Multidisciplinary Materials Chronicles

1. Introduction

Fatigue behavior in materials is a critical concern in engineering, as it significantly influences the longevity and reliability of components subjected to cyclic loading. The phenomenon of fatigue involves the initiation and propagation of cracks under repeated stress cycles, which can eventually lead to catastrophic failure without prior warning [1–3]. This unpredictable nature of fatigue-induced failures has historically resulted in severe accidents and substantial economic losses, particularly in high-stakes industries such as aerospace, automotive, and infrastructure.

Understanding the mechanisms of fatigue is essential for predicting the service life of components and for developing strategies to mitigate failure risks. Key factors influencing fatigue behavior include the material's microstructure, the presence of stress concentrators, surface finish, mean stress, frequency of loading, and environmental conditions [4, 5]. These variables interplay to affect the initiation and growth of fatigue cracks, making it imperative to comprehend their roles to improve material performance and safety.

The study of fatigue behavior encompasses several aspects: the fatigue stress cycle, the S-N curve, the existence of fatigue limits, and the fracture mechanisms under cyclic loading [6–8]. The fatigue stress cycle includes parameters such as maximum and minimum stress, mean stress, and stress amplitude, which are crucial for characterizing fatigue loading. The S-N curve, or Wöhler curve, illustrates the relationship between stress amplitude and the number of cycles to failure, providing valuable insights into material endurance under cyclic loading. The concept of a fatigue limit, where materials can withstand infinite cycles without failure below a certain stress threshold, is particularly significant for designing durable components [9, 10].

Additionally, the fracture mechanisms in fatigue involve stages of crack nucleation, growth, and eventual failure. Microstructural features, such as grain boundaries and dislocations, play a pivotal role in these stages. Understanding these mechanisms aids in identifying the factors that contribute to fatigue resistance and in developing materials with enhanced durability.

This paper aims to delve into the intricacies of fatigue behavior, exploring the fundamental principles and mechanisms that govern fatigue life in materials. By gaining a comprehensive understanding of fatigue processes, engineers can design more reliable and longer-lasting components, ultimately enhancing safety and performance across various applications.

2. Importance of understanding fatigue behavior

A thorough comprehension of fatigue behavior is crucial for various reasons. Firstly, fatigue-related failures, which can occur without prior warning, have historically resulted in catastrophic events, including aviation accidents and infrastructure collapses. The economic implications are also significant; unforeseen fatigue failures can lead to costly repairs, unexpected downtimes, and legal challenges [11]. Comprehending the fatigue process, including crack initiation, growth mechanisms, and impact of microstructure, is key to accurately predicting component fatigue life [2, 12, 13]. Knowledge of how factors like stress concentrations, surface finish, mean stress, frequency, and environment influence fatigue life is essential.

Fatigue-related failures, occurring unexpectedly, have historically led to catastrophic events like aviation accidents and infrastructure collapses, causing significant economic losses and reputational damage. Understanding fatigue processes, including crack initiation and growth, is crucial for predicting component life accurately [3]. Factors such as stress concentrations, surface finish, mean stress, frequency, and environmental conditions significantly affect fatigue life [14]. By comprehending fatigue behavior, engineers can design safer components, optimize materials, and extend product lifespans. This knowledge facilitates technological advancements, predictive maintenance, and risk reduction across industries like aerospace, automotive, and biomedical sectors. Fatigue analysis plays a pivotal role in ensuring structural integrity and safety standards.

3. Fatigue principle

3.1. Fatigue stress cycles

In fatigue testing, the term "Fatigue Cycle" refers to the smallest repetitive portion of a stress-time or strain-time cycle [15]. It is also called a stress, load, or strain cycle. The determination of the load cycle in fatigue testing involves considering various parameters, as presented in Figure 1 including the maximum stress (σ_max), minimum stress (σ_min), steady or mean stress (σ_m), stress amplitude, also referred to as alternating stress (σ_a), range of stress (σ_r) in a fatigue cycle, stress ratio (R), and amplitude ratio (A) [16, 17]. The amplitude ratio is calculated as

The time-dependent stress cycles, depicted in Figure 1 can be observed in the materials' fatigue performance. Figure 1 displays sinusoidal stress variations with different stress ratios that are typical in rotating and reciprocating machine components. Nonetheless, fatigue loading often involves a complicated stress cycle with irregular/random stress variation. An airplane wing subjected to storm-induced unexpected overloads may experience this irregular or random stress cycle. This type of stress can also be detected in intermittently working machinery due to natural vibrations with varying amplitudes through beginning and stopping. To analyze fatigue design, it is necessary to simplify the random stress cycle. This is usually achieved by dividing the actual stress cycle into several simplified sinusoidal variations, ensuring that each set contains the same number of oscillations with similar mean stress and stress amplitude [19].

Fatigue stress cycles can involve tension, compression, or shear stresses induced by axial, shearing, flexural, or torsional loading, or a combination of two of them. According to the ASTM Manual of Fatigue Testing, there are over thirty fatigue testing devices based on the load type and the way of its application [18]. The fatigue samples are loaded by applying a constant maximum load, moment, or maximum displacement or strain [20]. Fatigue cycles can be classified into two primary categories [21]:

1.   High-cycle fatigue, also called “stress-controlled fatigue,” involves conducting fatigue tests using a prescribed cycle of fixed load or stress limits. This type of fatigue generally occurs with high-stress cycle numbers, (<10⁴ cycles).

2.  Low-cycle fatigue, termed “strain-controlled fatigue,” involves conducting fatigue tests using a prescribed set of elastic and plastic strain limits within each cycle. This type of fatigue usually occurs at low numbers of stress cycles (> 10³ cycles).

3.  Ultra-low cycle fatigue. This section highlights the unique characteristics of ULCF, such as its occurrence under large cyclic loads leading to complete ductile fracture and the distinct mechanisms involved, which are different from those in HCF and LCF.

4.  Ultra-high cycle fatigue (UHCF), which refers to the phenomenon of fatigue failure that occurs at extremely high numbers of cycles, typically exceeding 10⁷ or even 10⁹ cycles. Unlike traditional HCF, which typically involves fatigue failure at lower numbers of cycles (e.g., 10⁴ to 10⁶ cycles), UHCF occurs under conditions of very low stress amplitudes and is often associated with components subjected to high-frequency loading or vibrations. The mechanisms governing UHCF differ from those of conventional fatigue, with factors such as material microstructure, surface conditions, and environmental influences playing significant roles. UHCF is characterized by the initiation and propagation of microstructural defects, such as microcracks, grain boundary cracking, and surface- initiated cracks, which gradually propagate over a large number of cycles until catastrophic failure occurs.

Worthy to note that high-cycle fatigue has gained engineering significance because it accounts for the vast common fatigue failures during service (<10⁴ cycles). The smooth and well-prepared surfaces of fatigue samples are essential to prevent stress concentrations and minimize tensile residual stresses. Although fatigue properties are a significant cause of safety failures in industries such as automotive and aerospace, they have not received adequate attention because of the complexity and time-consuming nature of fatigue testing, which requires numerous well- prepared specimens [21, 22].

Table 1 reveals the fatigue stress ratio (R) and amplitude ratio (A) associated with the fatigue loading modes. Lerch and Halford [23] studied the influence of control mode (strain and stress- controlled) and R-ratio at both R= 0 (zero-tension load) and R = -1 (fully reversed) on the fatigue performance of MMC samples. According to their results, it is found that loading condition affects damage mode. For instance, for R = 0, transverse fiber cracking is the dominant damage mechanism, and there is practically little cracking observed in the matrix. Whereas, for R = 1, transverse fiber cracking was uncommon during tension-compression loading, and the matrix was severely damaged. Both strain and stress-controlled conditions yield the same findings.

Table 1. The values corresponding to the fatigue stress ratio (R) and amplitude ratio (A) associated with the fatigue loading modes [23].

3.2. S-N curve and fatigue properties

The S-N curve, commonly called the fatigue curve or Wöhler curve, describes the material fatigue properties [24]. The S-N curve can be applied only for high-cycle (stress-controlled) fatigue, usually more than 10⁴ cycles. A series of fatigue samples are tested under identical conditions at different stress levels to establish the S-N curve. For each stress level, the samples are tested until a fracture occurs. Then, the stress amplitude (S) and the cycle numbers (N) at fatigue fracture are determined, and these obtained values of the S and N act for one point on the S–N curve [25]. Although the stress amplitude changes in each sample in a tested series, the mean stress remains constant. The S-N curve in Figure 2 (a) displays the correlation between S and N for two categories of metals: non-ferrous metals like aluminum and titanium alloys (curve A) and ferrous metals like steel (curve B).

The S-N curve for ferrous metals highlights the endurance limit or fatigue limit (σ_L), a specific stress level below which fatigue failure does not occur [26]. This is evident in the horizontal segment of the curve at increasing values of N. The fatigue limit, typically much lower than the yield strength, serves as a threshold below which fatigue failure is not expected, despite cycle numbers [27]. In contrast, nonferrous metals like aluminum, copper, and magnesium alloys lack a fatigue limit [28]. Their S-N curve never flattens out, exhibiting a decreasing trend and the absence of a distinct lower stress limit, making it challenging to identify a "safe" stress level for these materials.

In addition to the fatigue limit, fatigue properties such as fatigue strength can be determined from the S-N curve, as depicted in Figure 2 (a). Dynamically stressed components do not necessarily need a fatigue-limit design for economic purposes. In most cases, parts will only be subjected to stress cycles up to the number specified by their service life [16]. For instance, milling cutters are not designed to last indefinitely; instead, they are typically designated with a specific number of load cycles that they must endure without sustaining damage [29, 30]. This predetermined strength is now known as fatigue strength rather than the fatigue limit. Thus, the fatigue strength represents the stress amplitude that a material can endure under a specific mean stress for a given number of load cycles without encountering any fatigue damage.

Figure 2(b) illustrates the influence of strain aging on the fatigue limit. Strain aging refers to the phenomenon where materials undergo changes in their mechanical properties due to the interaction of dislocations with interstitial atoms (such as carbon or nitrogen) over time, particularly at elevated temperatures. This interaction can increase the strength and hardness of the material, thereby elevating the fatigue limit. The S-N curve in Figure 2(b) demonstrates how strain aging can result in a higher fatigue limit compared to materials that have not undergone strain aging.

The differences in the shapes of the S-N curves in Figure 2(a) and 2(b) are thus attributed to these underlying high-cycle fatigue mechanisms. The presence of a distinct fatigue limit in Figure 2(a) is indicative of materials that can endure cyclic stresses indefinitely below this limit. In contrast, the increased fatigue limit shown in Figure 2(b) displays the impact of strain aging, which enhances the material resistance to cyclic loading by impeding dislocation movement and reducing the rate of crack initiation and propagation.

3.3. Existence of fatigue limit

The existence or non-existence of a fatigue limit is a subject of ongoing debate. Canteli et al. [31] addressed this issue in their paper and proposed that various aspects of fracture mechanics must be considered. However, the reasons why certain materials have a fatigue limit are unknown, and there are many hypotheses regarding this important question. One of these hypotheses is associated with the stacking fault energy (SFE). Fatigue damage accumulates to cause fatigue failure.

Assuming flawless samples without defects or notches, fatigue damage initiates through the localized plastic strain, which can result in intergranular or transgranular microcracks. Grain boundaries can hinder the growth of these microcracks, and a material's SFE is a crucial factor in determining whether or not microcracks can surpass these barriers. With high SFE materials such as Al, Zn, and Mg, for instance, dislocations can easily traverse slip and overcome impediments, which is the same goes with microcracks. SFE can affect the structure of dislocations, but it does not directly impact crack growth. Then, the microcracks can easily propagate to form macrocracks, eventually resulting in fatigue failure [31].

Awatani et al. [32] studied the impact of SFE on fatigue crack growth through dislocation patterns. They declared that dislocation could transfer through cell boundaries in metals with high SFE, lessening crack growth rates. On the other hand, in materials with very low SFE, i.e., titanium and steel alloys, slip is limited, and crack growth is faceted in the case of low imposed load/strain amplitude. Thus, microcracks can be impeded by obstacles like grain boundaries, preventing the formation of macrocracks. Consequently, when the accumulated fatigue damage is inadequate to induce failure, the material will display a distinctive break on the S-N curve known as a fatigue limit. Another hypothesis ascribed the appearance of the fatigue limit to a metallurgical phenomenon called "strain aging", which can occur in some materials but not in others. Strain aging is a phenomenon were solute atoms trap dislocations, causing an increase in yield stress. This means that greater stresses are required to release these dislocations and produce deformation in materials like steel that exhibit strain aging [33].

It was reported that interstitial elements in materials determine a fatigue limit [34, 35]. For more clarification, if we considered a pure metal, the S-N curve would decrease gradually as the number of cyclic cycles increases (curve A) in Figure 2 (b). Suppose adding a solute element to the pure metal forms a solid solution. The S-N curve will be shifted to curve B (Figure 2 (b)) because of the solid solution strengthening. Suppose a sufficient number of interstitial elements are added to the pure metal. In that case, the interstitials can strengthen additional alloy through strain aging. The fatigue limit (curve C in Figure2 (b)) results from the equilibrium between the localized strengthening by strain aging and the fatigue failure induced through applied stress at a specific limiting stress since strain aging is relatively independent of applied stress. Worthy to note that the strain aging increases with higher fatigue temperatures or a higher number of interstitial elements. Curve D in Figure 2 (b) illustrates this augmented strain aging by showing that the fatigue limit is increased and an earlier onset of the horizontal 'knee' compared to curve C (Figure 2 (b)), as demonstrated by the reduced number of stress cycles.

Wilson [36] provided evidence suggesting that strain aging can result in the development of a fatigue limit in mild steel. Also, Nakagawa and Ikai [37] investigated the influence of strain aging on the fatigue limit. They demonstrated that the fatigue limit occurs when dislocation multiplication, work hardening, and strain aging reach a state of equilibrium at the maximum stress level. The S-N curve for most nonferrous metal slopes steadily downward with increasing cycle numbers. It never gets horizontal, so these metals lack a real fatigue limit. Thus, it is practically common to define the fatigue properties of the material by considering the fatigue strength at a specific number of cycles. Usually, the fatigue test is terminated for practical considerations at low stress. The fatigue limit for these materials is commonly expressed as the stress at which failure happens at a rate of roughly 10⁸ or 5x10⁸ cycles [38].

4. Fracture fatigue mechanism

Understanding the fracture mechanism in fatigue necessitates understanding the development of fatigue fracture. Fatigue fractures occur when stress is applied, removed, and reapplied cyclically. This repetitive process can continue for millions of cycles until the fatigue fracture reaches a critical point, leading to material failure [38].

4.1. Stages of crack formation

The fatigue failure process involves three main stages, which include: crack nucleation, crack growth and propagation, and eventually fatigue failure. The following is a brief discussion of these three stages.

4.1.1. Crack nucleation

Crack nucleation is a failure process that happens at the initial stage of fatigue. As a result of surface irregularities such as roughness, small cracks, and other imperfections, no material surface can be considered completely smooth at a microscopic level [2, 39–41]. These defects can act as heterogeneous sites for initiating crack formation, i.e., microscopic notches, resulting in the buildup of triaxial stresses that exceed the uniaxial nominal stresses. The yield point is, therefore, locally exceeded at these defects, resulting in microscopic deformations. During fatigue, metals undergo deformation by slipping on the same atomic planes and the same crystallographic directions [42].

Under fatigue conditions, the formation of slip lines and the gradual accumulation of small slip lines frequently occur within the initial few thousand stress cycles, leading to the development of cyclic slip bands. These tiny slips move back and forth within a slip band, producing ridges called slip-band extrusions and grooves or notches, known as slip-band intrusions, which exacerbate crack formation. Zhang et al. [43] confirmed that the slip bands formed during fatigue loading are crucial in initiating and growing cracks, as illustrated in Figure 3 depicting the fatigue crack initiation and propagation process in a nickel-based superalloy loaded to failure.

4.1.2. Crack growth and its propagation

This stage can be divided into two distinct steps: crack growth, characterized by the initial propagation of the crack along slip planes at an extremely slow rate on the order of nanometers per cycle [44]. During this phase, the fatigue crack extends gradually, spanning a few grain diameters until it reaches a critical length at which the stress field surrounding the crack tip becomes dominant. Figure 4 (a) shows a schematic depiction of the fatigue failure crack growth through the three main fatigue stages. The mechanical characteristics of the material and the applied stress level could define the critical size of the fatigue crack [45]. After reaching the critical length, the second step starts crack propagation perpendicular to the applied principal tensile stress direction at a rate of microns per cycle, significantly faster than the rate in step 1. The crack propagation step typically results in a pattern of ripples or fatigue striations, invisible to the naked eye [46]. It can only be seen under magnification; see Figure 4 (b).

The distance between these striations, often referred to as fatigue striation spacing, provides a measure of the crack growth rate per stress or strain cycle. Each striation represents the incremental advancement of the crack front during successive loading cycles. The presence of striations signifies the progressive movement of the crack front as it propagates through the material under cyclic loading conditions.

Fatigue crack propagation involves a cyclic process of plastic blunting and re-sharpening of the crack front, as depicted schematically in Figure 5 (a-f). During each loading cycle, the crack tip undergoes plastic deformation, resulting in the blunting of the crack and the formation of plastic zones around the crack tip. Subsequent unloading allows for crack closure, but residual plastic deformation remains. Upon reapplication of the load, the crack tip re-sharpens as it advances further into the material, forming new fatigue striations [46, 47].

This cyclic process of crack blunting and re-sharpening continues throughout the fatigue life of the material, leading to gradual crack propagation. The observation and analysis of fatigue striations provide valuable insights into the mechanisms of fatigue crack growth and are instrumental in predicting the fatigue life of structural components and materials.

4.1.3. Fatigue failure

In this stage, fatigue failure would occur if the crack developed in the second stage (growth and propagation stage) continues to grow due to existing sufficient energy. The ultimate mode of failure, which is a catastrophic fracture, can either be ductile or brittle or a combination of both, depending on various   factors like the material type, thickness, applied stress, temperature applied stress, etc. Fatigue failures often happen unexpectedly and suddenly, without prior warning. Fatigue-induced fractures in damaged parts consistently display smooth surfaces without evidenceof plastic deformation. Recent research studies for additional clarification on the stages of fatigue failure are found elsewhere [48–51].

4.2. Fatigue striations and beach marks

The presence and appearance of fatigue striations and beach marks are highly dependent on the type of material being analyzed. For instance, materials like aluminum and aluminum alloys tend to exhibit the formation of striations [46, 52]. On the other hand, steel may display cleavage as a dominant fracture mechanism [19, 53]. Regardless of the material, the formation of striations is typically attributed to the combination of cyclic stress, plastic strain, and stress frequency acting simultaneously. Typical fatigue striations, see Figure 4 (b) can appear in the material during the second stage, in which crack propagation occurs due to the cyclic progress of the crack with each load change. When the stress intensity level exceeds significantly or non-uniform dynamic stress is applied, these fatigue striations can convert into beach marks (shown in Figure 4 (b)) which become visible to the naked eye, unlike fatigue striations. These marks appear as concentric rings in the fatigue zone and resemble the tide marks seen on a beach. Beach marks are oriented normally to the orientation of the crack propagation direction. The main difference between fatigue striations and beach marks is that fatigue striations are only visible at a microscopic level. In contrast, beach marks are visible at a macroscopic level. Various components, for example, the automobile drive shafts, are rarely exposed to uniform dynamic stress; hence, such a multistage load is typically encountered in practice. Since the fatigue test is often a uniform dynamic load (a single-stage load with constant stress amplitude and not a multi-stage load), no beach marks can be detected on the fracture surface [54].

5.  Conclusion

This study highlights the critical importance of understanding fatigue behavior in engineering materials to prevent unforeseen failures, such as those in aviation and infrastructure, which lead to significant economic and safety impacts. The research emphasizes the mechanisms of crack initiation and growth, the role of microstructural features, and the influence of stress parameters on fatigue life. Analyzing S-N curves reveals that certain materials have a fatigue limit, which is crucial for designing components to endure cyclic stresses below this threshold. Understanding the stages of fatigue fracture, from crack nucleation to propagation and failure, is essential for mitigating fatigue-related failures. The insights from this study aid in designing safer components, optimizing material selection, and extending product lifespans across various industries. This knowledge facilitates technological advancements, predictive maintenance, and risk reduction, ensuring structural integrity and safety. Future work should focus on advancing fatigue testing methods, exploring new materials with enhanced fatigue resistance, and developing more accurate predictive models for fatigue life.