## 2 Materials and Methods
### FCG Assessment of the Energy Release Rate with VCCT  
In LEFM, the single scaling parameter is SIF (K) as the crack progresses in Equation (1) have been correlated with energy methods for stress function [25]. The crack propagation for planar crack is possible once the energy released for unit width and length of fracture surface \(\Delta\)a is proportional to the fracture toughness characteristics of the material as strain energy release rate, (G). The strain energy release rate range (\(\Delta\)G) is computed with VCCT using finite elements with the integration of the Paris law in Equation (4) using the fracture crack parameter (C and m) for crack growth simulation. \[\text{da}/\text{dN}=\text{C}\;(\Delta\text{G}_{\text{eq}})^{\text{m}}, \tag{4}\]  
VCCT is a nodal release method which calculates the strain energy release rate (G) by calculating a force, area of surfaces created and a displacement. Nodes at interfaces are released when the G is high enough to propagate the crack. The energy balance can be expressed using (G) as a fracture criterion as crack starts to grow when it reaches the critical  
Figure 2: Close-up of test rig for optical and crack gauge measurement procedures. energy release rate value in Equation (5), where \(\Pi\) is the strain energy and 'da' is the change in crack length which depends on the geometry of the structure and the current loading. \[G=-\frac{d\Pi}{da}, \tag{5}\]  
[PERSON] described (G) as the result of crack extension which is correlated to the strain energy release rate based on the crack closure integral theory [26]. The phenomenon has been for interface failure of stress components [27; 28]. For 3D VCCT, the modified Irwin equation is Equation (6):  
\[G_{I}=\frac{1}{2 wi\Delta a}\ \int_{Si+1}^{Sj+1}\int_{0}^{\Delta a}\sigma_{y}(r, \pi)\Delta v(\Delta a-r,0)drds, \tag{6}\]  
where (\(\mathrm{w_{i}}\)) is the element length and (s) is the distance along the crack front, the distance between the crack tip and the point of surface crack is r, the distance of crack extension is \(\Delta a\) and the displacement in the y-direction is \(\Delta v\) during the crack extension when the normal stress (\(\sigma_{y}\)) is acting along the closed crack surface (line). The results of the node forces (F) ahead of the crack front and the node displacements behind the crack front for the \(\mathrm{i_{th}}\) segment with contributions for elements on each side of the crack front. For mode 1 planar crack is estimated with node 8 element as describe by [29] in Equation (7). The X and Z, respectively are the tangential (x-axis) and normal (z-axis) components of the nodal forces, while \(\Delta u\) and \(\Delta v\) are the corresponding relative displacement variation of the nodes. \[G=\frac{1}{2\ \mathrm{wi\Delta a}}\ (X\Delta u+Z\Delta v), \tag{7}\]  
For 3D solid conditions, the evaluation is carried out separately for each node along the crack front of the model as illustrated in Figure 3. \(u\) is crack opening displacement at crack face. For the case of higher-order elements, the contributions from the mid-side nodes included as the obtained strain energy release rate \(\mathrm{G_{Imax}}\) and \(\mathrm{G_{Imin}}\) in fatigue loading are calculated based on Equations (8) and (9). The simplified representation of the numerical procedure implemented in program VCCT for SIF based on energy release rate solver is illustrated Figure 4.  
\[\Delta G=\left(\sqrt{G_{I\ max}}-\sqrt{G_{I\ min}}\right)^{2}, \tag{8}\]  
\[G_{I}=\frac{\mathrm{F_{1}u_{1}+\ F_{2}u_{2}+\frac{1}{2}(F_{3}\ u_{3}+\ F_{4}\ u_{4})}}{2a}, \tag{9}\]  
Figure 3: Three-dimensional mesh for VCCT for higher-order mesh elements. The crucial procedure for FEM calculation of a crack component is considering nodes coordinates, elements, topology with a peculiar mesh generator for cracks, the material properties and loading condition implemented in the VCCT program of MSC Marc/Mentat with visualization of the input model.