# Advances in Finite Element Modeling of Fatigue Crack Propagation
## 1 Introduction  
Fatigue crack growth arises when a material experiences repeated cyclic loading, leading to the initiation and growth of cracks. This phenomenon is a significant concern across various industries, including aerospace, automotive, civil engineering, and biomechanics, as it can compromise the structural integrity and safety of components [1; 2; 3]. In the aerospace industry, finite element modeling is crucial for assessing the fatigue life and structural integrity of various components, including wings, fuselage, engine parts, and landing gear. These components are subjected to cyclic loading conditions during flight, leading to the initiation and growth of fatigue cracks [4; 5; 6]. By accurately modeling crack propagation, engineers can determine the remaining useful life of components, optimize maintenance schedules, and develop strategies for crack mitigation and repair. This is particularly important for ensuring the safety of aircraft structures and meeting regulatory requirements [7; 8; 9]. In the automotive industry, finite element modeling is widely used to evaluate the fatigue life of vehicle components, such as suspension systems, engine parts, and chassis structures [10; 11]. These components experience cyclic loading due to road-induced vibrations, engine forces, and vehicle weight. Fatigue crack propagation analysis helps engineers understand the behavior of cracks in these components, enabling them to optimize designs, select suitable materials, and enhance manufacturing processes. Accurate modeling of crack growth behavior aids in reducing the risk of unexpected failures, improving vehicle safety, and increasing the durability of automotive structures. In civil and structural engineering, fatigue crack propagation modeling is essential for assessing the long-term durability of infrastructure such as bridges, buildings, and offshorestructures [12; 13]. These structures are exposed to cyclic loading conditions, including traffic loads, wind loads, and wave-induced loads. Finite element analysis enables engineers to predict crack initiation and growth, estimate the remaining service life of structures, and develop maintenance strategies. By understanding the behavior of cracks, engineers can ensure the safety and reliability of civil infrastructure, avoid catastrophic failures, and plan cost-effective maintenance and repair activities. In the energy and power generation sectors, fatigue crack propagation modeling is vital for assessing the structural integrity of components used in nuclear power plants [14; 15], wind turbines [16; 17], and oil and gas installations [18; 19]. These components, such as rotating machinery, pressure vessels, and pipelines, are subjected to cyclic loading due to operational conditions. Finite element analysis assists engineers in evaluating the fatigue life of these components, identifying potential failure points, and optimizing maintenance schedules. Accurate modeling of crack propagation behavior helps in minimizing the risk of failures, ensuring the continuous and safe operation of energy and power generation systems. In material science research, finite element modeling is widely used to study the fatigue behavior of various materials. Finite element models help in analyzing crack propagation mechanisms, evaluating crack growth rates, and developing predictive models for fatigue life estimation. This research contributes to the development of more reliable and durable materials for a wide range of applications, including aerospace, automotive, and energy industries. Fatigue cracks typically originate from stress concentrations like notches or surface defects and gradually propagate until they reach a critical size, potentially causing catastrophic failure [20; 21]. Accurate prediction of fatigue crack initiation and propagation is crucial for ensuring the safe operation of engineering structures, from aircraft fuselages to automotive suspension systems.