, meshless methods have never been employed to calculate fracture toughness and critical strain energy release rate for SCB specimens. It must be mentioned that the meshless methods are advantageous due to their simplicity in the formulations. Besides, it is feasible to use the FE mesh background to generate the nodal discretizations, which may lead to reducing the computational costs with the absence of finite elements. Other privileges of meshless methods can be related to their independence to the element shapes/types and the element distortion that may occur during the finite element analyses. Despite FEM, meshless methods do not require FE remeshing which is undesirable in terms of crack propagation problems and solution convergences on the nonlinearity in the engineering structures with complex material behaviors such as damage mechanics or porous models, c.f. [16; 17; 19; 20; 23; 24]. Nevertheless, owing to the aforementioned related works, it can be inferred that the literature lacks a comprehensive analysis of SCB specimens for fracture toughness characterization by meshless methods. Therefore, the main contribution of this work can be related to the development of meshless method formulations combined with LEFM theory. Therefore, it contributed to elastostatically simulating the cracked domain in order to attain the fracture toughness and critical strain energy release rate.