# Advances in Finite Element Modeling of Fatigue Crack Propagation
## 2 Fundamentals of Finite Element Modeling
### Material Modeling for Fatigue Crack Growth  
Modeling fatigue crack growth in materials through finite element analysis (FEA) has emerged as a prominent area of research in recent years. One key aspect is the material constitutive model used to capture the relevant deformation and damage mechanisms. Regarding deformations, fatigue phenomena can be categorized into low-cycle and high-cycle fatigue. The distinction between these two types is generally based on the number of cycles, typically set around \(10^{3}\) to \(10^{4}\) cycles. High-cycle fatigue usually involves elastic deformations with a focus on crack initiation, while low-cycle fatigue is characterized by significant plastic deformations, necessitating different approaches for analysis and design. For metals, elastic-plastic models with damage variables have been widely employed, accounting for phenomena like cyclic hardening/softening, ratcheting, and void growth. For polymers and composites, viscoelastic-viscoplastic models are often required to represent time-dependent and non-linear behavior. Accurate modeling of material behavior under cyclic loading is crucial for effectively capturing fatigue crack propagation. The selection of the material model is influenced by the type of material being analyzed and its specific response to cyclic loading. Linear elastic fracture mechanics (LEFM) are applicable for high-cycle fatigue (HCF). In HCF conditions, stress levels are generally low, enabling materials to exhibit elastic behavior. LEFM effectively predicts crack initiation and growth in these cases by focusing on the elastic stress fields surrounding cracks. However, it is crucial to ensure that the material remains within its elastic limits for accurate predictions. Conversely, the LEFM assumption may be inaccurate for cycles N \(<10^{3}\) to \(10^{4}\), depending on the material characteristics [40]. This limitation underscores the necessity for alternative methods that account for non-linear behavior, especially in situations involving considerable plastic deformation [41; 42]. These may include elasto-plastic models to capture the cyclic plasticity exhibited by many metallic materials. The J2 plasticity model [41; 43] and the Chaboche model [44; 45] are examples of commonly used models that consider the accumulation of plastic strain and cyclic hardening. In addition to elasto-plastic models, viscoelastic models are used to represent the time-dependent behavior of materials under cyclic loading [46; 47]. Viscoelastic models account for phenomena such as creep and stress relaxation, which can significantly influence fatigue crack propagation in certain materials. [PERSON] et al. [48] have observed that crack growth in viscoelastic solids under cyclic loading is generally faster than that under static loading at the same amplitude. This phenomenon, referred to as the \"dynamic effect\", is recognized as a key mechanism driving the fatigue fracture of soft viscoelastic polymers, although its underlying physical nature remains unclear. On the other hand, brittle materials, characterized by their low plastic deformation beforefailure, exhibit unique crack propagation behavior. Crack initiation typically occurs at stress concentrators, such as microstructural defects or surface irregularities. Once initiated, the crack propagates through the material with minimal plastic deformation, often following pre-defined crystallographic planes. LEFM is a foundational theory applied to analyze crack growth in brittle materials. The Paris law, which relates the crack growth rate to the range of stress intensity factors, has been widely used to describe FCG. However, it is often noted that LEFM may not fully capture the complexities involved in the crack growth mechanisms of brittle materials under all conditions. Cohesive Zone Models (CZMs) have become an essential framework for simulating crack growth in brittle materials, offering a more nuanced understanding of fracture mechanics.