# Fracture Toughness Determination on an SCB Specim by Meshless Methods
## 3 Mathematical Formulation  
As shown in Figure 1, a cracked straight through semi-circular bend specimen (CSTSCB) is considered as the case study following the testing method suggested by [PERSON] et al. [13]. It aims at the determination of the mode I fracture toughness for various crack growth stages. Figure 1: A Schematic view of the semi-circular bend specimen. From the theoretical point of view, Equation (3) is applicable for the computation of fracture toughness for mode \(I\)[13],  
\[K_{lc}=\frac{P\sqrt{\pi a}}{2 RB}Y_{I}. \tag{3}\]  
In which \(P\) is the maximum load, \(R\) is the specimen radius, \(B\) is the specimen thickness, and \(Y_{I}\) is the normalized SIF, which is dimensionless and dependent on the normalized crack length and normalized span. By considering a thin specimen, \(K_{lc}\) in Equation (3) provides the critical SIF value. Regarding the normalized term, \(Y_{I}\), [PERSON] et al. [32] used the classical displacement extrapolation method and obtained the following equation by FE analysis;  
\[Y_{I}=S/R\Big{(}2.91+54.39\ \alpha-391.4\ \alpha^{2}+1210.6\ \alpha^{3}-1650\ \alpha^{4}+875.9\ \alpha^{5}\Big{)}. \tag{4}\]  
Being \(\alpha=a/R\) indicates the ratio of the crack length to the radius. The low accuracy of the technique used to acquire Equation (4) in comparison to \(J\)-integral, encouraged [PERSON] et al. [13] to propose an alternative formulation. They used least-square for curve fitting where a second-order function of \(\alpha\) is adopted for the simple relationship expressed in Equation (5). This equation has provided more accurate solutions in comparison to the ones obtained by [PERSON] et al. [32] and [PERSON] and [PERSON] [33]. \[Y_{I}=-1.297+9.516\ \bigg{(}\frac{S}{R}\bigg{)}+\bigg{(}-0.47-16.457\bigg{(} \frac{S}{R}\bigg{)}\bigg{)}\alpha+\bigg{(}1.071+34.401\bigg{(}\frac{S}{R}\bigg{)} \bigg{)}\alpha^{2}. \tag{5}\]  
It must be noted that the recent formulation is valid if \(\alpha\geq 0.2\). Nevertheless, a fairly deep notch is necessary on the bending effect to provide a robust mode I stress field near the notch tip. Therefore, it is suggested to respect \(0.4\leq\alpha\leq 0.6\). Furthermore, it is recommended to consider \(0.5\leq S/R\leq 0.8\) in which it is desirable to consider \(S/R\) approaching to the maximum value [13]. Particularly in plane stress condition, the value of \(K_{lc}\) defined in Equation (3) at the point of crack extension is called the critical SIF value and expressed as \(K_{lc}\) in the literature [34]. It then defines the onset of crack extension. It does not necessarily indicate a fracture of the specimen; it depends on the crack stability. It is usually regarded as a material property and can be used to characterize toughness. However, in this work, Equation (5) together with Equation (3) was considered as a reference theoretical solution on \(K_{lc}\) to validate the results obtained by numerical studies resorting to FEM study by the standard 2D constant strain triangle elements and meshless methods. Considering \(G\) as the strain energy release rate per crack tip, that is, for a double-ended crack within an infinite solid, the rate of release in strain energy per crack tip for a linearly-elastic material is:  
\[G=\frac{K_{I}^{2}}{E}. \tag{6}\]  
Thus, critical strain energy release rate, \(G_{c}\), is then related to the mode I fracture toughness calculated through the following equation:  
\[G_{c}=\frac{K_{IC}^{2}}{E}. \tag{7}\]  
Being \(E\) the material elasticity modulus. It must be noted that Equation (7) will be used to obtain the theoretical solution and the numerical solutions on the critical fracture energy. Regarding the radius of the fracture process zone, \(r_{c}\), so-called a critical distance from the crack tip, [PERSON] et al. [6] adopted an equation to determine the size of \(r_{c}\) for brittle materials as follows:  
\[r_{c}=\frac{1}{2\pi}\Big{(}\frac{K_{IC}}{\sigma_{t}}\Big{)}^{2}. \tag{8}\]  
in which \(\sigma_{t}\) denotes the material tensile strength.