Fatigue Failure of Adhesive Joints in Fiber-Reinforced Composite Material Under Step/Variable Amplitude Loading—A Critical Literature Review

Abstract

Most fatigue-loading research has concentrated on constant-amplitude tests, which seldom represent actual service conditions. Because of the significant time and expense associated with variable-amplitude experiments, researchers often employ block/step-loading tests to evaluate the effects of variable-amplitude loading. These tests utilize various sequences of low-to-high and high-to-low loads to simulate real-world scenarios. Empirical investigations have shown inconsistencies in the damage accumulation under different load sequences. Although literature reviews exist for simulation and experimental methods, there is limited research examining the impact of step/variable-amplitude loading on adhesive joints in composite materials. This review aims to address this gap by comprehensively analyzing the effects of load sequence and block loading on fatigue damage progression in fiber-reinforced polymer composites. Additionally, the applicability of various step-loading fatigue damage accumulation models to adhesive materials is evaluated through numerical simulation to study its suitability in predicting fatigue failure. This review also explores recent theoretical advancements in this field over the past few years, examining more than 100 fatigue damage accumulation models categorized into seven subcategories: (i) linear damage rules, (ii) nonlinear damage curve and two-stage linearization models, (iii) life curve modification models, (iv) models based on crack growth concepts, (v) continuum damage mechanics-based models, (vi) material degradation models, and (vii) energy-based models. Finally, numerical simulations using the most common nonlinear cumulative fatigue damage accumulation models were conducted to predict fatigue failure in adhesively bonded joints under four step-loading tests, and the results were compared with the experimental data. Numerical simulations revealed the need and scope of further development of a fatigue failure model under step/variable loading. This comprehensive review offers valuable insights into the complex nature of fatigue failure in adhesive joints under variable loading conditions and highlights current state-of-the-art nonlinear fatigue damage accumulation models for adhesive materials.

1. Introduction

To accurately assess the fatigue characteristics of adhesive materials, it is crucial to replicate real-world fatigue loading conditions as closely as possible during testing. Adhesive joints under fatigue in real-life service usually encounter variable/step loading and rarely encounter a constant amplitude. However, fatigue testing is often performed at a constant amplitude owing to high cost and variable-amplitude experimental time, fatigue testing facility limitations, and the unclear nature of the in-service loading spectrum.

Despite its long history, cumulative fatigue damage remains an unresolved issue. In 1945, Palmgren and Miner introduced a mathematical model to address cumulative fatigue damage under variable loading, which can be expressed by Equation (1) [1], as follows:

$$D=\sum \left({\displaystyle \frac{n_i}{N_i}}\right)$$

(1)

where D is the total damage, $n_i$ is the number of fatigue cycles under the ith constant-amplitude loading level, and $N_i$ is the number of fatigue cycles until failure under the ith constant-amplitude loading level. Building on Palmgren and Miner’s damage accumulation theory, numerous researchers have created models to describe how fatigue damage accumulates under varying fatigue load conditions. First, fatigue damage caused by a specific stress level must be determined. Second, the cumulative fatigue damage incurred at multiple stress levels must be accounted for accurately. Finally, the critical damage threshold that triggers failure must be identified when the material or structure reaches its limit.

Several experimental investigations were conducted on the fatigue life of adhesively bonded joints under block/variable loading conditions [2,3,4,5]. These investigations have uncovered notable inconsistencies between empirical results and conventional models that assume uniform fatigue damage accumulation. The observed discrepancies can be attributed to several factors.

• Neglecting to factor in the load history;
• Inability to consider the loading sequence when determining fatigue failure;
• Disregarding the impact of load interaction effects.

Miner’s rule typically yields conservative fatigue life predictions for loading sequences that progress from low load to high load (${load}_1<{load}_2$). Conversely, when the loading sequence transitions from high load to low load (${load}_1>{load}_2$), the results tend to be nonconservative, as illustrated in Figure 1. The observed material behavior may be attributed to various factors, some of which are as follows:

• Substantial strain often leads to early development of initial microscopic cracks. These tiny fissures then expand under less intense cycles, causing accelerated deterioration compared with conditions with consistent stress levels.
• An increased number of load cycles on a specimen can lead to surface roughening during repeated plastic deformation. This roughening process generates additional locations where cracks could be initiated under lower load cycles.
• It should be emphasized that the accumulation of damage is not a linear process.

To address these problems, researchers have developed several theories regarding nonlinear damage accumulation. However, these theories fail to account for one or more of the factors listed above [6,7,8].

The second section of the study examines various nonlinear cumulative fatigue damage accumulation models created to address the nonlinear nature of damage accumulation during step/variable loading these models, which focus on cumulative fatigue damage accumulation, can be classified into seven categories: (i) linear damage rules [1], (ii) nonlinear damage curve and two-stage linearization models [9,10,11,12,13], (iii) life curve modification model [14,15,16,17,18,19], (iv) models based on crack growth concepts [20,21,22,23,24,25,26,27], (v) continuum damage mechanics-based models [28,29,30,31,32,33,34], (vi) material degradation models [7,35,36,37,38], and (vii) energy-based models [39,40,41,42,43,44].

In the third section, several nonlinear cumulative fatigue damage accumulation models (discussed in Section 2) are used to predict the fatigue failure of adhesively bonded fiber-reinforced composite (FRP) joints under variable fatigue loading. The authors used a double-lap joint with four different step-loading cases (two high-load to low-load transition cases and two low-load to high-load transition cases) and compared the results with experimental data. This study also provides a comprehensive discussion of the limitations and applicability of fatigue damage accumulation models.

2. Fatigue Damage Accumulation Models

Expanding on the work done by Fatemi and Yang [3], the fatigue damage accumulation rules can be categorized into seven groups:

• Linear damage rules (LDRs);
• Nonlinear damage curve and two-stage linearization models;
• Fatigue damage accumulation theories are based on life curve modification methods;
• Fatigue damage accumulation approaches are based on crack growth concepts;
• Fatigue damage accumulation theories are based on continuum damage mechanics;
• Material degradation-based model;
• Fatigue damage accumulation theory based on energy.

However, these categories are not clearly distinguishable from one another [9]. Some of the theories cannot be placed into these categories; these damage rules are placed in the miscellaneous category.

2.1. Linear Damage Rule (LDR) Models

The origins of fatigue damage modeling can be traced back to 1924, when Palmgren introduced the concept of linear summation of fatigue damage. In 1933, French conducted a notable study on the impact of overstress on endurance limits [45]. In 1938, Kommers suggested using changes in endurance limits as an indicator of damage [46]. In 1937, Langer suggested dividing fatigue damage into two phases, crack initiation and propagation [10], proposing a linear rule for each phase. These three early ideas—linear summation, endurance limit changes, and two-stage damage process–formed the basis for phenomenological cumulative fatigue damage models. In 1939, Gassner et al. 49 introduced the concept of linear damage accumulation in adhesive materials [47].

Miner introduced the first rule for fatigue damage accumulation under variable loading in 1945 [1]. Miner’s rule quantifies fatigue damage as a cycle ratio of applied number of cycles vs. total number of cycles to failure at any given load, which can be expressed as

$$D=\sum \left({\displaystyle \frac{n_i}{N_i}}\right)$$

(2)

where D is the total damage, $n_i$ is the number of applied fatigue cycles under the ith constant amplitude loading level, and $N_i$ is the number of fatigue cycles until failure under the ith constant amplitude loading level.

The linear damage rule (LDR) postulates a consistent amount of work performed per cycle and the properties of the work absorbed at the point of failure. Material degradation occurs because of the accumulation of energy, resulting in a linear addition of the cycle ratio, which is expressed as $D=\sum \left({\displaystyle \frac{n_i}{N_i}}\right)=1$. The progression of damage according to the LDR is illustrated in Figure 2.

Machlin introduced an alternative form of LDR in 1949 and proposed a metallurgically based cumulative damage theory [48]. During the 1950s, Coffin et al. expressed LDR in terms of the plastic strain range, linking it to the fatigue life through the Coffin–Manson relationship [49,50]. Later, Topper et al. utilized strain-based LDR to analyze experimental results [51,52,53,54,55]. Miller reviewed LDR applications in strain-controlled fatigue damage analyses in 1970 [56].

Jeans et al. first investigated the spectrum loading effects on adhesively bonded composite materials in 1983 [57]. Subsequently, Yang et al. explored the impact of service-loading spectra on fatigue damage, focusing on the reduction in fatigue life and residual strength in composites [58]. Jones et al. studied fatigue failure under adhesive bonding with variable loading by using LDR in 1989 [59]. In 2001, Erpolat improved the LDR by using a cycle mix fracture for adhesively bonded composite materials [60].

2.2. Non-Linear Damage Curve and Two-Stage Linearization Models

Richart and Newmark introduced the damage curve concept in 1948 to address the shortcomings of LDR by linking the damage to the cycle ratio [61]. They hypothesized that the stress levels influenced the D-r curves, where r denotes the cycle ratio n/N. Building on this foundation, Marco and Starkey proposed the nonlinear damage rule (NLDR) in 1954. This rule posits that the damage progression follows a power relationship with the number of cycles [9], as follows:

$$D=\sum {\left({\displaystyle \frac{n_i}{N_i}}\right)}^{x\left({\sigma }_i\right)}$$

(3)

where D is the total damage, $n_i$ is the number of applied fatigue cycles under the ith constant-amplitude loading level, $N_i$ is the number of fatigue cycles to failure under the ith constant-amplitude loading level, and $x\left({\sigma }_i\right)$ is a variable related to the ith load level. For LDR, the exponent $x\left({\sigma }_i\right)=1$.

A comparison of the linear damage rule, Freudenthal–Heller rule (also known as the two-stage linearization rule), and damage curve approach (DCA) rule for variable fatigue loading is illustrated in Figure 3.

Langer proposed that the accumulation of damage be evaluated using two LDRs for each load level [10]. In 1960, Grover introduced a two-stage linear damage rule based on observations of fracture mechanics [62]. The initial stage represents crack initiation, whereas the subsequent stage represents crack propagation. Although Grover’s approach was qualitative, it lacked a quantitative formula for separating the total life into the initiation and propagation phases. Manson proposed a double linear damage rule (DLDR) in 1966 based on Grover’s work [62,63]. In the following year, Manson et al. modified the DLDR to represent two effective stages of the fatigue process rather than literal crack initiation and propagation [64]. This modification made the DLDR dependent on the material and load, addressing some of the limitations of their original proposal. Experimental comparisons between DLDR and Miner’s rule by Manson and Halford revealed that Miner’s rule led to an average overestimation of 37%, while DLDR resulted in an average overestimation of only 12% [11,65].

Following the work of Marco and Starkey, Manson and Halford built upon the findings of DLDR to propose a damage curve approach (DCA) [9,11]. The damage exponent for multi-level loading sequences in NDLR was empirically formulated by Manson and Halford, which is represented by the equation [11]

$$D={\left[{\left[{\left[{\left({\displaystyle \frac{n_1}{N_1}}\right)}^{{\alpha }_{1,2}}+{\displaystyle \frac{n_2}{N_2}}\right]}^{{\alpha }_{2,3}}+{\displaystyle \frac{n_3}{N_3}}\right]}^{{\alpha }_{3,4}}+\cdots +{\displaystyle \frac{n_{i-1}}{N_{i-1}}}\right]}^{{\alpha }_{i-1,i}}+{\displaystyle \frac{n_i}{N_i}}=1$$

(4)

where

$${\alpha }_{i-1,i}={\left({\displaystyle \frac{N_{i-1}}{i}}\right)}^{0.4}$$

(5)

where D is the total damage, $n_i$ is the number of applied fatigue cycles under the ith constant-amplitude loading level, $N_i$ is the number of fatigue cycles to failure under the ith constant-amplitude loading level, and ${\alpha }_{i-1,i}$ is a variable related to the ith load level.

In the case of multilevel loading sequences, a minor discrepancy was observed between the DLDR and DCA approaches, despite the DLDR being considered a linear damage concept. It is crucial to recognize that, while DLDR does not consider the loading sequence, DCA incorporates its effects. Numerous studies have performed experimental comparisons between the NLDR, DLDR, and DCA methods. These investigations consistently demonstrated that DLDR yields the most precise results [11,34,66,67,68].

Although the NLDR, DLDR, and DCA models consider load-level dependency, they fail to account for the effects of load interactions and cycles with amplitudes below the fatigue threshold [6,7,8]. Corten and Dolan tackled this limitation by introducing a novel theory for fatigue damage accumulation. They hypothesized that fatigue damage results from the formation of microscopic voids, leading to crack initiation and growth [13]. This phenomenon can be mathematically expressed using the following Equation (6):

$$D=mr{n}^a$$

(6)

where n is the number of cycles applied; m is the number of damaged nuclei; r is the damage propagation rate coefficient, which is a function of the stress condition; and a is the material constant related to material failure. In the context of two-level block-loading experiments, Corten and Dolan proposed a stress-dependent ratio denoted as R, which is defined as $R={\sigma }_1/{\sigma }_2$. This concept led to the development of a novel empirical equation to determine the number of cycles until failure occurred, as follows:

$$D=\left({\displaystyle \frac{n_i}{N_i}}\right){\left({\displaystyle \frac{{\sigma }_i}{{\sigma }_{i-1}}}\right)}^d$$

(7)

where D represents the total damage and $n_i$ denotes the number of applied fatigue cycles under the ith constant-amplitude loading level. $N_i$ signifies the number of fatigue cycles leading to failure under the same ith constant-amplitude loading level. The load level at the ith load block is represented by ${\sigma }_i$, and ${\sigma }_{i-1}$ indicates the load level at the preceding load block. The variable D corresponds to the gradient of a linear curve in the $log\left(R^{1/a}\right)-log\left({\sigma }_2/{\sigma }_1\right)$ diagram.

Chen et al. examined the fatigue life of 304 stainless steel under various loading sequences, including axial/torsional, torsional/axial, in-phase/90° out-of-phase, and 90° out-of-phase/in-phase [69,70]. Their research also evaluated NDLR, DLDR, and DCA, which yielded non-conservative outcomes compared with experimental tests. To address the impact of the loading path under variable loading conditions, researchers incorporated a non-proportionality function J into the DCA model exponent $\mathsf{\alpha }$ based on experimental findings. For two-block loading, the exponent $\mathsf{\alpha }$ was defined as

$${\alpha }_{i-1,i}=\left({\displaystyle \frac{1}{1+\beta J}}\right){\left({\displaystyle \frac{N_{i-1}}{N_i}}\right)}^{0.4}$$

(8)

where

$$J={\displaystyle \frac{1.57}{T{\epsilon }_{1,max}}}{\int }_T^0\left|sin\xi \left(t\right)\right|\epsilon \left(t\right)dt$$

(9)

where ${\epsilon }_1\left(t\right)$ and $\xi \left(t\right)$ are the absolute value and the angle of the maximum principal strain at time t, respectively. T and ${\epsilon }_{1,max}$ are the time for a cycle and the maximum value of $\epsilon \left(t\right)$ in a cycle.

Based on Corten and Dolan’s work, Xu et al. acknowledged that DCA failed to account for the loading sequence [13,71]. They proposed modifying the parameter ${\alpha }_{i-1,i}$ to incorporate the load interaction and effective stress related to loading. Inspired by Corten and Dolan’s DCA research and Freudenthal and Heller’s studies on load interaction fatigue models, Gao et al. refined the parameter ${\alpha }_{i-1,i}$ by incorporating the minimum ratio of the applied stress amplitude [13,72,73,74], which can be expressed as

$${\alpha }_{i-1,i}={\left({\displaystyle \frac{N_{i-1}}{N_i}}\right)}^{0.4min\left\{{\displaystyle \frac{{\sigma }_{i-1}}{{\sigma }_i}},{\displaystyle \frac{{\sigma }_i}{{\sigma }_{i-1}}}\right\}}$$

(10)

where ${\alpha }_{i-1,i}$ is the exponent factor in the DCA equation; $N_i$ and $N_{i-1}$ are the numbers of cycles to failure in the current load and the previous load, respectively; and ${\sigma }_i$ and ${\sigma }_{i-1}$ are the current load and the previous load, respectively.

Drawing on earlier research by Gao et al., Yuan et al. proposed that the assessment of fatigue damage should also consider the deterioration of the material strength [74,75]. Consequently, the formula for total fatigue damage D was enhanced to

$$D=\gamma {\left({\displaystyle \frac{n_i}{N_i}}\right)}^{{\left({\displaystyle \frac{N_{i-1}}{N_i}}\right)}^{0.4min\left\{{\displaystyle \frac{{\sigma }_{i-1}}{{\sigma }_i}},{\displaystyle \frac{{\sigma }_i}{{\sigma }_{i-1}}}\right\}}}$$

(11)

where

$$\gamma =exp\left[\alpha \left({\displaystyle \frac{1}{A}}-1\right)\right]$$

(12)

where A is the residual strength of the degradation coefficient that reflects the relationship between the strength degradation and fatigue damage accumulation, and $\alpha$ is a material coefficient obtained from the experimental data.

In 2019, Zhou et al. introduced a novel fatigue damage accumulation model, drawing inspiration from the DCA and Corten and Dolan models [13,76], as follows:

$$F={\left\{{\left[\left({{\displaystyle \frac{n_1}{N_1}}}^{{\left({\displaystyle \frac{N_1}{N_2}}\right)}^{\alpha \left({\displaystyle \frac{{\sigma }_{max,1}}{{\sigma }_f}}\right)}}\right)+{\displaystyle \frac{n_2}{N_2}}\right]}^{{\left({\displaystyle \frac{N_2}{N_3}}\right)}^{\alpha \left({\displaystyle \frac{{\sigma }_{max,2}}{{\sigma }_f}}\right)}}+\cdots +{\displaystyle \frac{n_{i-1}}{N_{i-1}}}\right\}}^{{\left({\displaystyle \frac{N_{i-1}}{N_i}}\right)}^{\alpha \left({\displaystyle \frac{{\sigma }_{max,i-1}}{{\sigma }_f}}\right)}}+{\displaystyle \frac{n_i}{N_i}}$$

(13)

where ${\sigma }_{max,i}$ is the maximum stress at the ith stress level, ${\sigma }_f$ is the fatigue limit, and ${\alpha }_i$ is the damage exponent from the DCA equation.

2.3. Non-Linear Damage Accumulation Models Based on Wöhler (S-N) Curve

The Wöhler curve or S-N curve-based damage models modify damage evolution based on the evolution of the S-N curve. In 2011, Kwofie and Rahbar introduced a fatigue damage model based on the fatigue driving stress (FDS) based on the Basquin equation [14,77]. The authors stated that FDS is the primary factor in fatigue damage and can be utilized to estimate the remaining fatigue life under fluctuating loads. The FDS value calculated from the prior loads was used to forecast the remaining lifespan of the material. The FDS resulting from the applied cyclic stress ${\sigma }_i$ can be represented as follows:

$$S_{D_i}={\sigma }_i{N}^{{\displaystyle \frac{-b{n}_i}{N_i}}}$$

(14)

where $n_i$ is the number of cycles applied at ith load, $N_i$ is the number of cycles to failure at ith load, and b is the fatigue strength exponent. Based on this, Kwofie and Rahbar described the damage due to fatigue as

$$D=\sum {\displaystyle \frac{n_i}{N_i}}{\displaystyle \frac{ln\left(N_i\right)}{ln\left(N_1\right)}}$$

(15)

where $n_i$ is the number of cycles applied at the ith load; $N_i$ is the number of cycles to failure at the ith load, and $N_1$ is the number of cycles to failure in first load. The ratio $ln\left(N_i\right)/ln\left(N_1\right)$ accounts for load sequence and interaction effects.

Based on this Kwofie and Rahbar damage model, Zuo et al. and Zhu et al. developed a novel fatigue damage accumulation rule that incorporates load interaction effects using the FDS method [14,15,77,78]. This newly proposed rule enables the estimation of the remaining fatigue life and can be expressed as

$${\displaystyle \frac{n_i}{N_i}}=\left(1-{\displaystyle \frac{n_1}{N_1}}-{\displaystyle \frac{n_2}{N_2}}-\cdots -{\displaystyle \frac{n_{i-1}}{N_{i-1}}}\right){\left({\displaystyle \frac{ln\left(N_1\right)}{ln\left(N_i\right)}}\right)}^{{\displaystyle \frac{{\sigma }_{i-1}}{{\sigma }_i}}}$$

(16)

where $n_i$ is the number of cycles applied at the ith load, $N_i$ is the number of cycles to failure at the ith load, and ${\sigma }_i$ is the load in the ith cycle.

Similar to the Kwofie and Rahbar damage model, Si-Jian et al. introduced a damage model grounded in the Basquin equation, which only necessitates the Wöhler curve or S-N curve [17]. The damage accumulation parameter D is characterized as follows:

$$D=1-{\left[{\left[{\left({\displaystyle \frac{n_1}{N_1}}\right)}^{{\displaystyle \frac{{\sigma }_2}{{\sigma }_1}}}+{\displaystyle \frac{n_2}{N_2}}\right]}^{{\displaystyle \frac{{\sigma }_3}{{\sigma }_2}}}+\cdots +{\displaystyle \frac{n_{i-1}}{N_{i-1}}}\right]}^{{\displaystyle \frac{{\sigma }_i}{{\sigma }_{i-1}}}}+{\displaystyle \frac{n_i}{N_i}}$$

(17)

where $n_i$ is the number of cycles applied at the ith load, $N_i$ is the number of cycles to failure at the ith load, and ${\sigma }_i$ is the load in the ith cycle.

A damage model based on the Wöhler (S-N) curve was introduced by Aeran et al. without the inclusion of any additional parameters [16,79]. The model depends exclusively on the number of cycles to failure ($N_i$). The absolute value of the proposed damage index $D_i$ represents the fatigue damage D, expressed as $D=\left|D_i\right|$,

$$D_i=1-{\left[1-{\displaystyle \frac{n_i}{N_i}}\right]}^{{\delta }_i}$$

(18)

where

$${\delta }_i={\displaystyle \frac{-1.25}{ln\left(N_i\right)}}$$

(19)

Additionally, Aeran et al. introduced an alternative damage model incorporating a novel factor µ, which considers the effects of load sequence and load interaction [16,79]. This interaction factor is characterized as follows:

$${\mu }_{i+1}={\left({\displaystyle \frac{{\sigma }_i}{{\sigma }_{i+1}}}\right)}^2$$

(20)

where ${\sigma }_i$ and ${\sigma }_{i+1}$ represent consecutive stress levels and ${\mu }_{i+1}$ denotes the interaction of the loads between these levels. The authors used parameter ${\mu }_{i+1}$ to redefine the damage parameter D, which can be expressed through the following set of equations:

$$D_i=1-{\left[1-{\displaystyle \frac{n_{\left(i+1\right),eff}}{N_{i+1}}}\right]}^{{\displaystyle \frac{{\delta }_{i+1}}{{\mu }_{i+1}}}}$$

(21)

where

$$n_{\left(i+1\right),eff}=\left[1-{\left(1-D_i\right)}^{{\displaystyle \frac{{\mu }_{i+1}}{{\delta }_{i+1}}}}\right].N_{i+1}$$

(22)

This concept can be expanded by defining $n_{i+1}$ as the number of cycles for the stress state ${\sigma }_{i+1}$; the total number of cycles of the loading step i+1 is defined as

$$n_{\left(i+1\right),total}=n_{\left(i+1\right),eff}+n_{\left(i+1\right)}$$

(23)

$$D=\left|D_{i+1}\right|=\left|1-{\left[{\displaystyle \frac{n_{\left(i+1\right),total}}{N_{i+1}}}\right]}^{{\delta }_{i+1}}\right|$$

(24)

A novel approach to predicting fatigue life was introduced by Theil, which utilizes linearized damage growth curves on a double linear Wöhler (S-N) curve, with the S-N curve indicating failure [67]. This technique aims to estimate overloads that are approximately 0.2% of the yield strength and slightly exceed it by approximately 10%. The damage-calculation process involves iterative computations.

2.4. Life Curve or Iso-Damage Line Modification Models

Life curve modification models were developed to adjust the Wöhler (S-N) curve by considering various factors, including the load sequence and interaction effects. Modification of life curves or iso-damage lines can be divided into two primary categories.

• Models based on the Corten–Dolan and Freudenthal–Heller models;
• Models based on the iso-damage line.

2.4.1. Models Based on the Corten–Dolan and Freudenthal–Heller Models

Both the Corten–Dolan and Freudenthal–Heller models exhibit a common feature: the S-N curve rotates in a clockwise direction around a specific reference point [72]. Based on the Corten–Dolan model, Kaechele discovered that a larger number of damaged nuclei caused by high-stress amplitudes resulted in increased growth. This suggests that the nuclei were damaged at the lower stress levels [13,80]. Freudenthal and Heller set the low load cycle to high load cycle transition as a benchmark, which is between 10³ and 10⁴ cycles [72,73].

Owing to the intricate interaction effects described by Freudenthal and Heller and Corten and Dolan, the Freudenthal–Heller method is more challenging to align with experimental data than the Corten–Dolan model. Consequently, researchers tend to prefer the Corten–Dolan model. Spitzer and Corten determined the gradient of a modified N curve by averaging the slopes of several S-N curves derived from two-level block loading and tests [81]. Numerous studies have showcased the wide range of applications and high level of accuracy of the Corten–Dolan model [37,82,83,84,85].

In contrast to the exponent d in the Corten–Dolan equation as a material constant, Zhu et al. proposed expressing it as a function of the applied stresses [7,86,87]. They introduced a revised formulation for the parameter d, which is presented as follows:

$$d\left({\sigma }_i\right)=\mu {\displaystyle \frac{{\sigma }_1^{\lambda }{\delta }_f^{1-\lambda }}{{\sigma }_i}}$$

(25)

where $\mu$ represents a material constant and $\lambda$ is a coefficient that considers the effects of the load sequence, ranging from 0 to 1 (0 < $\lambda$ < 1). For constant-amplitude loading, $\lambda$ equals 0 because of the absence of load interaction. To simplify the calculations, $\lambda$ was approximated as n1/N1. The variable ${\delta }_f$ denotes the initial strength of the specimen, which was determined experimentally.

Expanding on a research by Zhu et al., Gao et al. enhanced the definition of the damage exponent d by incorporating a material parameter $\gamma$. This parameter was obtained from the experimental results and selected failure criterion, such as n/N = 1 [7,86,87,88].

$$d=exp\left[{\left({\displaystyle \frac{n_i}{N_i}}\right)}^{{\sigma }_i/{\sigma }_{max}}\right]+\gamma$$

(26)

where $n_i$ is the number of cycles applied at the ith load, $N_i$ is the number of cycles to failure at the ith load, ${\sigma }_{max}$ is the maximum load applied, and ${\sigma }_i$ is the load in the ith cycle.

The model of Gao et al., when compared with Zhu’s model, reduced errors by 50% [88]. Xue et al. further introduced the stress ratios of consecutive blocks [89], as follows:

$$d={\displaystyle \frac{{\sigma }_i}{{\sigma }_{i-1}}}\left[exp\left[{\left({\displaystyle \frac{n_i}{N_i}}\right)}^{{\sigma }_i/{\sigma }_{max}}\right]+\gamma \right]$$

(27)

where $n_i$ is the number of cycles applied at the ith load, $N_i$ is the number of cycles to failure at the ith load, ${\sigma }_{max}$ the maximum load applied, and ${\sigma }_i$ is the load in the ith cycle.

Liu et al. introduced nonlinearity to the effects in the stress ratios of consecutive blocks by introducing the parameter k, which is expressed as

$$d={\left({\displaystyle \frac{{\sigma }_i}{{\sigma }_{i-1}}}\right)}^k\left[exp\left[{\left({\displaystyle \frac{n_i}{N_i}}\right)}^{{\sigma }_i/{\sigma }_{max}}\right]+\gamma \right]$$

(28)

where $n_i$ is the number of cycles applied at the ith load, $N_i$ is the number of cycles to failure at the ith load, ${\sigma }_{max}$ is the maximum load applied, ${\sigma }_i$ is the load in the ith cycle, and k is a user-defined constant.

A novel approach was developed by Leipholz et al. by replacing the conventional S-N curve with an S-$\widehat{N}$ model. This new model accounts for life-reducing load interaction effects that occur between cycles of varying amplitudes [90,91,92,93]. The researchers formulated their model as follows:

$$N={\left[\sum _{i=1}^k{\displaystyle \frac{{\beta }_i}{{\widehat{N}}_i}}\right]}^{-1}$$

(29)

where

$${\beta }_i={\displaystyle \frac{n_i}{N_i}}$$

(30)

where N is the total accumulated fatigue life and ${\widehat{N}}_i$ is the modified life at the ith load. ${\widehat{N}}_i$ is obtained by comparing the virgin S-N data and the experimental data.

2.4.2. Life Modification Models Based on Iso-Damage Lines

The S-N curve is commonly employed to assess the fatigue life of a structure and determine its remaining lifespan after a certain number of fatigue cycles. The iso-damage line concept, derived from the S-N curve, posits that the curve represents 100% fatigue damage, whereas stress-cycle combinations (${\sigma }_i$,$n_i$) with equal damage values form smooth curves when graphed. In 1976, Subramanyan introduced a set of straight iso-damage lines on an S-log(N) diagram, converging near the knee point of the S-N curve, which is demonstrated in Figure 4 [18].

Damage was quantified as the ratio between the iso-damage line slope inclination and the limiting slope of the S−N curve. Expressing the damage as a logarithmic function of load cycles implies that the cycle count required for a specific damage amount increases with decreasing stress amplitude and decreases with increasing stress amplitude. This model accounts for the extended lifespan observed in high-to-low block-loading sequences and the shortened lifespan observed in low-to-high block-loading sequences.

$$D_i={\displaystyle \frac{tan\left({\theta }_i\right)}{tan\left({\theta }_f\right)}}={\displaystyle \frac{log\left(N_f\right)-log\left(N_i\right)}{log\left(N_f\right)-log\left(n_i\right)}}$$

(31)

where $N_f$ represents the cycle count to failure at the knee point of the S-N curve, $N_i$ is the cycle count to failure at the ith load, $n_i$ denotes the cycle count at the ith load, and ${\sigma }_i$ is the load in the ith cycle. The iso-damage line concept can be utilized to calculate the equivalent cycle count $n_{i,i+1}$ in the i+1-th load ${\sigma }_{i+1}$ cycle. The equivalent cycle count $n_{i,i+1}$ can be determined as follows:

$$n_{i,i+1}=N_{i+1}{\left({\displaystyle \frac{n_i}{N_i}}\right)}^{{\alpha }_i}$$

(32)

where

$${\alpha }_i={\displaystyle \frac{log\left(N_f\right)-log\left(N_{i+1}\right)}{log\left(N_f\right)-log\left(N_i\right)}}={\displaystyle \frac{{\sigma }_{i+1}-{\sigma }_f}{{\sigma }_i-{\sigma }_f}}$$

(33)

Form the above equations, the residual fatigue life $N_{\left(i+1\right)R}$ at the (i+1)-th load cycle can be defined as

$$N_{(i+1)R}=N_i-n_{i+1}$$

(34)

Based on iso-damage lines, Hashin and Rotem proposed a novel method for predicting damage. According to the authors, the iso-damage line converges at a point on the S-axis instead of the N-axis, which is in contrast with the Subramanyan model [94]. Figure 5 demonstrates Hashin and Rotem life modification models.

When both models are compared, the Subramanyan model becomes more favorable since the Hashin–Rotem model becomes invalid at low-stress levels [18,94,95]. However, the Subramanyan model does not consider cycles within the fatigue limit [18]. Multiple studies have been carried out in the Subramanyan and Hashin–Rotem models, and it was found that although both models outperformed Miner’s rule model, the results were still non-conservative in nature [96,97,98,99,100].

Research conducted by Hu et al. examined the use of 2D and 3D Hashin models to forecast the damage behavior and failure mechanisms in carbon fiber-reinforced aluminum laminates (CARALLs) [101]. Researchers found that both 2D and 3D Hashin models closely predicted the tensile and bending properties. In contrast, the 3D Hashin model considers out-of-plane stress components and employs an additional element deletion technique to calculate the damaged area. This approach leads to enhanced precision and prevents substantial mesh distortion.

Rege and Pavlou enhanced the non-conservative predictions of the Subramanyan model by introducing nonlinearity through the parameter $q\left({\sigma }_i\right)$, which varies based on stress amplitude [18,102]. This modification aims to address the limitations of the original Subramanyan model.

$$D_i={\left[{\displaystyle \frac{tan\left({\theta }_i\right)}{tan\left({\theta }_f\right)}}\right]}^{q\left({\sigma }_i\right)}={\left[{\displaystyle \frac{log\left(N_f\right)-log\left(N_i\right)}{log\left(N_f\right)-log\left(n_{i-1,i}+n_i\right)}}\right]}^{q\left({\sigma }_i\right)}$$

(35)

where

$$log\left(n_{i-1,i}\right)=log\left(N_f\right)-{\displaystyle \frac{log\left(N_f\right)-log\left(N_i\right)}{D_{i-1}^{1/q\left({\sigma }_i\right)}}}$$

(36)

where $N_i$ represents the number of cycles until failure at the ith load level, $N_f$ denotes the number of cycles at the fatigue limit, $n_i$ indicates the number of cycles at the ith load level, and $q\left({\sigma }_i\right)$ is a function of the stress amplitude, which can be expressed as follows:

$$q\left({\sigma }_i\right)={\left(a{\sigma }_i\right)}^b={\left(2{\displaystyle \frac{{\sigma }_i}{{\sigma }_s}}\right)}^b$$

(37)

where ${\sigma }_s$ represents the stress amplitude value at the point where the S-N curve intersects the stress axis. The variable b is specific to the material, with steel having a b value of −0.75. There was no significant improvement in Rege and Pavlou’s fatigue model for steel compared with Subramanyan’s model [18,102].

To address these inconsistencies, a new formula for $q\left({\sigma }_i\right)$ was introduced by Zhu et al. [103], as follows:

$$q\left({\sigma }_i\right)=llog{n}_i+slog{\epsilon }_{a,i}$$

(38)

where ${\epsilon }_{a,i}$ represents the amplitude of the strain for the ith load, whereas l and s denote the coefficients for the load sequence and load weighting, respectively.

A novel fatigue damage model termed the S-N fatigue damage envelope was introduced by Pavlou and incorporated alterations to the iso-damage line [104]. The authors suggest that the area bounded by the S-axis, N-axis, and S-N curves represents the macroscopic effects of the damage mechanism for any given S-N curve pair. In this model, dimensionless S-N axes are established by defining ${\sigma }_i^{*}$ and $n_i^{*}$, as follows:

$${\sigma }_i^{*}={\displaystyle \frac{{\sigma }_i-{\sigma }_f}{{\sigma }_u-{\sigma }_f}}$$

(39)

$$n_i^{*}={\displaystyle \frac{n_i}{N_i}}$$

(40)

where ${\sigma }_i$ represents the amplitude of the stress for the ith block of loading, ${\sigma }_f$ denotes the material’s fatigue threshold, and ${\sigma }_u$ indicates the maximum tensile stress that the material can withstand before failure.

A significant limitation of life curve modification models is their difficulty in calculating the damage during frequent load amplitude fluctuations. Furthermore, these models employ bilinear S-N curves, which are not suitable for all scenarios. Several alternative approaches have been developed to overcome these challenges.

In 2013, El and Khaled introduced the virtual target life curve (VTLC) model, which proposes that materials possess a virtual expected life exceeding their actual life under a constant load amplitude [105]. The model suggests that as the number of cycles increases, the expected life of the material decreases, and the load amplitude influences the damage life. In addition, the VTLC model accounts for the overloading effects on the materials. Dan et al. compared the fatigue life of asphalt under two-stage loading with theoretical fatigue models, including VTLC [106]. Liu et al. developed a novel fatigue damage model by integrating VTLC and Manson models to predict material fatigue life [63,105,107]. However, this new model failed to demonstrate significant improvements in the fatigue life estimation for 300 CVM steel compared with the previously discussed models.

To address these problems, Batsoulas introduced a hyperbolic iso-damage curve damage model [108]. The hyperbolic curves can be defined as

$$log{\displaystyle \frac{\sigma }{{\sigma }_f^{\prime }}}=c{\left(log{\displaystyle \frac{N}{N_e}}\right)}^{-1}$$

(41)

where ${\sigma }_f^{\prime }$ represents the fatigue strength coefficient, $N_e$ denotes the cycle count necessary for damage initiation, and c is a constant value.

For each individual curve, the damage accumulation can be defined by

$$D_i={\displaystyle \frac{log\left(\frac{{\sigma }_i}{{\sigma }_f^{\prime }}\right)log\left(\frac{n_i}{N_e}\right)}{log\left(\frac{{\sigma }_i}{{\sigma }_f^{\prime }}\right)log\left(\frac{N_i}{N_e}\right)}}$$

(42)

Based on the above equation, the rule for accumulating damage under multiple levels of loading can be formulated as

$$D_i={\left({\left({\left({\displaystyle \frac{n_1}{N_1}}\right)}^{{\phi }_{2,3}}+{\displaystyle \frac{n_2}{N_2}}\right)}^{{\phi }_{2,3}}+\dots +{\displaystyle \frac{n_i}{N_{i-1}}}\right)}^{{\phi }_{i-1,i}}+{\displaystyle \frac{n_i}{N_i}}$$

(43)

$${\phi }_{i-1,i}={\displaystyle \frac{log\left({\sigma }_{i-1}/{\sigma }_f^{\prime }\right)}{log\left({\sigma }_i/{\sigma }_f^{\prime }\right)}}$$

(44)

Batsoulas and Giannopoulos used this model to compute the fatigue life of composite material under step/variable loading [109].

Xie et al. modified the damage accumulation exponent ${\phi }_{i-1,i}$ [110]. They integrated the Batsoulas model with the Ye and Wang model, drawing from the material degradation model [35,108,109]. The resulting modified exponent ${\phi }_{i-1,i}$ can be described as follows:

$${\phi }_{i-1,i}=\left({\displaystyle \frac{log\left({\sigma }_{i-1}/{\sigma }_f^{\prime }\right)}{log\left({\sigma }_i/{\sigma }_f^{\prime }\right)}}\right)\left({\displaystyle \frac{log{N}_f-log{N}_i}{log{N}_f-log{N}_{i-1}}}\right)$$

(45)

2.5. Crack-Growth-Based Fatigue Damage Accumulation Models

In 1921, A. A. Griffith first described the concept of linear elastic fracture mechanics (LEFM) leading to material failure [111]. During the 1950s and the 1960s, several fatigue crack growth theories based on LEFM were introduced and gained widespread acceptance owing to the direct relationship between cracks and damage [112,113,114,115,116,117]. According to Makkonen, the total fatigue life of a material can be divided into three stages [118].

• Crack initiation;
• Stable crack growth;
• Unstable crack growth.

Crack initiation typically accounts for 40–90% of the total fatigue life of a material [118].

In 1963, Paris and Erdogan developed a formulation for stable fatigue crack growth [115], which is expressed as

$${\displaystyle \frac{da}{dN}}=C{\left(\Delta K\right)}^m$$

(46)

where a represents the depth of the crack, K denotes the material’s stress intensity, and C and m are constants specific to the material properties.

Wheeler introduced a modification to the Paris law’s constant amplitude crack growth rate in 1972 by adding a retardation factor $R_i$ [115]. The authors proposed that crack propagation occurs owing to the interaction of plastic zones near the crack tip, which develops as a result of residual compressive stresses caused by overloading. This modified law expresses the crack growth rate as follows:

$${\displaystyle \frac{da}{dN}}=R_i\left[C{\left(\Delta K\right)}^m\right]$$

(47)

$$R_i={\left({\displaystyle \frac{r_{pi}}{r_{max}}}\right)}^p$$

(48)

where $r_{pi}$ represents the size of the plastic zone corresponding to the ith loading cycle, and $r_{max}$ denotes the distance between the current crack tip and the largest previous elastic-plastic zone generated by the overload. The variable p is an empirical parameter that shapes the equation based on the material characteristics and spectrum of the applied loading.

The Willenborg–Wood model, an extension of the Wheeler model, incorporates a retardation mechanism that employs the effective stress intensity factor at the crack tip ($\Delta K_{eff}$) [20,21]. This factor diminishes the applied crack-tip stress intensity factor ($\Delta K$) owing to the increased compressive stress at the crack tip resulting from the overloads. The reduction in the applied $\Delta K$ is determined by comparing the current plastic zone size at the ith load cycle with the maximum plastic zone size generated by the overload. Unlike its predecessor, the Willenborg model eliminates the need for an empirical shaping parameter.

Elber proposed that fatigue-induced cracks can close because of the formation of a compressive residual stress zone at the crack tip [22,23]. This phenomenon reduces the driving force for crack propagation, resulting in decreased stress intensity at the crack tip, which is influenced by the effective stress range.

$$\Delta S_{eff}=S_{max}-S_{op}$$

(49)

where $\Delta S_{eff}$ represents the effective stress, $S_{max}$ denotes the maximum stress, and $S_{op}$ signifies the crack-tip opening stress.

Newman et al. formulated a model for fatigue crack growth under variable loading conditions, focusing on short-crack closures [119,120,121,122,123,124,125]. Their approach, reminiscent of Wheeler’s model, involved redefining the effective stress intensity to characterize crack propagation [126,127]. Newman proposed that stress intensity could be expressed as

$$\Delta K_{eff}=\left(S_{max}-S_{op}\right)\times \sqrt{\left(\pi c\right)}\times F\left(c/w\right)$$

(50)

where $\Delta K_{eff}$ represents the effective stress intensity factor and $S_{max}$ denotes the maximum stress. The term $S_{op}$ indicates the stress at which the crack tip opens, and F is a factor that corrects for the boundary conditions.

Using the finite element method (FEM), Newman et al. developed an iterative approach to estimate the crack opening stress through cycle-by-cycle closure analysis [126,127]. Ritchie and Suresh accounted for environmental factors such as corrosion and fracture surface roughness during crack fracture growth [128,129]. Furthermore, several variable loading fatigue models based on crack-closure fatigue have been developed [130,131,132,133]. However, these models have limitations, including the exclusion of plasticity or nonlinear behavior, difficulty in application to complex variable amplitude loading, and a limited scope for small crack growth theories.

To address these concerns, Miller and Zachariah introduced a double exponential crack growth fatigue law that established an exponential relationship between the crack length and elapsed life for each phase [134,135]. Later, Miller and Ibrahim enhanced this model by identifying stages I (crack initiation) and II (stable crack growth) within the context of the linear elastic fracture mechanics (LEFM) theory [135,136].

$${\displaystyle \frac{da}{dN}}=\varphi {\left(\Delta {\gamma }_p\right)}^{\alpha }a$$

(51)

where $\varphi$ and $\alpha$ are constants specific to the material, $\Delta {\gamma }_p$ denotes the range of the plastic shear, and a indicates the current length of the crack. A power function was utilized to establish the relationship between the number of cycles $\left(N_i\right)$ and instantaneous crack length $\left(a_i\right)$ with their corresponding values in the plastic strain range $\left(\Delta {\gamma }_p\right)$. The equation for damage is expressed as follows:

$$D={\displaystyle \frac{a}{a_f}}={\left({\displaystyle \frac{a_1}{a_f}}\right)}^{\left(1-r\right)\left(1-r_I\right)}$$

(52)

where $a_f$ is the final crack length, r is the cycle ratio, and $r_I$ is the cycle ratio in stage I.

However, the double exponential crack growth model, which was developed within the framework of LEFM, does not consider plasticity in fatigue crack growth.

To address this issue, Miller et al. introduced a novel fatigue damage accumulation model focused on crack growth [137,138,139,140,141,142,143]. Their research highlighted that, in metal fatigue, crack initiation occurs at the onset, and the entire fatigue lifespan consists of crack propagation starting from an initial flaw size $a_0$.

Miller et al. combined linear elastic fracture mechanics (LEFM) and elastic-plastic fracture mechanics (EPFM) theories to develop crack growth fatigue failure theory [137,138,139,140,141,142,143]. The authors defined two different elastic-plastic fracture mechanics phases: (i) microstructurally short cracks (MSCs) and (ii) physically short cracks (PSCs). The crack growth model for MSCs is defined as follows:

$${\displaystyle \frac{da}{dN}}=A{(\Delta {\gamma }_p)}^{\alpha }(d-a)$$

(53)

where, $a_0<a<a_t$.

Similarly, the crack growth model for MSCs is defined as

$${\displaystyle \frac{da}{dN}}=B{\left(\Delta {\gamma }_p\right)}^{\beta }a-C$$

(54)

where, $a_t<a<a_f$,

where A, B, $\alpha$, and $\beta$ are determined by fitting experimental data to the model. $\Delta {\gamma }_p$ represents the range of shear strain, while $a_t$ denotes the crack length at which the transition from microstructurally small cracks (MSCs) to physically small cracks (PSCs) occurs. The variable $a_f$ is defined as the crack length at failure, and C indicates the rate of crack growth under threshold conditions.

Dugdale and Barenblatt introduced a traction–separation law to model the crack initiation and propagation along ductile material interfaces, drawing from the short-crack regime and Paris’s law [144,145]. Turon derived the overall damage accumulation rate d by examining the relationship between the total damage parameter $d=d_s+d_f$ and the area encompassed by the cyclic loading curve in the traction–separation law [146]. Harper established the fatigue damage accumulation rate $d_f$ by suggesting the concept of fatigue crack length across an element, and Landry utilized the same idea to derive the fatigue damage accumulation rate $d_f$ [147]. The research conducted in these investigations offers a significant understanding of how FCGR and the bilinear cohesive law are interconnected. Tabiei and Zhang presented a new bilinear law that connects FCGR to the damage parameter, and after obtaining the damage parameter, it is used to reduce the critical energy release rate in the cohesive zone model [148,149]. The damage accumulation under bilinear traction separation law regime can be defined as follows:

For mode I:

$${\displaystyle \frac{\partial d_f}{\partial N}}={\displaystyle \frac{b}{A_{cz,I}}}{\displaystyle \frac{\partial a}{\partial N}}={\displaystyle \frac{32T^2}{9\pi E{G}_{IC}}}C{\left(\Delta G\right)}^m$$

(55)

For mode II:

$${\displaystyle \frac{\partial d_f}{\partial N}}={\displaystyle \frac{b}{A_{cz,II}}}{\displaystyle \frac{\partial a}{\partial N}}={\displaystyle \frac{S^2}{E{G}_{IIC}}}C{\left(\Delta G\right)}^m$$

(56)

where b is the mix ratio of mode I and mode II loads; $A_{cz,I}$ and $A_{cz,II}$ are the cohesive zone size ($A_{cz}$) for mode I and mode II; Tand S are tractions in the normal (mode I) and shear (mode II) direction; $E_n$ and $E_t$ are the elastic modulus in normal (mode I) and shear (mode II) directions; $\Delta G$ is the energy release rate; and C and m are Paris law parameters, which are experimentally obtained for mode I and mode II. Figure 6 demonstrates the damage accumulation in Bilinear Traction Separation Law.

Ma and Laird expanded upon the short-crack regime theory by introducing the concept of crack population, P, which is similar to MSC, and is linearly related to the strain damage amplitude and used life [25,150]. This concept can serve as a damage indicator, as the authors defined cumulative damage accumulation as

$$D=\sum \left({\displaystyle \frac{P_i}{P_{crit}}}\right)=K\sum n_i\left[{\left(\Delta {\gamma }_p/2\right)}_i{\alpha }_i-{\left(\Delta {\gamma }_p/2\right)}_{limit}\right]$$

(57)

where ${\left(\Delta {\gamma }_p/2\right)}_{limit}$ represents the strain at which fatigue limit occurs. $K=C/P_{crit}$, where C is a constant found in the strain-life equation, and $P_{crit}$ denotes the critical crack population when failure happens. Additionally, ${\alpha }_i$ signifies the factor related to loading history at the ith level of load, and is defined as follows:

$${\alpha }_i={\left(\Delta {\gamma }_p/2\right)}_i/{\left(\Delta {\gamma }_p/2\right)}_{maxinpreloadinghistory}$$

(58)

This model considers the loading sequence, that is, the high-to-low load transition and low-to-high load transition.

Valek and Polak proposed a new two-stage damage model based on experimental observations and analysis [151]. The crack initiation regime is described as follows:

$${\displaystyle \frac{da}{dN}}={\nu }_i$$

(59)

where $a_0<a<a_c$.

Meanwhile, Stable crack growth regime was defined by

$${\displaystyle \frac{da}{dN}}={\nu }_i+k\left(a-a_0\right)$$

(60)

where $a_c<a<a_f$,

where ${\nu }_i$ represents the rate at which cracks grow, independent of the applied cycle. The constant k is utilized, while $a_0$, $a_c$, and $a_f$ denote the initial, critical, and final crack lengths, respectively.

Based on the above equations, the damage accumulation for stages I and II were defined through

$$D=2D_{c}r$$

(61)

For initiation: $D\le r\le 1/2$

$$D=D_c+{\displaystyle \frac{D_c}{m}}\left[e^{m\left(2r-1\right)}-1\right]$$

(62)

For propagation: $1/2\le r\le 1$

where r is the n/N, $D_c$ is the critical damage ($a_c/a_f$), and m is a constant that can be defined as follows:

$$m=k\left({\displaystyle \frac{N_f}{2}}\right)$$

(63)

In 2005, Noroozi et al. introduced the UniGrow fatigue crack growth rate model based on the EPFM crack growth theory for cumulative fatigue damage accumulation. This model builds on the work of Walker and Kujawski [27,46,152,153,154,155]. The UniGrow model presents a novel method for calculating the stress intensity factor, resulting in a new definition of crack propagation. The research concluded that crack propagation can be characterized as

$${\displaystyle \frac{da}{dN}}={\rho }^{*}/N_f$$

(64)

where

$${\rho }^{*}={\displaystyle \frac{{\left({\psi }_{y,1}\right)}^2}{2\pi }}{\left({\displaystyle \frac{\Delta K_{th}}{\Delta {\tilde{\sigma }}_{th}^a}}\right)}^2$$

(65)

The UniGrow fatigue crack growth rate model was developed by Mikheevskiy et al., who analyzed the elastic-plastic stress/strain behavior near the crack tip [26,156]. This model underwent modifications to accommodate various variable loading spectra and later incorporated the environmental effects on cumulative fatigue damage under variable loading. Abduallah et al. enhanced the UniGrow model developed by Noroozi et al. by integrating additional crack growth rate models [27,46,152,153,154,155,157,158,159]. The impact of plasticity-induced crack closure using the UniGrow model was investigated by Pedrosa et al. [160]. Bang et al. employed the UniGrow model to explore fatigue damage accumulation under variable loading for different aluminum alloys in both short and long fatigue crack regimes [161,162,163].

Other models have been created that employ statistical or probabilistic techniques to determine the cumulative fatigue damage under crack growth conditions. One such model was developed by Manjunatha, who proposed using the root mean square of the stress intensity factor ($\Delta K_{rms}$) in the fatigue crack growth rate regime to account for cumulative fatigue damage under spectrum loading [164]. Several authors have developed probabilistic models for cumulative fatigue damage under a crack growth regime. Nonetheless, these models have several limitations, including the need for experimental data, sensitivity to assumptions, limited applicability across various problems, and unreliable estimation parameters [165,166,167,168,169,170,171,172,173,174].

2.6. Continuum Damage Mechanics-Based Models

Continuum damage mechanics (CDM) is a branch of mechanics that investigates the initiation and evolution of material damage. In CDM, damage is considered a measurable and quantifiable physical phenomenon. This theory emerged from the studies of Kachanov and Rabotnov on creep damage issues [175,176,177]. Chaboche pioneered the application of CDM to fatigue life prediction by introducing a nonlinear continuum damage (NLCD) model that describes progressive deterioration prior to macroscopic crack formation [28,29,178,179]. This model extends the nonlinear damage mechanics models of Marco and Starkey and the DCA model of Manson and Halford, all of which are underpinned by the CDM principles [9,11,65,180]. Subsequently, Chaboche and Lense examined the CDM model for uniaxial loading [181], and damage was expressed as

$$D=1-{\left[1-{\left({\displaystyle \frac{n}{N}}\right)}^{{\displaystyle \frac{1}{1-\alpha }}}\right]}^{{\displaystyle \frac{1}{1+\beta }}}$$

(66)

where

$$\alpha =1-a〈{\displaystyle \frac{{\sigma }_a-{\sigma }_{l_0}\left(1-b\overline{\sigma }\right)}{{\sigma }_u-{\sigma }_{max}}}〉$$

(67)

and

$$N_F={\displaystyle \frac{1}{\left(1+\beta \right)\left(1-\alpha \right)}}{\left[{\displaystyle \frac{{\sigma }_{max}-\overline{\sigma }}{M_0\left(1-b\overline{\sigma }\right)}}\right]}^{-\beta }$$

(68)

where $\alpha$, $\beta$, and $M_0$ represent material constants. The variable ${\sigma }_{l_0}$ denotes the fatigue limit under fully reversed conditions, whereas ${\sigma }_u$ indicates the ultimate tensile strength. The applied stress is represented by ${\sigma }_a$, where ${\sigma }_{max}$ signifies the maximum stress experienced.

Chaboche’s NLCD model presents multiple benefits compared with alternative theories regarding cumulative fatigue damage [28,29,178,179]. It considers damage growth beneath the initial fatigue threshold in damaged materials, incorporates interaction effects by including a variable for strain hardening, directly connects damage to mean stress, and allows for the easy determination of material parameters from standard S-N curves. Sun et al. [182] demonstrated these advantages.

Numerous fatigue damage accumulation models have been developed based on Chaboche’s work that encompass various variables, damage rate equations, and boundary conditions [183,184,185,186,187,188,189]. Bhattacharaya and Ellingwood created a continuum damage mechanics (CDM)-based model for predicting fatigue life based on fundamental thermodynamic principles and a stochastic ductile damage growth model [31,190]. This approach differs from other CDM-based methods by deriving damage accumulation equations from the first principles of mechanics and thermodynamics rather than starting with a dissipation potential function or a kinetic equation of damage growth. Damage accumulation is defined as follows:

$$dD\left({\epsilon }_p\right)=A\left({\epsilon }_p\right)\left(1-D\left({\epsilon }_p\right)\right)d{\epsilon }_p+B\left({\epsilon }_p\right)dW\left({\epsilon }_p\right)$$

(69)

where A and B are coefficients that depend on ${\epsilon }_p$,${\sigma }_f$, M, K, and ${\epsilon }_0$. ${\epsilon }_p$ is the plastic strain, ${\sigma }_f$ is the failure stress, M and K are the hardening coefficients.

A thermo-elastoplastic damage model was created by Oller et al., who linked the remaining strength of materials to the evolution of damage thresholds [32,191,192]. To represent the nonlinear damage behavior observed during fatigue, they introduced a novel fatigue state variable, called the reduction function $f_{red}\left(N,{\sigma }_{max},R,\theta \right)$. This model differs from the approach adopted by Chaboche and Lense, as it does not explicitly define the number of cycles within the damage model [181].

Dattoma et al. introduced an innovative nonlinear fatigue damage model for uniaxial structures subjected to multilevel sequence loading, expanding on Chaboche’s initial framework [33]. The researchers proposed a new approach to damage accumulation for multi-level sequence loading, which is described as

$$D_i=1-{\left[1-{\left({\displaystyle \frac{n_i+N_i}{N_{f_i}}}\right)}^{{\displaystyle \frac{1}{1-{\alpha }_i}}}\right]}^{{\displaystyle \frac{1}{1+\beta }}}$$

(70)

where $N_{f_i}$ represents the number of cycles until failure at the ith load level, and $N_i$ denotes the equivalent number of cycles applied at the same load level. The latter produces an identical amount of damage to $n_{i-1}$ at the previous load level (i−1). Furthermore, the authors redefined $\alpha$ as follows:

$$\alpha =1-{\displaystyle \frac{1}{H}}{〈{\displaystyle \frac{{\sigma }_a-{\sigma }_f}{{\sigma }_u-{\sigma }_a}}〉}^a$$

(71)

where a and H are experimentally obtained parameters.

According to Dattoma et al., the quantity $N_{f_i}$, representing the number of cycles until failure for the ith load, was established as follows [193]:

$$N_{f_i}={\displaystyle \frac{1}{1-{\alpha }_i}}{\displaystyle \frac{1}{1+\beta }}{\left[{\displaystyle \frac{{\sigma }_a}{M_0}}\right]}^{-\beta }$$

(72)

This approach provided an accurate estimate of the linear correlation between log S and Log N31. Nevertheless, the primary limitation of Dattoma’s model is its requirement for S-N curve data specific to each geometry and various loading conditions [193]. Zhang and colleagues developed an enhanced version of the Chaboche and Lense model by integrating a strengthening function, $f_s$, for cycles with low amplitude [194]. This function accounts for the inconsistencies found in low-amplitude cycles where the Chaboche model failed prematurely.

Further modification to the Chaboche and Lesne model was introduced by Zhang et al., who incorporated a strengthening function ($f_s$) to account for low-amplitude cycles [195]. This function was designed to address the shortcomings identified in the Chaboche model for low-amplitude cycles, where it would prematurely fail [196,197,198,199]. The strengthening function ($f_s$) is defined as follows:

$$f_s=\left\{\begin{array}{c}1,{\sigma }_i\in \left[0,{\sigma }_f\right]\\ e^{\left(m^{\prime }{\sigma }_i\right)},{\sigma }_i\in \left[{\sigma }_f,{\sigma }_0\right]\end{array}\right.$$

(73)

where $m^{\prime }$ represents the coefficient of strengthening, which depends on the characteristics of the material and can be determined through experimental methods. Additionally, ${\sigma }_f$ denotes the minimum stress required for strengthening.

Li et al. developed a novel fatigue model to evaluate damage accumulation in bridges subjected to traffic loads, building on the Lemaitre CDM fatigue model [184,200]. This enhanced model incorporates the concepts of microplastic strain, strain energy density release rate, and current damage state. Within this framework, damage is characterized as

$$D_{i+1}=1-{\left\{{\left(1-D_i\right)}^{{\alpha }_1+1}-{\displaystyle \frac{\left({\alpha }_1+1\right)N_{bl}^i}{B(\beta +3)}}\sum _{j=1}^{m_{rb}}\left[{\left[\left({\sigma }_j+2{\sigma }_{mj}\right){\sigma }_j\right]}^{{\displaystyle \frac{\beta +3}{2}}}{\left(1-D_i\right)}^{{\alpha }_1-{\alpha }_j}\right]\right\}}^{{\displaystyle \frac{1}{1+\alpha }}}$$

(74)

where

$${\alpha }_i=k_{\alpha }{\sigma }_i+{\alpha }_0$$

(75)

where the material parameters B, $\beta$, $k_{\alpha }$, and ${\alpha }_0$ can be derived from the S-N curve. $m_{rb}$ represents the cycle count where the peak stress in the representative block exceeds the fatigue threshold. $N_{bl}$ denotes the total number of loading blocks. For the jth cycle, ${\sigma }_j$ indicates the stress level, whereas ${\sigma }_{mj}$ signifies the mean stress.

Several studies have been conducted using this method to assess the damage in long-span bridges [200,201,202,203,204,205]. However, the primary drawback of the model is that it was specifically designed to address high-cycle fatigue and is not reliable under low-cycle conditions.

A novel damage stress model (DSM) based on cumulative fatigue damage accumulation was introduced by Mesmacque et al. This model postulates that the fatigue life of a material is solely determined by the loading conditions when the physical damage state remains constant [34], which is demonstrated in Figure 7. The authors defined the damage accumulation using the residual life ($N_{iR}$) associated with the damage stress (${\sigma }_{ed,i}$) following $n_i$ loading cycles, which was derived from the Wöhler (S-N) curve.

The remaining lifespan ($N_{iR}$) can be calculated as $N_{iR}=N_i-n_i$, where $n_i$ represents the number of cycles applied to the ith load. For multiple levels of loading, damage accumulation is defined as

$$D_i={\displaystyle \frac{{\sigma }_{ed,i}-{\sigma }_i}{{\sigma }_u-{\sigma }_i}}={\displaystyle \frac{{\sigma }_{equiv}-{\sigma }_{i+1}}{{\sigma }_u-{\sigma }_{i+1}}}$$

(76)

where ${\sigma }_u$ is the ultimate stress and ${\sigma }_{equiv}$ is the equivalent damage stress.

Several studies have examined the use of DSM to address multistep loading fatigue problems [100,206,207,208,209]. These studies have shown that the results are considerably more consistent than those obtained using Miner’s rule. The application of DSM to predict the fatigue life of riveted railway bridge components and corroded bridge elements was investigated by Siriwardane et al. [210,211]. Researchers have advised the use of DSM when stress histories are available. Nevertheless, a study by Rege and Pavlou on fatigue life estimations using DSM revealed poor performance when the S-N curve exhibited linearity in a semi-log plot [102].

Researchers have expanded DSM through various models. Pitoiset and Preumont introduced a method utilizing von Mises stress for multiaxial loading scenarios, and they shifted from the time domain to the frequency domain, drawing on the Crossland criterion [212,213]. Subsequently, Aid et al. formulated a novel fatigue damage model that integrated the material’s S-N curve, DSM, and Miner’s rules [214]. This model also incorporates the von Mises stress approach for multiaxial loading cases, as proposed by Pitoiset and Preumont, along with Sines and Crossland’s criterion [212,213,215]. Multiple iterations of the DSM have been developed by Socie et al., Shen et al., Benkabouche et al., Wang-Brown, and Lagoda-Macha, each employing distinct criteria for assessing cumulative fatigue damage accumulation [100,216,217,218,219,220,221].

2.7. Material Degradation Based Models

Numerous theoretical and practical investigations have shown that materials display varying static mechanical characteristics, such as Young’s modulus, yield stress, and ultimate tensile stress, under both their initial and post-fatigue conditions [222,223,224,225,226]. Consequently, researchers have formulated various fatigue damage accumulation models to explain the deterioration behavior of materials.

Through experimentation, Ye and Wang discovered that the internal energy of a material increases with each subsequent fatigue cycle, leading to an increase in dislocations and internal flaws [225]. Researchers introduced a novel fatigue damage model that considers damage as the irreversible dissipation of cyclic plastic strain energy, ultimately resulting in fatigue fracture. The newly proposed damage model is expressed as follows [35]:

$$D=-{\displaystyle \frac{D_{N_f-1}}{ln\left(N_i\right)}}ln\left(1-{\displaystyle \frac{n_i}{N_i}}\right)$$

(77)

where

$$D_{N_f-1}=1-{\displaystyle \frac{{{\sigma }_a}^2}{2E{U}_{T0}}}$$

(78)

where E represents Young’s modulus and $U_{T0}$ denotes the static toughness of the material before damage occurs. The applied stress amplitude is denoted by ${\sigma }_a$, where $N_i$ denotes the number of cycles until failure at the ith load level. Additionally, $n_i$ represents the number of cycles applied at the ith load and $D_{N_f-1}$ represents the critical value of the damage variable.

Lv et al. further refined the Ye and Wang model by incorporating a load interaction factor, as proposed by Corten and Dolan and Morrow, to consider the load sequence effects [7,13,227]. The damage accumulation parameter becomes

$$D=-{\displaystyle \frac{D_{N_f-1}}{ln\left(N_i\right)}}ln\left(1-{\displaystyle \frac{n_i}{N_i}}\right)\left({\displaystyle \frac{{\sigma }_i}{{\sigma }_{max}}}\right)$$

(79)

Researchers have employed this model to predict the fatigue life of C45 and 16Mn steels under two distinct block-loading sequences. Unlike the Ye and Wang model, the Lv et al. model exhibited superior accuracy in predicting fatigue life, underscoring the importance of considering load sequence effects [7].

Böhm et al. introduced an alternative approach to material degradation. They formulated a fatigue damage accumulation model based on the psychological theory, specifically utilizing the Ebbinghaus forgetting curve. This exponential function describes how memory retention decreases over time when it is not actively maintained [36]. To represent the material degradation due to fatigue, the authors adapted the time function into a cycle-based function. The concept of material memory is defined as follows:

$$m={\left(a-c\right)}^e+{\displaystyle \frac{N_f}{d}}+c$$

(80)

where m represents the material’s memory performance, a denotes the rate of memorization, d stands for the inverse of the forgetting rate expressed as the number of cycles, and c indicates the function’s horizontal asymptote. Based on this equation, researchers have developed models for damage assessment.

$$D=\sum _{t=1}^f{\displaystyle \frac{N_f-e^{\frac{n_t}{d}}}{d\left(1-e^{\frac{n_t}{d}}\right)N_i}}$$

(81)

In 2018, Peng et al. introduced a novel method that merges the material memory concept with a residual S-N curve [36]. Traditionally, the residual S-N curve illustrates the remaining lifespan of a material or component after exposure to multiple stress cycles. This curve typically maintains the same slope as the S-N curve of the virgin material but with a different S-intercept. However, the researchers proposed that the slope of the residual S-N curve, $\Delta$b, is influenced by the loading history. At the outset, the slope of the residual S-N curve matches that of the virgin material (b); however, as fatigue damage progresses, the slope $\Delta$b increases. The ratio of the slopes, b/$\Delta$b, is theorized to indicate the accumulation of fatigue damage. Drawing from the material memory degradation concept, a decay coefficient $\alpha$ is established that correlates with the slope ratio b/$\Delta$b. This memory degradation concept is employed to model the alterations in the residual S-N curve, yielding a more precise representation of the fatigue life of the material for the ith loading block.

$${\displaystyle \frac{b}{\Delta b}}={\alpha }_i={\displaystyle \frac{e^{-\frac{n_i}{N_i}}-e^{-1}}{1-e^{-1}}}$$

(82)

Using this relation, Peng et al. redefined the accumulative fatigue damage accumulation rule [36], as follows:

$$D=\sum _1^i{\displaystyle \frac{n_i}{N_i}}\times \prod _{j=1}^{i-1}{\left({\displaystyle \frac{N_i}{N_{i+1}}}\right)}^{\left({\prod }_{k=1}^j{\displaystyle \frac{e^{-\frac{n_k}{N_k}}-e^{-1}}{1-e^{-1}}}\right)-1}$$

(83)

An evaluation of fatigue life prediction outcomes from two-block and multi-block models, using Miner’s rule and Kwofie and Rhabar’s rule, indicated that the model by Peng et al. yielded more precise results [1,36,77].

Zhou et al. introduced a novel fatigue damage accumulation rule, building upon the material memory degradation rule proposed by Bohm et al. [36,38]. The damage model is expressed as follows:

$$D_i={\left({\displaystyle \frac{1-e^{-\frac{n_i}{N_i}}}{1-e^{-1}}}\right)}^{\zeta {\left(\sqrt{{\sigma }_{max,i}{\sigma }_{a,i}}\right)}^{\delta }}$$

(84)

where $\zeta$ and $\delta$ are the fitting parameters of Zhou’s fatigue model. The authors used this model for fatigue life prediction of 30NiCrMoV12 steel; however, the residual life $\zeta$ was omitted without proper reasoning. Moreover, because this model uses fitting parameters, it cannot be generalized.

In 2020, echoing the work of Zhou et al., Chapman et al. introduced a model for the cumulative fatigue damage in anisotropic composite materials based on material degradation [38,228]. Their research postulates that the material strength ($R\left(n\right)$) decreases as a function of the number of cycles to failure (n), with the rate of this decline approximated using a power-law relationship.

$${\displaystyle \frac{dR\left(n\right)}{dn}}=-A\left(\sigma \right)/m{\left[R\left(n\right)\right]}^{m-1}$$

(85)

where $A\left(\sigma \right)$ is a function dependent on the peak cyclic stress, and m is a constant. Furthermore, this study introduces elastic property degradation through the degradation parameter ($\eta$).

$$E\left(n\right)=\eta E_s$$

(86)

where

$$\eta ={\left[1-{\left({\displaystyle \frac{logn-log0.25}{log{N}_f-log0.25}}\right)}^{\lambda }\right]}^{{\displaystyle \frac{1}{\gamma }}}\left(1-{\displaystyle \frac{\sigma }{E_s{\epsilon }_f}}\right)+{\displaystyle \frac{\sigma }{E_s{\epsilon }_f}}$$

(87)

where $\lambda$ and $\gamma$ are the experimental curve fitting parameters, $E_s$ is the static stiffness, and ${\epsilon }_f$ is the average strain to failure.

2.8. Energy-Based Damage Rule

In the 1960s, Morrow and Halford initially established a failure criterion based on hysteresis energy and fatigue behavior, which Inglis introduced [11,39,229]. Subsequently, Zuchowski, Budiansky-O’Connell, and Glinka developed preliminary failure models that incorporated the strain energy [41,230,231]. Researchers have developed strain-energy-density-based fatigue damage rules for high-cycle fatigue, drawing on the observation that plastic energy dissipation ($\Delta W_p$) during cyclic loading can serve as a criterion for low-cycle fatigue failure [42,232]. However, this approach is restricted to low-cycle fatigue, because the plastic component $\Delta \epsilon$ diminishes. Consequently, the plastic strain energy density $\Delta W_p$ also approaches zero, rendering this method less effective for high-cycle fatigue.

To address this limitation, Ellyin et al. proposed a novel failure rule that combines high- and low-cycle fatigue by introducing the concept of total energy ($\Delta W_t$), which is the sum of elastic energy ($\Delta W_e$) and plastic energy ($\Delta W_p$) [232,233,234,235]. The researchers introduced a double logarithmic curve of SED versus fatigue life cycle ($log\Delta W-logN$), analogous to the Wöhler (S-N) curve. This curve is founded on a power law, which can be expressed as

$$\Delta W_t=\Delta W_e+\Delta W_p=\kappa N^{\alpha }+C$$

(88)

where $\kappa$, $\alpha$, and C are material constants. The constant C denotes the proportion of elastic energy input in the tensile mode and is associated with the material’s fatigue limit strain energy density $\Delta W_{lim}$. The elastic component of a material under tensile stress promotes crack propagation, whereas the plastic component ultimately leads to material degradation. This theoretical framework is applicable to both the Masing and non-Masing materials. For either type, a master curve can be generated by shifting the loop along its linear response segment (in the case of non-Masing materials). The cyclic plastic strain for the non-Masing behavior can be represented as

$$\Delta W_p={\displaystyle \frac{1-n^{*}}{1+n^{*}}}\left(\Delta \sigma -\delta {\sigma }_0\right)\Delta {\epsilon }_p+\delta {\sigma }_0\Delta {\epsilon }_p$$

(89)

and the for the ideal Masing behavior can be represented as

$$\Delta W_p={\displaystyle \frac{1-n^{\prime }}{1+n^{\prime }}}\Delta \sigma \Delta {\epsilon }_p$$

(90)

where $n^{*}$ and $n^{\prime }$ are the cyclic exponents of the mater curve and the Masing material, respectively. Further developments have been made based on this theory, accounting for both the onset of crack formation and crack propagation.

Therefore, Ellyin et al. proposed the following fatigue damage law for a multi-block loading sequence [232,233,234,235]:

$$D={\left\{{\left[{\left({\displaystyle \frac{n_1}{N_1}}\right)}^{{\displaystyle \frac{log\left(\frac{N_2}{N^{*}}\right)}{log\left(\frac{N_1}{N^{*}}\right)}}}+{\displaystyle \frac{n_2}{N_2}}\right]}^{{\displaystyle \frac{log\left(\frac{N_3}{N^{*}}\right)}{log\left(\frac{N_2}{N^{*}}\right)}}}+\cdots +{\displaystyle \frac{n_{i-1}}{N_{i-1}}}\right\}}^{{\displaystyle \frac{log\left(\frac{N_i}{N^{*}}\right)}{log\left(\frac{N_{i-1}}{N^{*}}\right)}}}+{\displaystyle \frac{n_i}{N_i}}$$

(91)

where $N^{*}$ corresponds to $W^{*}$, which is extrapolated from the $log\Delta W-logN$ curve and is referred to as the reduced fatigue limit.

Kreiser et al. introduced a novel energy-based nonlinear cumulative fatigue damage model. This model is particularly effective for materials and structures subjected to low-cycle, high-amplitude loading conditions where significant plastic strain is observed [236]. The authors employed Ellyin-Golos’s model, which considers loading sequence history, and integrated it with Coffin–Mason relations, which account for plastic strain in both stable and unstable hysteresis [234]. The proposed damage model is described as follows:

$$\phi =f\left(\psi ,p_m\right)={\displaystyle \frac{1}{log10\left(\Delta W_p/\Delta W_e\right)}}$$

(92)

where $\phi$ is the cumulative fatigue damage, $\psi$ is the cumulative damage parameter, and $p_m$ is the material parameter. Using the above equation, Manson derived the fatigue life of a material using the following equation:

$$\Delta \epsilon =\Delta {\epsilon }_e+\Delta {\epsilon }_p=3.5{\displaystyle \frac{{\sigma }_B}{E}}{N_f}^{-0.12}+{{\epsilon }_f}^{0.6}{N_f}^{-0.6}$$

(93)

where ${\epsilon }_f$ is the fracture ductility, and ${\sigma }_B$ is the tensile strength of the material. Various studies have used this model to predict the fatigue life of materials under various loading conditions.

Leis introduced a nonlinear damage model based on energy that assumes the rate of damage is influenced by cycle-dependent alterations in the overall microstructure [43]. The damage rule he put forward is expressed as follows:

$$D={\displaystyle \frac{4{\sigma }_f^{\prime }}{E}}{\left(2N_f\right)}^{2b_1}+4{\sigma }_f^{\prime }{\epsilon }_f^{\prime }{\left(2N_f\right)}^{2b_1+c_1}$$

(94)

where

$$c_1={\displaystyle \frac{-1}{1+5n_1}}$$

(95)

$$b_1={\displaystyle \frac{-n}{1+5n_1}}$$

(96)

where ${\sigma }_f^{\prime }$ is the fatigue strength of the material, ${\epsilon }_f^{\prime }$ is the ductility coefficient of the material, and $n_1$ is defined as a function of the accumulation of the plastic strain. Xiaode et al. after conducting fatigue tests under a constant strain amplitude and discovering a new cyclic stress–strain relationship, researchers further developed Leis’s model [43,237].

Niu et al. observed that the strain hardening coefficient changed during cyclic loading, whereas the strain hardening exponent changed only negligibly [238,239]. The authors introduced a new cyclic stress-strain relation as follows:

$${\displaystyle \frac{\Delta \sigma }{2}}=K{\left({\displaystyle \frac{\Delta {\epsilon }_p}{2}}\right)}^{n^{*}}{r}^{\beta }$$

(97)

where $K^{*}$ and $n^{*}$ are the cyclic strain hardening coefficients, and $\beta$ is the cyclic hardening rate. $\beta$ is found through

$$\beta =a{\left({\displaystyle \frac{\Delta \sigma }{2}}{\displaystyle \frac{\Delta {\epsilon }_p}{2}}\right)}^b$$

(98)

where a and b are constants.

Based on this, the damage is defined as

$$D={\Phi }^{1/\left[\left(n^{\prime }+\alpha \right)\left(1+\beta \right)\right]}=r^{1/\left(n^{\prime }+\alpha \right)}$$

(99)

where

$$\alpha ={\left({\displaystyle \frac{\Delta \sigma \Delta {\epsilon }_p}{4}}\right)}^{2b}\sqrt{a}$$

(100)

where $n^{\prime }$ is the strain hardening parameter.

Radhakrishnan proposed that the crack growth rate is proportional to the plastic strain energy density, which accumulates linearly until failure [44,240]. For m-level load variation, an expression for predicting the remaining life fraction at the last load step is formulated as follows:

$$D_m=1-\sum _{i=1}^{m-1}{\displaystyle \frac{W_{fi}}{W_{fm}}}{D}_i$$

(101)

where $W_{fi}$ is the total plastic strain at failure for the ith load level, and $W_{fm}$ is the total plastic strain at failure for the mth load level. Subsequently, a similar approach was adopted by Kliman for the harmonic loading cycles.

Lagoda et al. showed in 2001 that the normal strain energy density on the critical plane effectively predicts high-cycle fatigue life under random and non-proportional cyclic loading [241,242]. They also introduced a novel strain energy density parameter that distinguishes between positive strain energy density in tension and negative strain energy density in compression. Researchers have applied this model to assess the fatigue life of steel and cast iron, yielding a strong experimental correlation.

Park et al. modified the study by Ellyin et al. by introducing the plastic strain energy consumed by mean stress [232,233,234,235,243]. This formulation for the total work conducted is expressed as

$$\Delta W_t=\Delta W_e+\Delta W_t+{\left\{{\displaystyle \frac{f\left({\epsilon }_m\right)}{N_f}}\right\}}^{\gamma }=\kappa N^{\alpha }+C$$

(102)

where $f\left({\epsilon }_m\right)$ represents the plastic strain energy associated with the mean strain and $\gamma$ is the material constant. This model was employed to estimate the fatigue life of AZ31 alloy with anisotropic properties owing to rolling, and the predictions were in good agreement with the experimental findings.

A novel approach for predicting fatigue life based on energy was introduced by Jahed and Varvani-Farahani [244,245]. Their method utilizes the upper and lower limits of the experimentally determined lifespan of a material. The equation for calculating the number of cycles until failure ($N_F$) is as follows:

$$N_F={\displaystyle \frac{\Delta E_A}{\Delta E}}{N}_A+{\displaystyle \frac{\Delta E_T}{\Delta E}}{N}_T$$

(103)

where $\Delta E$ is the total elastic-plastic energy, and $\Delta E_A$ and $\Delta E_T$ are the energies due to purely tensile loading and purely torsion loading, respectively. Furthermore, the authors extended this to non-proportional loading. Gu et al. used this model to predict the fatigue life of welded frames for mining trucks with a good experimental correlation [246].

Ozaltun et al. proposed a new energy-based fatigue damage accumulation model that considered the aging effects caused by cyclic loading on the fatigue strength of structural components in gas turbine engines [247]. The model also accounts for plastic strain energy dissipation. Fatigue life (N) was estimated as follows:

$$N=C{\displaystyle \frac{{\sigma }_n\left({\epsilon }_n-\frac{{\sigma }_0}{2E}\right)+{\epsilon }_0{\sigma }_0\left[cosh\left(\frac{{\sigma }_n}{{\sigma }_0}\right)-1\right]}{2{\sigma }_c\left\{\frac{{\sigma }_a}{{\sigma }_c}sinh\left(\frac{2{\sigma }_a}{{\sigma }_c}\right)-\left[cosh\left(\frac{2{\sigma }_a}{{\sigma }_c}\right)-1\right]\right\}}}+{\displaystyle \frac{\frac{{\beta }_1}{2}\left({{\epsilon }_f}^2-{{\epsilon }_n}^2\right)+{\beta }_0\left({\epsilon }_f-{\epsilon }_n\right)}{2{\sigma }_c\left\{\frac{{\sigma }_a}{{\sigma }_c}sinh\left(\frac{2{\sigma }_a}{{\sigma }_c}\right)-\left[cosh\left(\frac{2{\sigma }_a}{{\sigma }_c}\right)-1\right]\right\}}}$$

(104)

where ${\sigma }_n$ is the necking stress, ${\epsilon }_n$ is the necking strain, ${\epsilon }_f$ is the fracture strain, ${\sigma }_a$ is the generalized cyclic stress, ${\epsilon }_a$ is the generalized strain, ${\beta }_0$ and ${\beta }_1$ are the d intercept of the stress–strain relationship in the necking region, ${\sigma }_0$, ${\sigma }_c$, and C are the curve fitting parameters.

An energy-based variant of DSM by Mesmacque et al. [34,248,249] was introduced by Djebli et al. In this new approach, the cumulative damage accumulation parameter ($D_i$) is established as

$$D_i={\displaystyle \frac{W_{ed,i}-W_i}{W_u-W_i}}$$

(105)

where $W_{ed,i}$ represents the energy associated with the damage stress, $W_i$ denotes the energy related to the applied stress, and $W_u$ denotes the energy corresponding to the ultimate stress of the material. Researchers have employed this model to estimate the fatigue lives of metals and polymers [248,249].

Peng et al. introduced a novel method called the fatigue driving energy (FDE) approach, which integrates Kwofie and Rhabar’s fatigue driving stress model with strain energy density (SED) [14,250]. This model for fatigue damage accumulation is expressed as

$$D={\displaystyle \frac{W_D-W_{D_0}}{W_{D_C}-W_{D_0}}}={\displaystyle \frac{N^{-2b\frac{n}{N}}-1}{N^{-2b}-1}}$$

(106)

where,

$$W_D={\displaystyle \frac{1}{2E}}{\sigma }^2{N}^{{\displaystyle \frac{-2bn}{N}}}$$

(107)

$$W_{D_0}={\displaystyle \frac{{\sigma }^2}{2E}},{\displaystyle \frac{n}{N}}=0$$

(108)

$$W_{D_c}={\displaystyle \frac{A^2}{2E}},{\displaystyle \frac{n}{N}}=1$$

(109)

where W is the strain energy density, $W_D$ is the fatigue driving energy, $W_{D_0}$ is the stress-energy density for the initial stage without damage, $W_{D_c}$ is the critical stress-energy density at failure, A is the material constant, and b is the exponent of the Basquin equation. For multi-block loading, the residual fatigue life of the material can be described as

$$\begin{array}{cc}\hfill {\displaystyle \frac{n_i}{N_i}}& =1-{\displaystyle \frac{1}{-2bln\left(N_i\right)}}\times ln\left(\left({N_i}^{-2b}-1\right){\left({\displaystyle \frac{{N_{i-1}}^{-2b\left[\frac{n_{i-1}}{N_{i-1}}+1-{\left(\frac{n_{i-1}}{N_{i-1}}\right)}_{mp}\right]}-1}{{N_{i-1}}^{-2b}-1}}\right)}^{{\displaystyle \frac{{\sigma }_{i-2}}{{\sigma }_{i-1}}}\times {\displaystyle \frac{{\sigma }_i}{{\sigma }_{i-1}}}}+1\right)\hfill \end{array}$$

(110)

Peng’s and Miner’s models provided a more accurate fatigue life estimation for five different experimental datasets with different geometries.

Gan et al. proposed a new energy-based cumulative fatigue damage accumulation model based on a series of energy-life curves as an alternative to incremental plasticity theories [251]. The authors modified the multiaxial Wöhler (S-N) curve to redefine a new multiaxial S-N curve based on axial and torsion S-N curves, called the modified Wöhler curve method (MWCM).

$$N_f=N_A{\left[{\displaystyle \frac{\Delta {\tau }_A\left(\rho \right)}{\Delta \tau }}\right]}^{k\left(\rho \right)}$$

(111)

where $\Delta \tau$ is the shear stress range; $N_A$ is a reference fatigue life usually taken in the interval 10⁶–10⁷ cycles; $\Delta {\tau }_A\left(\rho \right)$ and $k\left(\rho \right)$ are the reference shear stress amplitude and inverse slope related to the multiaxial S-N curve, respectively, which can be defined by

$$\begin{array}{c}\hfill \Delta {\tau }_A\left(\rho \right)=\left[\Delta {\tau }_A\left(1\right)-\Delta {\tau }_A\left(0\right)\right]\rho +\Delta {\tau }_A\left(0\right),\\ \hfill k\left(\rho \right)=\left[k\left(1\right)-k\left(0\right)\right]\rho +k\left(0\right)\end{array}$$

(112)

where $\rho$ is the stress ratio associated with the multiaxiality and non-proportionality of the local stress field, which can be expressed as

$$\rho ={\displaystyle \frac{{\sigma }_{max}}{2\Delta \tau }}$$

(113)

Lu et al. proposed a new energy-based cumulative fatigue damage accumulation model for low-cycle fatigue life prediction under multiaxial irregular loading [252]. The model combines both cyclic plastic work and the non-proportional (NP) hardening effect. The model defines the damage accumulation parameters (D) by defining the effective plastic work ($W_{eff}$), as follows:

$$D=N_f\sum _{i=1}^m{\displaystyle \frac{{W^i}_{eff}}{{W^i}_f}}$$

(114)

where the effective energy parameter $W_{eff}$ is defined as

$$W_{eff}=\left(1+{\alpha }_w{F}_{NP}\right)W_{eq}$$

(115)

where $W_{eq}$ is the equivalent uniaxial plastic work, ${W^i}_f$ is the fatigue toughness under the ith loading path in a cycle, ${\alpha }_w$ is a material constant reflecting the material sensitivity to an NP path, $F_{NP}$ accounts for the path dependency of NP cyclic hardening, and $N_f$ is the number of cycles to failure.

This study utilized this model to predict the fatigue life of composite materials under multiaxial loading with multi-block loading sequences, and it showed an improved correlation with experimental data compared with Miner’s rule.

3. Fatigue Life Prediction of Adhesive Joint in FRP Composite Under Step/Variable Fatigue Loading

3.1. Numerical Model

A double-lap joint testing model is used to test the applicability of the aforementioned theories. Khabbaz et al. have carried out experimental studies to determine the fatigue life of adhesively bonded FRP structural joints under constant and variable loading [2].

A double-lap joint model shown in Figure 8 was used for the numerical simulation. The total length of each specimen is 350 mm. The outer and inner GFRP laminates were 200 mm long. The blue section of Figure 8 of the outer GFRP laminates was fixed and 50 mm long, whereas the load was applied to the yellow section (50 mm) at the end of the inner GFRP laminate.

The inner and outer tabs were made of pultruded GFRP laminates consisting of E-glass fibers, and an isophthalic polyester resin was used to fabricate the bonded joints. The table below (i.e.,

Table 1. Mechanical properties of GFRP laminates used in this study.

Property	Value
Density	2560 kg/m3
Young’s Modulus	25 GPa
Tensile Strength	307.5 MPa
Poisson’s Ratio	0.23

) shows the material properties of the GFRP laminates.

Sikadur 330 (Sika AG, Zugerstrasse 50, CH-6340 Baar (ZG), Switzerland), a two-component epoxy adhesive system, was used for the adhesive joint [253]. The material properties for the Sikadur 330 are listed in

Table 2. Mechanical properties of the adhesive (Sikadur 330) used in this study.

Property	Value
Flexural Strength	60.6 MPa
Modulus of Elasticity in Flexure	3489 MPa
Tensile Strength	33.8 MPa
Elongation at Break	1.2%
Heat Deflection Temperature	50 °C

.

The fatigue test was performed under load control at a constant frequency of 10 Hz for constant loading. The load ratio, which was the ratio of the maximum applied cyclic load to the minimum applied cyclic load, was 0.1.

Four different two-block loading experiments were performed. The experiments consisted of two low-high-cycle loadings and two high-low-cycle loadings. A constant loading frequency of 10 Hz was used for the block loading, and the load ratio was maintained at 0.1.

Table 3. Experimental cases used in this study to check the theoretical damage accumulation models.

	Fmax1 [kN]	n1	N1	D1 = n1/N1	Fmax2 [kN]	n2	N2	D2 = n2/N2	D
Low-High Sequence	14.4	48,649	142,978	0.3403	21.6	483	1070	0.4514	0.7917
	12	456,038	1,291,993	0.3530	19.2	2592	4434	0.5846	0.9376
High-Low Sequence	21.6	287	1070	0.2682	14.4	207,559	142,978	1.4517	1.7199
	19.2	1330	4434	0.3000	12.0	4,832,688	1,291,993	3.7405	4.040

lists the two-block loading experiments.

where F_max1 is the max load applied in first loading block, F_max2 is the max load applied in second loading block, n₁ is the number of cycles with the first loading block, n₂ is the number of cycles with the second loading block until failure, N₁ is the number of cycles to failure in the first loading block, N₂ is the number of cycle to failure in the second loading block, D₁ is the damage caused in the first loading block, D₂ is the damage caused in the second loading block, D is the total damage according to Miner’s rule (D₁ + D₂).

3.2. Simulation Results

To assess the validity of the theoretical models for fatigue damage accumulation under step/variable loading, as discussed in the previous section, 12 models were selected based on four criteria: popularity, relevance to adhesive materials, recency, and uniqueness. A comparison of the selected models with experimental data for fatigue damage accumulation under step/variable loading is presented in

Table 4. Comparison of the experimental number of cycles to failure with selected theoretical models.

	Case 1	Case 2	Case 3	Case 4
Experiment	49,132	458,630	207,846	4,834,018
Miner	49,355	458,906	104,915	905,784
Corten–Dolan	48,963	457,158	236,000	2,317,476
Freudenthal–Heller	49,558	459,860	57,277	301,967
Damage Curve Approach	49,516	459,558	127,992	1,109,683
Kwofie	49,064	457,749	178,565	1,517,491
Subramanyan	48,650	456,038	142,978	1,291,993
Dattoma	48,650	456,038	142,978	1,291,993
Xie	49,018	457,465	138,338	1,242,244
Radhakrishnana	48,963	457,158	236,000	2,317,476
Carpenteri	49,551	459,848	82,304	682,574
Iso-Damage line modification	50,688	470,081	22,160	97,819
Morrow	218,377	2,107,692	722	2901
Bilinear Cohesive model	51,400	300,000	107,180	913,610

.

Table 5. Comparison of REP for selected theoretical models.

	Case 1	Case 2	Case 3	Case 4
Experiment	0	0	0	0
Miner	0.4539	0.06018	49.5227	81.2623
Corten Dolan	0.3440	0.3210	13.5456	52.0590
Fruedenthal Heller	0.8671	0.2682	72.4426	93.7533
Damage curve approach	0.7816	0.2023	38.4198	77.0443
Kwofie	0.1384	0.1921	14.0878	68.6081
Subramanyan	0.9810	0.5651	31.2096	73.2729
Dattoma	0.9810	0.5651	31.2096	73.2729
Xie	−0.2320	0.2540	33.4421	74.3020
Radhakrishnan	0.3440	0.3210	13.5456	52.0590
Carpenteri	0.8528	0.2656	60.4015	85.880
Iso-damage line modification	3.1670	2.4968	89.3383	97.9764
Morrow	344.470	359.5626	99.6526	99.940
Bilinear cohesive model	4.616	34.588	48.433	81.100

shows a comparison through relative error of prediction (REP), which can be defined as

$$REP={\displaystyle \frac{N_{exp}-N_{sim}}{N_{exp}}}\times 100$$

(116)

The majority of the models demonstrated lower error percentages than Miner’s rule predictions for the number of cycles to failure, particularly for low-to-high load cycle scenarios (Cases 1 and 2), with most models exhibiting REP of approximately 1%. However, the Morrow’s damage accumulation model deviated from this trend, showing REP of approximately 2.5% for both cases. However, Kwofie’s fatigue damage accumulation model achieved the most favorable results for these scenarios, with REP of approximately 0.15% in both cases.

For the high-to-low load cycle cases (Cases 3 and 4), most models also showed lower error percentages than Miner’s rule predictions. Nevertheless, all theoretical damage accumulation models had a substantial degree of error when compared with the experimental data. Radhakrishnan’s damage accumulation model had the least REP, with 13.5% for Case 3 and 52% for Case 4 when compared to the experimental data.

The reason for the low percentage error in the low-to-high load cycle scenarios (i.e., Cases 1 and 2) was that the number of cycles to failure post-load transition was much smaller than the number of pre-load transition cycles.

The impact of the loading sequence on the fatigue life of the examined joints was related to the crack growth rate during the applied loading blocks. The high-to-low load cycle cases (i.e., Cases 3 and 4) resulted in a retardation of the crack growth rate, whereas acceleration was observed under the low-to-high load cycle cases (i.e., Cases 1 and 2).

An analysis of the damage accumulation rate per cycle revealed that the experimental results showed a faster rate for low-to-high loading cycle scenarios (Cases 1 and 2), and then predicted the damage accumulation rate. By contrast, high-to-low loading situations demonstrated a reduced damage accumulation rate per cycle, which was aligned with the experimentally observed trends [2].

A primary factor contributing to this discrepancy is the fatigue crack propagation, which is affected by the energy release rate. Many existing models do not account for the energy release rate when calculating failure, resulting in a linear and steady failure progression [144,145]. However, empirical evidence has demonstrated that the energy release rate increases exponentially as fractures begin to form in a material, which leads to material instability and rapid failure [7,13,227]. Even models that consider the energy release rate and crack propagation as the primary mechanisms of fatigue failure do not account for loading history. This limitation applies to fatigue damage accumulation models that are based on crack growth.

These findings suggest that most existing fatigue failure models are insufficient for accurately predicting fatigue failure in adhesive joints. These models often overlook crucial aspects, such as load history factors, the importance of loading sequences in determining fatigue failure, and the effects of load interaction. Additionally, many models fail to account for the swift and progressive nature of crack propagation during fatigue failure cycles. Consequently, there is a need for further research to develop an improved failure model that incorporates all of these critical factors when predicting cumulative fatigue failure in adhesive joints. Such a model would provide a more comprehensive and reliable approach to understanding and predicting the fatigue failure in these materials.

4. Conclusions

In this paper, a critical review of cumulative fatigue damage accumulation under step/variable fatigue loading experiments on adhesive joints in fiber-reinforced polymer composites is presented. This study investigated experimental observations of fatigue failure under step/variable loading and attempted to explore the theoretical reasoning behind the experimental observations. Cumulative fatigue damage has been extensively studied, and many approaches have been developed to treat it. This study investigated both linear and nonlinear accumulations, which can be grouped into eight categories, exploring over 100 cumulative fatigue damage accumulation models. A double-lap joint test model with four different step-load cases (two high-to-low cases and two how-to-high cases) was used to examine the applicability of the cumulative fatigue damage accumulation models. However, the cumulative fatigue damage accumulation model could not provide accurate predictions of failure in the aforementioned experimental cases.

This study emphasizes the significance of accounting for the loading history, load interaction effects, and material behavior when predicting fatigue life. Additionally, numerical simulations have demonstrated the limitations of conventional cumulative fatigue damage models, as they frequently yield conservative results under both low-to-high and high-to-low loading conditions. This reveals the necessity for more comprehensive models that integrate all the relevant factors to better predict the fatigue life of adhesive joints under variable loading scenarios. Therefore, additional research is necessary to fully understand the consequences of step/variable fatigue loading on the accumulation of fatigue damage in adhesive joints in fiber-reinforced polymer composites.