# Numerical Modeling of Crack Growth under Mixed-Mode Loading
## 2 ANSYS Mechanical APDL 19.2  
ANSYS Mechanical APDL 19.2 XFEM was applied to model the crack propagation direction for the three-point bending beam with three holes for various configurations. There are three types of cracks that can be modeled within ANSYS: arbitrary, seminal, and pre-meshed cracks. The pre-meshed crack process includes the front of the crack which is used by the \"Smart Crack Growth\" simulation tool, in which the failure criterion is the SIFs. Instead of using the enrichment area (splitting) of the XFEM, SMART automatically updates the mesh from crack geometry changes due to crack growth on each solution stage, thereby eliminating long pre-processing sessions. The previously generated node sets were assigned to the front of the crack and the top and bottom sides of the crack. The new feature introduced in ANSYS is the tetrahedron mesh-based crack called the smart crack growth, which introduces the \"pre-meshed crack\" requirement after completion in which the user can choose the crack growth option type. The mesh around the crack tip should be refined using the sphere of influence methodology around the geometrical edge moving across the thickness. SMART can be used with a higher order tetrahedron (entire domain of the model) mesh only. The named geometric regions to be defined are the edge, top surface, and the bottom surface of the crack. Three procedures were primarily used to illustrate the fatigue assessment of materials, which include the fracture mechanics method developed by [37], the method of strain-life introduced by [38], and the stress-life method introduced by [39]. For mixed-mode fatigue crack growth analysis, the maximum circumferential stress criterion was implemented as a crack propagation criterion by the ANSYS software. The following are the equations for the direction of crack propagation in ANSYS [40; 41]:  
\[\theta=\cos^{-1}\left(\frac{3K_{II}^{2}+K_{I}\sqrt{K_{I}^{2}+8K_{II}^{2}}}{K_{ I}^{2}+9K_{II}^{2}}\right) \tag{1}\]  
where \(K_{I}\) = Max \(K_{I}\) during cyclic loading and \(K_{II}\) = Max \(K_{II}\) during cyclic loading. In ANSYS Mechanical APDL 19.2, it can be found that when using XFEM, the crack growth simulation is restricted to region \(II\) of the typical fatigue crack growth behavior graph, which can be defined in the modified formula of Paris law equation as:  
\[\frac{da}{dN}=C(\Delta K_{eq})^{m}. \tag{2}\]  
From Equation (2), the number of life cycles for fatigue can be estimated with a crack growth increment 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 SIF formula can be formulated as follows [40]:  
\[\Delta K_{eq}=\frac{1}{2}\cos\frac{\theta}{2}[(\Delta K_{I}(1+\cos\theta))-3 \Delta K_{II}\sin\theta] \tag{4}\]  
where: \(\Delta K_{I}\) = range of SIF in mode \(I\) loading and \(\Delta K_{II}\) = range of SIF in mode \(II\) loading. There are many approaches formulated for evaluating the SIFs based on numerical analysis, including the use of XFEM and other numerical methods. The approaches utilizing the XFEM include the method of displacement extrapolation (DEM), interaction integral method (IIM), stress extrapolation method (SEM), and node displacement method (NDM) [42]. The interaction integral technique is usually the most accurate method and it is able to estimate \(K_{I}\) and \(K_{II}\) separately. ANSYS provides two approaches for the determination of SIFs, which are the DEM and IIM. The second approach was utilized because it is numerically simpler to execute and thus offers greater preciseness and less mesh requirements. This approach depends on the domain integral approach [43] where an auxiliary field is used to separate \(K_{I}\) from \(K_{II}\), as this ability is missing in the domain integral itself. According to the mixed-mode stress intensity factors \(K_{I}\), \(K_{II}\), and \(K_{III}\), the energy release rate was formulated as [43; 44]:  
\[J(s)=\frac{K_{I}^{2}+K_{II}^{2}}{E^{*}}+\frac{1+\
u}{E}K_{III}^{2}\ \ E^{*}=\left[