## 1 Introduction  
The assessment of fatigue analysis is of utmost importance in determining the structural integrity and long-term durability of diverse engineering components that experience cyclic loading. A thorough comprehension of the crack growth mechanisms under diverse loading conditions is crucial for the precise prediction of fatigue life and for ensuring the safe operation of critical structures [1; 2; 3]. In order to mitigate significant damage, it is crucial to calculate the three-dimensional fatigue crack growth (FCG) and determine the permissible lifespan of structures that have cracks. The stress ratio (R) is computed by comparing the minimum stress to the maximum stress experienced during each loading cycle. Based on experimental data, it has been observed that the FCG is significantly affected by both the stress ratio (\(R\)) and the range of SIFs [4; 5]. The stress ratio (\(R\)) is a crucial parameter in fatigue analysis as it quantifies the ratio of the minimum stress to the maximum stress encountered during a loading cycle. This parameter plays a pivotal role in characterizing the loading conditions and exerts a notable influence on the fatigue behavior and lifespan of a component. The selection of an appropriate stress ratio is crucial as it significantly influences the fatigue life of a material or component. To accurately estimate fatigue crack growth, several factors need to be taken into account, including stress level, load frequency, load ratio, and material type. The load ratio is a factor that exerts a noteworthy influence on the growth of fatigue cracks and exhibits a distinct effect on the occurrence of threshold behavior [6; 7; 8; 9; 10; 11]. Furthermore, the range of stress intensity factors represents the disparity between the maximum and minimum stress intensity factors experienced during each loading cycle. These factors play a crucial role in determining the rate of fatigue crack growth and are vital considerations in evaluating the durability and structural integrity of materials subjected to cyclic loading [12; 13; 14]. In this context, the investigation of stress ratios, deformation levels, and stress intensity factors (SIFs) becomes imperative to comprehensively analyze the fatigue life of materials. The strong correlation between FCG and LEFM forms a fundamental framework for understanding and analyzing crack behavior in materials, specifically within the linear elastic regime [15; 16]. In linear elastic fracture mechanics (LEFM), three commonly used criteria exist to analyze crack propagation: the Maximum Tangential Stress (MTS) Criterion, the Maximum Shear Stress Ratio (MSSR) Criterion, and the Extension of the Maximum Tangential Stress (Ex-MTS) Criterion. These criteria and their respective explanations can be found in detail in references [17; 18]. Numerous numerical techniques have demonstrated success in modeling and simulating engineering challenges, particularly in the field of fracture mechanics, where finding optimal solutions can be difficult due to the singularity of the stress field near the crack tip. These techniques include the Extended Finite Element Method (XFEM) [19; 20; 21; 22], Finite Element Method (FEM) [16; 23; 24; 25], Discrete Element Method (DEM) [26; 27; 28], a mesh-free method [29], and Boundary Element Method (BEM) [30]. Numerous numerical methods have been proposed for analyzing the FCG in the modified compact tension specimen under constant amplitude loading conditions. Recent research conducted by [PERSON] et al. [31] has shown the effectiveness of artificial neural networks in predicting the path and life of FCG in different geometries of the MCTS. The study highlights the importance of incorporating additional experimental data to further improve the training process. [PERSON] and [PERSON] [32] conducted a study where they employed both the dual boundary element method and finite element method using two software tools, namely BemCracker2D and Quebra2D. These methods were utilized to simulate two-dimensional crack growth in the MCTS.