Prediction of Fatigue Crack Growth Rate and Stress Intensity Factors Using the Finite Element Method

This study investigates crack growth and stress intensity factors via ﬁnite element methods in linear elastic fracture mechanics. The procedure involves estimating stress intensity factors (SIFs), crack trajectory, and fatigue life, using two diﬀerent softwares in both two and three-dimensional analyses. Crack modeling was done in a variety of ways depending on the software. ANSYS Mechanical R19.2 and FRANC2D/L software were used to prognosticate fatigue crack growth, fatigue life, and associated stress intensity factors under plane stress state. Fatigue analysis was governed by Paris’s law and crack growth direction by the theory of maximum circumferential stress. The results show that the fatigue growth was attracted to the hole and either changes its direction to reach the hole or ﬂoats by the hole and grows as the hole is missed. The ﬁndings of the study agree with other experimental and numerical crack propagation studies presented in the literature.


Introduction
Fatigue accounts for one of the more well-known mechanical and structural failures prevalent in industries, including aerospace and automotive components. Structural failure impairs components and poses health risks to people. Consequently, in obtaining an accurate solution, the study of such failures may involve significant complexity in evaluating a component's expected service life, which is the sum of the number of loading cycles needed to initiate a fatigue crack and the number of cycles needed to propagate to the critical size [1,2]. e assessment of fatigue crack growth includes identifying the direction of crack, equivalent stress intensity factor range (∆Keq), and crack growth rate per cycle number (da/dN). It is possible to ascertain the failure of the crack members subjected to static or fatigue loading by comparing the ∆Keq value to a certain characteristic of each material (known as fracture toughness or threshold stress intensity factor, respectively). Crack initiation and propagation should be associated to SIFs in a complicated state [3][4][5][6]. To address fatigue crack propagation, the linear elastic fracture mechanics (LEFM) method was developed. e LEFM techniques have been mostly accepted for long cracks with small-scale yielding characteristics at the crack front, i.e., the Paris regime [7,8]. e stress intensity factors (SIFs) under the concept of LEFM were used in identifying the status of crack propagation, i.e., whether the crack propagates or not. Internal crack propagation is important in engineering practice because it has a significant effect on the performance and stability of structural engineering. As a result, crack propagation path prediction is essential for engineering structural reliability. Preliminary investigations are necessary in estimating fatigue in aerospace, manufacturing, and aviation industry, this being cost-prohibitive, an alternative, but accurate estimation is necessary in predicting crack growth, orientation, and fatigue life in static and dynamic loads [9][10][11]. e finite element method (FEM) is a powerful computative approach in predicting the performance of complicated geometries and structures in engineering practice. e remeshing technique allows FEM to model fatigue crack propagation and simulates irregular crack growth by piecemeal mesh regeneration [12][13][14][15].
is work compares the fatigue crack growth under constant amplitude load between two softwares, ANSYS Mechanical R19.2 and FRANC2D/L, as an alternative tool for modeling fatigue crack growth problems in mixed-mode loads. Four different configurations of the modified compact tension were simulated in both softwares. e crack growth trajectory, SIFs, stresses distribution, and fatigue life cycles were predicted using both softwares, and the outcome from other researchers validated the results.
Fatigue assessment of materials can be mostly illustrated by three approaches: the fracture mechanics method proposed by Paris and Erdogan [37], the method of strain-life independently obtained by Coffin [38], and the stress-life method proposed by Wöhler [39]. e first method was employed (in the present work) in estimating fatigue life, and the crack tip was characterized independently by the SIFs. , and the crack growth in which mesh-based is a tetrahedron. e premeshed method requires the crack front used by the SMART crack growth analysis tool, where the stress intensity factor is the criterion of failure. Rendered node sets are distributed in the front, top, and bottom of the crack. Surrounding mesh at the crack tip is refined through the sphere of the influence process around the geometric sides that passes across the width. e geometric regions to be described are crack tip, top, and bottom surfaces of crack. Subsequently, these regions are linked with a node set for analysis. ANSYS Mechanical R19.2 software considers mixed-mode loading where the maximum circumferential stress criterion is implemented. e following are the formulas for the direction of crack propagation in ANSYS [40,41]:

Materials and Methods
where K I � Max K I during cyclic loading and K II � Max K II during cyclic loading. In ANSYS Mechanical R19.2, by using XFEM, crack growth simulation is restricted to region II, and typical fatigue crack growth graph is represented as where a is the length of the crack, N is the number of cycles, C is the Paris constant, and m is the Paris exponent. Both C and m are dependent strongly on the cyclic hardening  exponent and mechanical properties [42], and they are obtained experimentally. Several authors [43,44] identified a very consistent empirical relationship between Paris' law parameters, which was supported by experimental results, such as [45,46].
From equation (2), the number of fatigue life cycles for crack increment is e equivalent range for SIFs formula is represented as [47] ΔK eq � 1 2 cos where ΔK I represents the SIFs range in mode I loading and ΔK II represents the SIFs in mode II loading. ANSYS flowchart is shown in Figure 1.

FRANC2D/L Procedure.
e Cornell Fracture Mechanics Group at Cornell University developed the free twodimensional fracture analysis software FRANC2D/L, which was funded by the US National Science Foundation, NASA, the US Navy, and other agencies [19]. e FRANC2D/L analysis is carried out in two stages, with CASCA which is used as a mesh generator with different types of mesh and associated with FRANC2D/L. In the second part, the boundary constraints, problem characteristics, stress    computation, input crack singularity, crack growth, and problem outcomes were determined [48,49]. SIFs were computed using three approaches in FRANC2D/L: method of displacement correlation, method of modified crack closure integral, and method of J-integral. e crack orientation was estimated via maximum circumferential stress criterion, while fatigue crack growth rate was calculated based on Paris law equation with the same procedure used in ANSYS as represented in equations (1) and (2), respectively. e step-by-step procedure of the FRANCD2D/L software is shown in Figure 2. Figure 3 shows the modified CT specimen's geometrical dimensions in two and three-dimensional representations. According to this figure, the hole's diameter is 7 mm from the crack's initiation position, which is located on horizontal and vertical distances K and C; for isotropic and linear elastic material at 0.1 load ratio, the test was performed under fatigue load. A total of four specimens were simulated in both programs. e material properties are given in Table 1, as well as for the initial mesh for the first specimen, CT01 in both softwares is shown in Figure 4. Table 2 provides a total number of nodes and elements for every specimen with different values in two softwares. e size of the mesh element was set as 1 mm.

Modified Compact Tension Specimens (MCTS).
Simulation from FRAN2D/L and ANSYS was juxtaposed with experimental [50] and numerical fatigue crack growth paths [32] using the boundary elements method (BEM) with BemCracker 2D software and FEM with Quebra2D software for CT01, CT02, CT03, and CT04 as shown in Figures 5-8, respectively. e crack pathways in this work in all cases using FRANC2D/L and ANSYS align with the experimental and numerical results of [32,50]. Moreover, the growth of the fatigue crack was consistently captivated by the hole; it curves and grows into the hole (sinks in the hole behavior), as shown in Figures 6 and 8, or deflect and propagate after missing the hole (missing the hole's behavior), as shown in Figures 5 and 7. e fatigue life cycles in individual structures significantly vary, depending on crack path, though hole location is only slightly differentiated. e fatigue crack was still captivated to the hole, according to the simulations, so it could curve its path and grow toward the hole or merely deflect and propagate.
Maximum and minimum principal stresses of the four specimens are compared in both programs as shown in e predicted values of stress intensity factors for mode I and mode II (K I and K II ) associated with the crack length for CT01, CT02, CT03, and CT03 are shown in Figures 13-16, respectively.
It has worth noting that when there is a hole in the crack path, K I decreases to a certain amount while K II increases.
is is visible in the crack path's trajectory; when the crack continues in a straight line, K I increases, but as the direction  [50], (d) BemCracker2D software [32], and (e) Quebra2D software [32]. used.
e number of steps used in ANSYS for specimen simulation ranges from 11 to 20, with the number of increments controlled by the software. In FRANC2D/L, the steps range from 32 to 92 with 0.1-0.13 increment. Figures 17-20 show comparisons of FRANC2D/L and ANSYS fatigue life simulations with experimental outcomes [32] for CT01, CT02, CT03, and CT04 MCTS geometries, respectively. e simulated fatigue crack growth life using FRANC2D/L and ANSYS agrees very well with the experimental results by Gomes and Miranda [32]. In addition, the current research results were far more precise in predicting fatigue life compared to the numerical findings obtained with [32] using the two softwares, VIDA and BemCracker2D. e results of this analysis are closely associated with Quera2D rather than BemCracker2D, as shown in these figures. ese findings further demonstrate the influence of the hole position on the fatigue life of the specimens.

Conclusion
According to the predicted results, the fatigue crack grows approach the hole due to the unequal stresses at the crack extremities as a result of the hole. Furthermore, different positions of the hole had a remarkable impact on the fatigue crack growth path. e crack direction bends more severely as the hole's core distance approaches the initial crack. e position of the hole influences the crack and modifies its path, either changing its pathway and growing into it or deviating from it and growing. e predicted fatigue crack growth paths, as well as the fatigue life cycles, were validated by the experimental and numerical findings performed by other researchers. Due to the available computing resources, the two-dimensional analysis offers a faster computational time than three-dimensional analysis while also allowing for a finer mesh to be generated on the geometry. In 3D models, the time increases dramatically as the mesh density increases. It is not always possible to simulate a complex geometry in 2D, necessitating the use of 3D software. As a result, the 2D and 3D analytical results were nearly identical for plane stress geometries. For all geometries, the representations of stresses distribution in both software were visualized.

Data Availability
e data used to support this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest.