## 3 Finite Element Modeling Information
### FEM Modeling
#### 3.2.3 Crack Propagation  
Crack propagation was examined via ABAQUS extend finite element method (XFEM). When ABAQUS XFEM is used, it is assumed that crack is already initiated, and that the location of crack initiation is accurate. Specifically, ABAQUS XFEM depicts the crack propagation line that connects between virtual nodes, which are not on the element. The advantage of the method is that it conserves the stiffness of the structure because it does not delete the element. The basic enrichment function of the XFEM is in Figure 7 and Equations (10)-(12) [25].  
\[u=\sum_{I=1}^{N}N_{I}(x)\bigg{[}u_{I}+H(x)a_{I}+\sum_{a=1}^{4}F_{a}(x)b_{I}^{a }\bigg{]} \tag{10}\]  
Figure 5: Boundary and loading conditions of the finite element method (FEM).  
Figure 6: Damage contour of fatigue simulation in Optistruct.  
\[H(x)\Bigg{\{}\begin{array}{ll}1,&if\ (x-x^{*})*n\geq 0\\ &-1,\ \ \text{otherwise}\end{array}\ \ \ x^{*}\ \text{is the point on the crack closet to x} \tag{11}\]  
\[F_{\alpha}(x)=\bigg{[}\sqrt{s}sin\frac{\theta}{2},\ \ \sqrt{r}cos\frac{ \theta}{2},\ \sqrt{r}sin\theta sin\frac{\theta}{2},sin\theta cos\frac{ \theta}{2}\bigg{]}, \tag{12}\]  
where \(N_{1}(x)\) is the usual nodal shape functions and \(u_{I}\) is the displacement vector of the elements in the part. \(H(x)\) is the discontinuous function which can be presented like equation 11 [25]. \(F_{\alpha}(x)\) is the stress asymptotic function of crack tip can be represented as Equation (12) [25]. \((r,\theta)\) in Equation (12), is polar coordinate from origin crack tip. \(a_{I}\) and \(b_{I}^{\alpha}\) in Equation (10), are nodal enriched degrees of freedom vectors of \(H(x)\) and \(F_{\alpha}(x)\).  
Equation (10) consists of the near tip asymptotic functions that capture singularity around the crack tip and a discontinuity that represents the jump in displacement across the crack surfaces. \(u_{I}\) is applicable for all nodes. \(H(x)a_{I}\) is cut by crack. \(\sum\limits_{\alpha=1}^{4}F_{\alpha}(x)b_{I}^{\alpha}\) is used only for nodes whose shape function is cut by the crack tip.  
To decide the direction parameter, (r, \(\theta\)), the XFEM crack looks for the location which satisfies damage initiation criterion. Without this criterion, the crack will just extend without a crack path change (\(\theta=0\)). If the Paris law is used without the Darveuax coefficient, the onset of delamination will follow Equation (7), which satisfies \(\Delta G>G_{1C}\). However, if the damage initiation criterion is used, such as the Darveuax method, it will change the damage variable of the element, D, used in Equation (8). This means that \(\Delta G\) changes due to degradation of the element. This degradation can change the crack path by changing stress distribution near the crack tip based on Equation (9).  
To count the cycle, the direct cycling step in ABAQUS was used. Based on the results of the static step, the program calculates the cycle to propagate. The important assumption made in order to run ABAQUS XFEM is that the location of the initial crack is exact. In the study, the crack initiation point is same with the location of Optistruct results. Thus, the crack initiation point is based on the results of Optistruct as presented in Figure 6. Constants for the empirical Paris law are considered in XFEM. To apply the Darveux coefficient, a material keyword, * Damage initiation/evolution, should be used. A basic elastic-plastic curve and boundary condition are identical to those of Optistruct. The effect of the Darveuax method is presented in the next chapter, by comparing the model which used the Darveuax method in its material keyword and the model that did not. If the Darveux method is not used, the crack will be only be propagated with the fracture criterion keyword using the critical energy release rate without strain energy change.