# Fracture Toughness Determination on an SCB Specim by Meshless Methods
## 1 Introduction  
The capability to endure a substantial amount of damage is a demand for current engineering structures; therefore, it has become progressively imperative to improve methodologies to anticipate failure phenomenon in damaged components. Damage tolerance analyses can be fulfilled within the linear elastic fracture mechanics (LEFM) concepts where the stress intensity factor (SIF) plays a substantial role [1]. The fracture mechanics theorem in conjunction with crack growth laws, i.e., Paris' law, is characteristically engaged to examine and forecast crack growth and fracture behavior of structural integrity components [2]. To study crack growth and to evaluate the remaining lifetime of a structural component, demanding computational analyses must be performed and the SIFs shall be thereby evaluated. There exist several developed formulations to determine the fracture parameters such as mode I and mixed-mode SIFs, strain energy release rate, T-stress, etc. The fracture toughness, which is the critical SIF of a sharp crack where propagation of the crack suddenly becomes unlimited and fast, remains a very imperative specification of a material that can demonstrate the material resistance against the structural cracks governing by the work required to fracture a material, known as work of fracture. It meant that fracture toughness is a quantitative definition to express the material's resistance to a brittle fracture phenomenon in the presence of cracks. If the material possesses high fracture toughness, it would be more disposed to the ductile fracture [3]. On the otherhand, brittle fracture is a characteristic of materials with less fracture toughness. There exist general factors which can affect the material toughness including strain rate, temperature, presence of any stress concentration on the specimen surface and the relationship between the material strength and its ductility [4]. The critical strain energy release rate describes the resistance of the material against the crack growth. In this regard, [PERSON] et al. [5] developed numerical damage models calibrated with experimentally based fracture toughness parameters on the purtuded glass fiber reinforced polymers. [PERSON] et al. [6] studied the fracture strength of brittle materials by the effect of stress concentrations through SIF calculation. [PERSON] [7] studied the fracture toughness of a center cracked plate by the Finite Element Method (FEM) in which the critical SIF variation through specimen thickness was characterized. Amongst all testing specimens in the fracture mechanics studies, the semi-circular bend (SCB) specimen received considerable attention emerging in 1984 to test brittle materials in particular rocks contributing to determining the mode I fracture toughness [8]. [PERSON] et al. [9] proposed the SCB specimen in 1987, due to the method of rock and geometrical specimens' extraction being in a circular shape, simple geometry and demanding a common loading configuration. Subsequently, the SCB was adopted and improved on distinct solicitations in the field of solid and fracture mechanics. The SCB specimens can be simply manufactured to be adopted in the experimental tests and numerical simulations. There are other suggested testing methods for mode I fracture toughness of rock by the International Society for Rock Mechanics (ISRM) such as short rod specimen method (SR) [10], cracked notched Brazilian disk (CCNBD) [11], chevron-bend specimen method (CB) [12] and cracked straight through semi-circular bend specimen (CSTSCB) [13]. In this study, several numerical methods are used to characterize the mode I fracture toughness near the crack tip due to the remote load. The FEM and meshless methods including radial point interpolation method (RPIM) and natural neighbor radial point interpolation method (NNRPIM) are employed to determine the fracture toughness to be compared with the calculated values by CSTSCB method [13].