# Computational Simulation of 3D Fatigue Crack Growth under Mixed-Mode Loading
## 2 Mixed Mode Fatigue Life Evaluation Procedure Using ANSYS  
ANSYS can model three kinds of cracks: arbitrary, semi-elliptical, and pre-meshed. The smart crack growth analysis tool uses the crack front in the pre-meshed crack approach, and the stress intensity factor is the failure criteria. The rendered nodes will distribute to the front, bottom, and top of the crack, and the front of the crack is included in the pre-meshed crack process, which is used by the \"Smart Crack Growth\" simulation tool. This is used instead of employing the XFEM's enrichment area (splitting), which eliminates the re-mesh process around the crack tip and assumes that the discontinuities cut the element entirely. In this case, the displacement formulation does not account for the presence of singularity. As the crack propagates, the newly introduced crack segments are always assumed to have cohesive zone behavior. SMART automatically updates the mesh from crack geometry changes due to crack propagation on each solution stage, reducing the need for long pre-processing sessions. The sphere of influence was used to refined the crack tip mesh and around the geometric edge that passes through the thickness. The geometric regions 'contain the crack tip, the crack's top surface, and the bottom surface of the crack. However, each of these regions is connected to a set of nodes that will be utilized for analysis. In order to analyze a mixed-mode fracture, it is essential to have a proper understanding of the crack growth direction. The direction of crack extension is defined by an angle \(\theta\) measured from the initial crack plane [29; 30; 31]. The direction of a mixed-mode crack is definitely determined by the ratio of modes stress intensity at the crack tip. The ANSYS software considers mixed-mode loading where the maximum circumferential stress criterion is implemented. The following are the formulas for the direction of crack propagation in ANSYS [32; 33]:  
\[\theta=\cos^{-1}\left(\frac{3K_{II}^{2}+K_{I}\sqrt{K_{I}^{2}+8K_{II}^{2}}}{K_{ I}^{2}+9K_{II}^{2}}\right) \tag{1}\]  
where:  
\(K_{I}=\) Max \(K_{I}\) during cyclic loading and \(K_{II}=\) Max \(K_{II}\) during cyclic loading. In the present simulation using ANSYS, the simulation of crack propagation is restricted to region II of the typical crack propagation under fatigue loading that can be represented as:  
\[\frac{da}{dN}=C(\Delta K_{\text{eq}})^{m} \tag{2}\]  
From Equation (2), for a crack increment, the number of life cycles of fatigue may be predicted as:  
\[\int\limits_{0}^{\Delta a}\frac{da}{C(\Delta K_{eq})^{m}}=\int\limits_{0}^{ \Delta N}dN=\Delta N \tag{3}\]  
The equivalent range for the stress intensity factor formula can be found as follows [19]:  
\[\Delta K_{eq}=\frac{1}{2}\cos\frac{\theta}{2}[(\Delta K_{I}(1+\cos\theta))-3 \Delta K_{II}\sin\theta] \tag{4}\]  
where:  
\(\Delta K_{I}=\) the range of SIF in mode I loading and \(\Delta K_{II}=\) the range of SIF in mode II loading. Based on numerical analysis, there are many methods formulated for estimating the SIFs. The integral interaction method is usually the most accurate method that can estimate \(K_{I}\) and \(K_{II}\) separately. ANSYS proposes two methods, the Displacement Extrapolation Method (DEM) and the Integral Interaction Method (IIM), to determine SIFs. The second method was adopted because it is numerically easier to implement and has better precision and less mesh requirement. This approach depends on the domain integral approach [34], where an auxiliary field uses to separate \(K_{I}\) from \(K_{II}\), as this ability is missing in the domain integral itself. The energy release rate is expressed in terms of mixed-mode stress intensity factors \(K_{I}\), \(K_{II}\), and \(K_{III}\), which were proposed by [34; 35] as follows:  
Equation (5) becomes for the superimposed state:  
\[J(s)=\frac{K_{I}^{2}+K_{II}^{2}}{E^{*}}+\frac{1+\
u}{E}K_{III}^{2}\ \ E^{*}=\left[ \begin{array}{cc}\frac{E}{(1-\
u^{2})}&\text{Plane strain}\\ E&\text{Plane stress}\end{array} \right. \tag{5}\]  
where  
\[