# Advances in Finite Element Modeling of Fatigue Crack Propagation
## 2 Fundamentals of Finite Element Modeling
### Crack Growth Criteria and Models  
There are numerous crack growth criteria and models employed in the finite element analysis of fatigue crack initiation and propagation. Defining crack initiation in a finite element method (FEM) meshed component involves several key considerations related to element size and the crack growth threshold. It is crucial to use a mesh sensitivity procedure to generate a finer mesh (smaller element size) near potential crack initiation sites for better resolution of stress gradients and more accurate crack initiation predictions [62]. Larger elements might fail to capture localized stress concentrations that lead to crack formation. Ideally, the element size should be smaller than or comparable to the expected crack size to accurately represent the stress distribution near the crack tip. The crack growth threshold is often determined by a specific SIF or strain energy release rate that must be exceeded for crack growth to start. This threshold helps identify when a crack begins to propagate. Additionally, the mesh size should allow for accurate calculation of the stress intensity factor or energy release rate, as larger elements might average out stress values and obscure whether the threshold has been reached. Some of the recent and widely used approaches include:  
1. Stress Intensity Factor (SIF) Approach: The SIF quantifies the stress field around the crack tip and is determined by factors such as the geometry of the crack, the loads applied, and the material characteristics. According to the SIF criterion, the rate of crack growth is primarily influenced by the stress intensity factor, which defines the stress conditions at the crack tip. [63; 64]. This criterion does not explicitly consider the effects of crack closure, which can happen due to the presence of surface roughness, oxide layers, or compressive residual stresses [65]. When the stress intensity factor (SIF) reaches a critical threshold, crack growth begins, often described by the Paris law and its variants, such as the Modified Forman and Walker models. Paris law is a widely utilized crack growth model that relates the crack growth rate (da/dN) to the range of the stress intensity factor (AK). Finite element methods are employed to compute stress intensity factors and integrate the Paris law equation to predict crack propagation life. Variations of the Paris law, including the Modified Forman and Walker models, have been incorporated into finite element frameworks to account for factors like load ratio and other influences. 2. Strain Energy Density (SED) Approach: This criterion relates the energy release rate to crack growth, capturing the effect of the material's resistance to crack propagation. According to this criterion, a crack will propagate if the strain energy density at the crack tip exceeds a critical value. The SED criterion was initially developed for linear-elastic materials, and its application to elastic-plastic materials can be more complex and may require additional considerations [66; 67]. 3. Crack Tip Opening Displacement (CTOD) Approach: The CTOD criterion assumes that the crack growth rate is linked to the displacement at the crack tip, which reflects local deformation [68; 69]. The CTOD criterion was initially developed for elastic-plastic materials, but its applicability may be limited for materials with more complex deformation behavior, such as strain-hardening or strain-softening [70]. 4. Cohesive Zone Models (CZMs): CZMs simulate the gradual degradation of the material at the crack tip by introducing cohesive elements or surface-based cohesive interactions. These models can capture the effects of crack tip plasticity, crack closure, and microstructural features on crack growth. The CZM criterion assumes that crack growth occurs when the cohesive forces at the crack tip are no longer able to hold the two crack faces together.