# Advances in Finite Element Modeling of Fatigue Crack Propagation
## 4 Conclusions
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\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Method** & **Case Study** & **Advantages** & **Disadvantages** \\ \hline \multirow{6}{*}{Virtual Crack Closure Technique (VCCT)} & & This study found that combining the finite element approach with & \(\bullet\) Limited to linear elastic materials [136]. \\  & [PERSON] et al. [189] & XFEM and VCCT techniques, along with dividing the welding & \(\bullet\) This technique requires the presence of an initial crack, which cross-section into areas with varying mechanical and fatigue properties, accurately estimates the fatigue crack growth path and life. \\ \hline \multirow{6}{*}{Adaptive Mesh Refinement (AMR)} & [PERSON] et al. [62], developed a computationally efficient phase-field model for fatigue crack nucleation and growth. & \(\bullet\) Improves computational efficiency. & \(\bullet\) Not all finite element software packages support AMR, which can limit its applicability. \\  & Using adaptive mesh refinement and cycle jumps, their approach accurately simulates complex crack behavior (branching, kainking, coalescence, nucleation) while significantly reducing computation time. & \(\bullet\) Inverses accuracy in complex geometries. & \(\bullet\) Effective use of AMR requires significant expertise in both the underlying physics and the numerical methods, which may not be readily available [134]. \\ \hline \multirow{6}{*}{Dual Boundary Element Method (DBEM)} & [PERSON] et al. [50] employed the DBEM to model crack growth with an inclined anisotropic principal axis. & \(\bullet\) Lower computational costs for modeling crack initiation, propagation, and coalescence [149, 150]. & \multirow{6}{*}{Modeling interactions between multiple cracks can be complicated, and the method may require significant adjustments to handle such scenarios.} \\ \hline \multirow{6}{*}{Phase Field Modeling Method (DBEM)} & [PERSON] et al. [184] presented a simulation of brittle fatigue crack nucleation and growth based on several case studies. & The capability to simulate complex crack patterns and interactions seamlessly eliminating the necessity for remeshing [160, 161]. & Due to the complexity of fatigue in materials—affected by factors like load ratio, strain rate, and temperature—requires further enhancements to the current phase field model for a more comprehensive formulation [184]. \\ \hline \multirow{6}{*}{Multi-Scale Modeling Phase Field Modeling} & [PERSON] et al. [168] presented case studies demonstrating crack propagation in SiC cross micro-, meso-, and macro-scales. & This approach bridges the length scales from the microscopic to the macroscopic level, enabling the investigation of crack growth behavior at different scales [168, 166, 167, 168]. & \multirow{6}{*}{MSM requires enhancements to refine the model for accurate application to multiphase materials [168].} \\ \hline \multirow{6}{*}{Probabilistic Approaches} & [PERSON] et al. [176] presented case studies focused on a probabilistic evaluation of the positions and sizes of defects under fatigue loading conditions. & This method enables the identification and measurement of uncertainties in input variables (like material properties, loading intensities, and residual stresses) that may affect crack propagation [174, 175, 176]. & \multirow{6}{*}{Further enhancements and verifications of the simulation model are needed to create a flexible and effective tool for the joint optimization of mechanical design and heat treatment process conditions [176].} \\ \cline{1-1}  & [PERSON] et al. [184] reported three case studies: the compact tension shear (CTS) steel specimen, the fatigue life estimation of a hole plate, and the fatigue behavior of a titanium plate with an internal oblique crack. & The primary advantage was that the computational mesh is updated only when the finite elements become excessively distorted due to the movement of the mesh nodes, significantly reducing computational burdens [184]. & \(\bullet\) This technique requires the presence of an initial crack, which may not always be available or easy to define in some practical scenarios [130]. \\ \hline \multirow{6}{*}{Moving Mesh Techniques} & [PERSON] et al. [184] reported three case studies: the compact tension shear (CTS) steel specimen, the fatigue life estimation of a hole plate, and the fatigue behavior of a titanium plate with an internal oblique crack. & \(\bullet\) It was suggested that a natural progression for the proposed method could involve extending it to a 3D framework, thereby enhancing the overall reliability of the strategy. This refined approach could then be applied to investigate fatigue and fracture phenomenon in more complex materials and structures, such as laminates and microstructured fiber or particle-reinforced composites [184]. \\ \hline \hline \end{tabular}
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