Entropy-based damage model for assessing the remaining useful fatigue life

Abstract

A reliable approach based on an entropy-damage model for assessing remaining useful fatigue life is presented. Two damage models are presented and evaluated to assess their effectiveness in predicting remaining useful life. The first model focuses on reduced toughness caused by fatigue degradation, while the second is based on accumulating entropy during fatigue loading. The entropy-based approach employs infrared thermography to anticipate entropy accumulation and damage status. Outcomes reveal that the entropy-driven technique offers enhanced precision. Moreover, its damage growth rate remains consistent, regardless of the number of cycles leading to failure, ensuring a more stable tracking of damage evolution. It successfully predicts the remaining useful life and can treat variable load sequencing without knowing the loading history. An extensive set of experimental results with carbon steel 1018 are presented to illustrate the utility of the approach.

Keywords

Remaining useful life (RUL)non-destructive evaluation (NDE)damage mechanics fracture fatigue entropy (FFE)thermography fatigue variable amplitude loading

Introduction

Mechanical components deteriorate under cyclic loading (Ye et al., 2014), making them susceptible to fatigue failure (Todinov, 2007). Estimation of fatigue life, damage evolution, and remaining useful life has captured the attention of numerous studies (Ferreira et al., 2022; Hou et al., 2023; Huo et al., 2022; Santecchia et al., 2016). Research has demonstrated that the fatigue behavior of a material under cyclic loading can be influenced by a multitude of parameters (Bouchedjra et al., 2021; Nihei et al., 1986; Zhu et al., 2022) that tend to complicate the prediction of fatigue performance. Considerable efforts have been directed towards forecasting the remaining useful life for components experiencing fatigue loading (Gan et al., 2023; Kralovec and Schagerl, 2020), particularly in scenarios involving variable amplitude fatigue loading (Si-Jian et al., 2018; Zuo et al., 2015).

It is well known that due to external actuation, components experience self-heating that manifests itself in the rise in temperature due to unrecoverable heat generation. Well-documented experimental results reveal that the associated increase in surface temperature is directly proportional to the applied load: The higher the fatigue load, the hotter is the surface temperature (Fargione et al., 2002; Meneghetti, 2007). Thus, the temperature variations during cyclic loading have been widely utilized to quantify the severity of fatigue degradation (Amiri and Khonsari, 2010a; Huang et al., 1984; Jiang et al., 2001; La Rosa and Risitano, 2000). Recent advances in infrared (IR) thermography have made this approach even more attractive for quantifying the elastic-plastic deformation and the related changes in the microstructure (Khonsari and Amiri, 2012). Furthermore, thermography has been used —as a non-destructive evaluation (NDE) technique—for structural health monitoring (SHM) (Bagavathiappan et al., 2013), damage detection (Meola et al., 2017; Tao et al., 2023), and life assessment (Tinsley et al., 2017; Wang et al., 2015).

Typically, the surface temperature of a material undergoing fatigue increases rapidly at the beginning of the fatigue process and reaches a steady state for most of the fatigue duration before it experiences a sudden increase near the fracture (Huang et al., 1984). This trend has been put to use by many researchers to study the behavior of metal fatigue. For example, Jiang et al. (Jiang et al., 2001) used the steady-state temperature rise to develop an empirical relation to estimating fatigue life. Fargione et al. (Fargione et al., 2002) proposed a constant parameter for the area under temperature cycles and obtained the entire fatigue curve based on this parameter. Amiri and Khonsari (Amiri and Khonsari, 2010b) developed a thermographic method to relate the number of cycles to the evolution of surface temperature. They also showed that the temperature slope at the beginning of the fatigue process can be used to predict fatigue life. Another important contribution of thermography is its role in treating fatigue as an irreversible thermodynamic process involving the accumulation of damage. Recent research shows the applicability of the thermodynamic framework to treat fatigue degradation in metallic and non-metallic materials (Mahmoudi and Khonsari, 2022a; Amiri and Modarres, 2014; Basaran and Yan, 1998; Fatemi and Yang, 1998; Huang et al., 2020; Mohammadi and Mahmoudi, 2018; Naderi and Khonsari, 2010; Ontiveros et al., 2017).

Temperature rise affects the behavior of components undergoing fatigue, particularly at high frequencies (Ghadimi et al., 2021; Guennec et al., 2014). While many fatigue models do not consider the effect of temperature rise, thermodynamic models developed based on entropy accumulation enable one to treat problems that involve high frequencies and high temperatures (Lee and Basaran, 2022). Entropy has been shown to be a powerful index for different types of degradation (Amiri and Modarres, 2014; Bryant et al., 2008). Naderi et al. (Naderi et al., 2010) showed that the accumulation of entropy generation up to failure, called fracture fatigue entropy (FFE), remains constant under different types of loading. Further research showed that FFE is a material property independent of loading parameters, geometry, and environmental conditions (Amooie and Khonsari, 2023; Amooie et al., 2023; Lee et al., 2022a; Mahmoudi and Khonsari, 2022b; Osara and Bryant, 2020). The entropy framework can be used for accelerating the testing procedure to predict the fatigue life and construct the S-N curves (Amooie and Khonsari, 2023). Furthermore, these models can predict the temperature evolution during fatigue (Mahmoudi and Mohammadi, 2019b) and demonstrate how the temperature rise can be utilized to predict accumulated damage during fatigue. More recently, Jang et al. (Jang et al., 2020) presented a methodology wherein the rate of temperature rise at the beginning of the fatigue is used to estimate the remaining fatigue life.

In the present study, we report the development of a robust method that directly uses the temperature rise to quantify damage accumulation during cyclic loading. Two damage models are introduced to gain further insight into the efficacy of predicting the accumulated fatigue degradation via temperature rise. The remaining useful life predictions (RUL) are authenticated via extensive experimental results. The RUL approach covers low- and high-cycle fatigue, constant- and variable-loading sequences and does not require the knowledge of the prior loading history of a pre-fatigued specimen.

Theory and formulation

Thermodynamics of fatigue

Fatigue is an irreversible degradation process that results in the dissipation of heat. Thus, a thermodynamic framework is required to investigate the internal heat generation and changes in temperature during cyclic loading. Figure 1 shows a typical trend of temperature evolution during the fatigue process. The temperature rises rapidly at the beginning of the cyclic loading (Phase I). It stabilizes for most of the fatigue process (Phase II) before experiencing a sudden increase just prior to fracture (Phase III). Figure 1 also provides the rate of temperature rise at different stages of the fatigue process. The slope of temperature rise decreases as temperature increases in Phase I, then it stabilizes in Phase II before a sharp rise in Phase III. In Phase II of the fatigue process, the temperature is nearly constant at low-stress levels or increases gradually with a low constant rate at high-stress levels.

To investigate irreversible fatigue degradation, the second law of thermodynamics can be used. The interested reader is referred to references (Lemaitre and Chaboche, 1994; Naderi et al., 2010) for the detailed description of formulations. For a material undergoing fatigue, the 2^nd law is expressed by the following equation (Lemaitre and Chaboche, 1994):

\dot{γ} = \frac{σ : {\dot{ε}}_{p}}{T} + \frac{A {\dot{V}}_{k}}{T} - \frac{\vec{q} . \nabla T}{T^{2}} \geq 0

(1)

where

\dot{γ}

is the non-negative entropy generation rate and

T

is the temperature in Kelvin-scale.

σ

denotes the applied stress and

ε_{p}

is the plastic strain.

A

is the thermodynamic force associated with the internal state parameter

V_{i}

\vec{q}

represents the heat flux across the boundary. For metals, the energy associated with the internal parameters (

A {\dot{V}}_{k}

) is only 5-10% of plastic strain energy (

{\dot{W}}_{p} = σ : {\dot{ε}}_{p}

) (Halford, 1966) and can be neglected for simplicity. Also, entropy generation owing to heat conduction, the last term in equation (1), is negligible (Naderi et al., 2010). Therefore, the entropy generation associated with the damaging energies can be obtained as:

γ_{f} = \int_{0}^{t_{f}} \frac{{\dot{W}}_{p}}{T} d t = \int_{0}^{t_{f}} \frac{σ : {\dot{ε}}_{p}}{T} d t

(2)

It has been shown that the accumulated entropy up to fracture for a material undergoing fatigue is nearly constant and is independent of loading conditions, and the fracture occurs when $γ_{f}$ reaches the entropy threshold of a material, i.e., the so-called fracture fatigue entropy (FFE) (Lee et al., 2022b; Mahmoudi and Mohammadi, 2019a; Naderi et al., 2010).

Damage parameters for remaining life assessment

Toughness-based damage parameter, $ω_{T, N}$

The first damage parameter is developed based on the toughness reduction during fatigue. Duyi and Zhenlin (Duyi and Zhenlin, 2001) defined a damage parameter to describe the fatigue behavior of metals based on changes in static fracture toughness during cyclic loading:

ω_{T, N} = - 1 - \frac{U_{T_{N}}}{U_{T_{0}}}

(3)

where

ω_{T, N}

is the accumulated damage to cycle

N

th,

U_{T_{0}}

is the static fracture toughness of the pristine specimen at the beginning of the fatigue test, and

U_{T_{N}}

is the static fracture toughness of fatigued specimen after

N

cycles. They showed that the damage parameter can be defined as:

ω_{T, N} = - \frac{ω_{c r}}{\ln (N_{f})} l n (1 - \frac{N}{N_{f}})

(4)

where

ω_{c r}

can be calculated using:

ω_{c r} = 1 - \frac{σ_{a}^{2}}{2 E U_{T_{0}}}

(5)

It has been shown by Liakat and Khonsari (Liakat and Khonsari, 2014) that for a wide range of stress amplitude ( $σ_{a})$ , $ω_{c r} \approx 1$ . They further developed this damage parameter based on the rate of temperature changes ( $R_{θ}$ ) at different accumulated cycles and suggested a test procedure described as follows. After reaching a steady state, the fatigue test is stopped at different stages and the specimen is allowed to cool down to the ambient temperature, and then the fatigue test is resumed. They showed that at each stage, the slope of temperature rise changes and that this change in the rate of temperature rise grows linearly with an increase in the number of cycles. Accordingly, they modified the damage parameter in equation (4) as (Liakat and Khonsari, 2014):

ω_{T, N} = - \frac{1}{\ln (N_{f})} l n (1 - \frac{R_{θ_{N}} - R_{θ_{0}}}{R_{θ_{f}} - R_{θ_{0}}})

(6)

where

R_{θ 0}

R_{θ f}

and

R_{θ N}

are temperature gradients at the beginning of the fatigue process, the onset of fracture, and after N fatigue cycles, respectively. It, therefore, follows that the damage parameter can also be defined based on the temperature rise, rather than

R_{θ}

, at the stabilized phase of fatigue as follows:

ω_{T, N} = - \frac{1}{\ln (N_{f})} l n (1 - \frac{θ_{N} - θ_{0}}{θ_{f} - θ_{0}})

(7)

where

θ_{0}

is the stabilized temperature of the undamaged specimen.

θ_{N}

is the temperature rise of a specimen subjected to N load cycles. And

θ_{f}

is the temperature rise of a specimen at its final stage of fatigue before a sudden increase in temperature.

Entropy-based damage parameter, $ω_{E, N}$

The second damage parameter introduced in this study is the normalized entropy generation based on the FFE concept defined as follows.

ω_{E, N} = \frac{γ_{N}}{FFE}

(8)

where

ω_{E, N}

and

γ_{N}

are the accumulated damage and entropy at the N^th cycle

.

In equation (8),

ω_{E, N}

represents the expired capacity of the material caused by entropy generation. This parameter increases linearly during the fatigue process from 0 to 1 (Naderi et al., 2010). As the stabilized temperature also increases approximately linearly during the fatigue process, the damage parameter in equation (8) can be written as:

ω_{E, N} = \frac{θ_{N} - θ_{0}}{θ_{f} - θ_{0}}

(9)

Experiments and methods

To assess the efficacy of the proposed approach, a set of fatigue tests at different stress amplitudes was performed on carbon steel 1018. The material composition, test method, equipment, and experimental procedure are described next.

Material and test equipment

The chemical composition and mechanical, physical, and thermal properties of carbon steel 1018 are shown in Tables 1 and 2. Cylindrical dog-bone specimens are designed and manufactured based on ASTM E 466-15 for fatigue testing. The surface of the specimens is polished using sandpaper. The gauge section of specimens is also sprayed with a thin black layer to increase the thermal emissivity. Figure 2 shows the designed, polished, and painted specimens with the corresponding dimensions.

Table 1.

The chemical composition of carbon steel 1018.

Material	C	Fe	Mn	S	P
CS 1018	0.14–0.2	98.81–99.26	0.6–0.9	0.05 Max	0.04 Max

Table 2.

Mechanical, physical, and thermal properties of carbon steel 1018.

Material	Density ( $ρ$ )	Thermal conductivity (k)	Specific heat capacity (C)	Young’s modulus (E)	Ultimate Strength ( $σ_{u}$ )	Yield strength ( $σ_{y}$ )
	(Kg/m³)	(W/m.K)	(J/kg.K)	(GPa)	(MPa)	(MPa)
CS 1018	7870	51	486	199	510	395

Fatigue tests are carried out using a testing machine with a maximum 25 kN axial load capability. The tail ends of the specimens are gripped with sufficient gripping pressure to avoid slippage between the specimen and the gripping jaws of the machine. The MIKRON M7500 infrared (IR) camera is utilized to record the surface temperature of the specimens during fatigue tests. The IR camera has a temperature range capability between 0-500°C, a sensitivity of 0.08 °C at 30 °C, an accuracy of ±2% of reading, and a resolution of 320 × 240 pixels.

Test procedure

Load-controlled uniaxial fatigue tests are performed at different stress levels, load ratio of −1, and a frequency of 10 Hz. The experiments follow a repeating Run-Stop-Cooldown (RSC) process as described next. Referring to Figure 1, the specimen at ambient temperature is subjected to the fatigue load to reach a steady state (Phase II). Our test results show that the rate of temperature rise stabilizes after running 5000 fatigue cycles. Therefore, the intervals of 5000 cycles are selected to record the temperature rise. At the end of each interval, the fatigue test is stopped, and the specimen is allowed to cool down to the ambient temperature. Then the same fatigue load is applied to the specimen and the experiment is continued until a new stabilized temperature is reached. This procedure is repeated several times up to fracture to evaluate the relation between the stabilized temperature and the number of fatigue cycles. A schematic diagram of the surface temperature of specimens undergoing repeating RSC procedures is shown in Figure 3. It is worth mentioning that the temperature is assumed to be uniform in the radial direction. This assumption is grounded in the presence of uniform heat generation within the gauge section, owing to the absence of stress concentration and the high thermal conductivity of steel materials. These combined factors substantiate the rationale for considering a uniform radial temperature.

Results and discussion

In this section, the temperature measurements are utilized to investigate the behavior of CS 1018 specimens undergoing tension-compression fatigue loading. The evolution of stabilized temperature during cyclic loading is used to obtain the damage evolution throughout the fatigue process. Section ‘Remaining life assessment’ shows how the entropic concept and the concept of damage continuity can be used to assess the remaining useful life of a pre-fatigued specimen without knowing the loading history. Also presented in this section are a series of predictions and experimental verifications to illustrate the efficacy of the approach to variable loading sequences.

Evolution of Steady-State temperature

Figure 4 shows the results of temperature measurements at different stress levels. As described earlier, following the RSC procedure, the fatigue tests are stopped at the steady-state phase, and the temperature rise is measured. Then, the specimens are allowed to cool down to the ambient temperature before resuming the fatigue procedure. The energy associated with the plastic strain—the main source of energy dissipation in metal fatigue (Lemaitre and Chaboche, 1994)—is directly related to the stress level. As can be seen in Figure 4, the higher the temperature rises, the shorter the fatigue life becomes, indicating that at higher loads, the material reaches its maximum capacity of entropy generation faster than at lower loads. Furthermore, the progressive temperature rise occurs as the fatigue loading persists. This phenomenon can be attributed primarily to the increased generation of plastic strain energy resulting from material softening, leading to greater energy dissipation. In Figure 5, the expansion of the hysteresis loop, which serves as an indicator of energy dissipation, is depicted at three distinct stages of fatigue under the stress level of 360 MPa. The temperature rise during fatigue of CS 1018 specimens can be estimated using the following linear equation.

θ_{N} = C N + θ_{0}

(10)

where

θ_{N}

is the temperature rise of a specimen subjected to N load cycles.

C

is a constant and

θ_{0}

is the temperature rise of an undamaged specimen.

C

and

θ_{0}

depend on the material property, loading conditions, and geometry. The values of parameters

C

and

θ_{0}

for the presented test setup are given in Table 3, along with the fatigue life at different stress levels.

Table 3.

Fatigue life and parameters $C$ , $θ_{0}$ and $θ_{f}$ for temperature rise at different stress levels.

Stress level (MPa)	Number of cycles to failure ( $N_{f}$ )	$C$	$θ_{0}$ (°C)	$θ_{f}$ (°C)
340	825,200	1.9147 × 10⁻⁶	0.02	1.6
360	219,800	1.1507 × 10⁻⁵	0.05	2.55
370	49,900	7.3146 × 10⁻⁵	0.11	3.65
380	21,500	4.0792 × 10⁻⁴	0.25	8.85
390	15,300	6.0784 × 10⁻⁴	0.36	9.35

Evolution of damage parameters

Figures 6 and 7 show the evolution of damage parameters, $ω_{T, N}$ and $ω_{E, N}$ , throughout the fatigue process at stress levels of 340, 360, 370, 380 and 390 MPa. As shown in Figure 6, the toughness-based damage parameter ( $ω_{T, N})$ , as defined in equation (7), has a gradual increase in the first 70% of fatigue life to reach the value of 0.1. Then, it sharply increases in the rest of life to fracture to reach the value of 1. In this damage parameter, the rate of evolution of damage at different stress levels depends on the normalized life ( $N / N_{f}$ ). As an example, at 360 MPa, the damage reaches 0.1 after 72% of life and at 390 MPa, the same value is reached after 65% of expended life. On the other hand, the entropy-based damage parameter ( $ω_{E, N}$ in Figure 7), defined based on equation (9), has a nearly linear gradual increase to fracture. In the entropy-based damage parameter, the rate of changes of damage at different stress levels is independent of normalized life ( $N / N_{f}$ ), which means the damage at different stress levels reaches the same values if the expended normalized lives are equal.

Figures 6 and 7 show that quantifying the toughness-based damage ( $ω_{T, N})$ is more sensitive compared to the entropy-based damage ( $ω_{E, N}$ ). As an example, at stress levels of 360 MPa, the damage values at expended life of 50,000 and 100,000 cycles (22.5% and 45% of life) are 0.015 and 0.03 for $ω_{T, N}$ and 0.225 and 0.45 for $ω_{E, N}$ , respectively. This means that even 1 percent of change in $ω_{T, N}$ translates to large difference in estimation of the expended life, which can cause a significant error in the estimated remaining useful life. In contrast, the same amount of error in predicting $ω_{E, N}$ causes only 1% of error in estimated remaining useful life.

Figures 8 and 9 show the iso-damage curves on the temperature-life graph at different stresses. The dashed curves connect the points with equal damage values at different stress levels. Figure 8 indicates that if the toughness-based damage ( $ω_{T, N}$ ) is used, the same damage occurs at different normalized life ( $N / N_{f}$ ). For example, at 360 MPa the damage of 0.05 occurs after 121,000 cycles which is equivalent to 55% of fatigue life, and the same damage occurs after 404,000 cycles at 49% of fatigue life at 340 MPa. While Figure 9 shows that in the entropy-based damage parameter ( $ω_{E, N}$ ), the same damage values are corresponding to the same normalized life. As an example, the damage values of 0.5 occurs after expending approximately 50% of life at 360 and 340 MPa.

Figures 6 to 9 and the explanations presented in this section show that quantifying the damage using entropy-based damage ( $ω_{E, N}$ ) is more convenient and less prone to inaccuracy. $ω_{E, N}$ has a nearly linear gradual increase up to fracture which means the same expended normalized life at different stress levels corresponds to approximately equal damage values. Also, contrary to the $ω_{T, N}$ , the small error in prediction of $ω_{E, N}$ does not cause large differences between estimated and actual remaining useful life. Therefore, in this study, the entropy-based damage parameter ( $ω_{E, N})$ is used to estimate the damage values and corresponding remaining useful life at different fatigue cases.

Remaining life assessment

In this section, we present a methodology for predicting the remaining life of a pre-fatigued specimen. The approach presented is applicable regardless of whether the history of loading is known or not. The concept of continuity of damage is used in this section to find the remaining useful life of a material experiencing fatigue. In this concept, the damage is idealized as a continuous state variable that accumulates until fracture. In this approach, knowing the history of loading is not necessary. The damage in the specimen can be evaluated by conducting a single fatigue test at any of the characterized stress levels presented in previous sections. The following steps are needed to obtain the remaining useful life of a pre-fatigued material:

Measure the temperature rise during the steady-state phase ( $θ_{s}$ ) at one of the specific fatigue stress levels presented in section ‘Evolution of Steady-State temperature’.

Calculate the damage value corresponding to the current condition of the material using equation (7) or (9) (in this section, we use equation (9) for $ω_{E, N}$ , the entropy-based damage parameter, which has some advantages as discussed in section ‘Evolution of damage parameters’. The information provided in Table 3 is used to calculate the damage value).

Obtain the equivalent expended life of the material at any stress level using the damage parameter ( $N_{e q} = ω_{E, N} * N_{f}$ ).

Calculate the remaining useful life (RUL) of the material ( $RUL = N_{f} - N_{e q}$ )

Seven different fatigue cases, three constant amplitudes, and four variable amplitudes, including high-to-low, low-to-high, low-to-high-to-low, and high-to-low-to-high stresses, are presented with experimental verifications to investigate the application of this method. A summary of the results is presented in Table 4.

Table 4.

Summary of the results of remaining useful life (RUL) for different load cases.

Load case	Stress (MPa)	Estimated RUL	Experimental RUL	Error (%)
I. Constant Amplitude	370	26856	24900	7.8
II. Constant Amplitude	360	132484	124800	6.1
III. Constant Amplitude	340	627152	585200	7.2
III. Constant Amplitude	340 (damage evaluation at 360 MPa)	643500	585200	9.9
IV. High-to-Low	370 to 360	74732	74074	0.9
V. Low-to-High	360 to 370	31098	34144	−8.9
VI. Low-to-High-to-Low	360 to 380 to 370	35339	38121	−7.3
VII. High-to-Low-to-High	380 to 360 to 370	40944	44121	−7.2

Case I. Constant loading at 370 MPa stress amplitude (low-cycle fatigue)

We begin by illustrating how the remaining useful life of a fatigued specimen subjected to a constant amplitude load can be estimated. Consider a pre-fatigued specimen that operated for 25,000 cycles at 370 MPa and the fatigue test is stopped, and the specimen cooled down to the ambient temperature. To estimate the remaining life, a single fatigue test at the same stress is carried out to reach the steady-state temperature. The squares in Figure 10 are the same points in Figure 4 at the single stress of 370 MPa obtained via the RSC procedure. The rise of stabilized temperature after 5,000 cycles is 2.1 °C, which is equal to the damage of $ω_{E, N} = 0.562$ based on equation (9) and data provided in Table 3 for $θ_{0}$ and $θ_{f}$ at 370 MP. It means that the remaining useful life of the specimen is 43.8% of fatigue life at 370 MPa. Considering the number of cycles needed to obtain $θ$ , the estimated remaining useful life is 26,856 cycles.

Verification: The experiment was continued to fracture, and results showed that the prediction is slightly higher than the experimental results for this case, 24,900 cycles, with an error of less than 8%. Note that the information on the pre-fatigued history (i.e., 25,000 cycles) was not used in predictions of RUL.

Case II. Constant loading at 360 MPa stress amplitude (mid-cycle fatigue)

Now consider another example where a specimen is subjected to 95,000 cycles at 360 MPa before stopping the fatigue test. When the temperature of the specimen stabilizes, a single fatigue test at the same stress level is carried out. The circles in Figure 11 are the same stabilized temperatures at the single stress 360 MPa in Figure 4. The temperature stabilized after 5,000 cycles with a rise of 1.1 °C. Using equation (9) and data provided in Table 3 for $θ_{0}$ and $θ_{f}$ at 360 MP, the damage is calculated to be $ω_{E, N} = 0.42$ . This suggests that 58% of life is remaining. Considering the number of cycles needed to obtain temperature rise, the estimated remaining useful life is 132,484 cycles.

Verification: To confirm the validity of predictions, the experiment was allowed to run to fracture and the number of cycles to failure was recorded. The obtained results from experiments show that the RUL is 124,800 cycles. Therefore, the error between the estimated and experimental results is less than 7%.

Case III. Constant loading at 340 MPa stress amplitude (high-cycle fatigue)

In high-cycle fatigue, the temperature rise is small, and measuring the exact change in temperature can be difficult. Therefore, two different approaches can be used to find the damage value and the remaining useful life. In the first approach, the damage evaluation process is done at the same stress levels as the fatigue test and in the second approach, a stress level with higher temperature rises is utilized to evaluate the damage. Both approaches are discussed here for a specimen undergoing a high-cycle fatigue test at 340 MPa. The test is stopped after 240000 load cycles and the specimen is cooled down to the ambient temperature.

Using the first approach, a single fatigue test at the same stress level (340 MPa) is carried out. The squares in Figure 12 are the results of steady-state temperature rise shown in Figure 4 for single stress 340 MPa. The stabilized temperature rises 0.4 °C after 5,000 cycles. Based on equation (9) and the data provided in Table 3 for $θ_{0}$ and $θ_{f}$ at 340 MP, this values of temperature rise is equal to the damage of $ω_{E, N} = 0.24$ . Thus, the remaining useful life of the specimen is 76% of fatigue life at 340 MPa, which is equal to 627,152 cycles.

Verification: The experiment was continued until the specimen fractured. The number of cycles to fracture was 585200, showing an error of less than 8% between the experiments and predicted results.

The second approach uses a higher stress level to find the damage value and its corresponding RUL. Here, the stress level of 360 MPa is used as reference stress to evaluate the damage. Figure 13 shows the steady-state temperature rise versus normalized life at 360 MPa. The fatigue test is carried out at 360 MPa and the temperature rises 0.6 °C after 5,000 cycles, which is equivalent to the damage value of $ω_{E, N} = 0.22$ based on equation (9) and the data provided in Table 3 for $θ_{0}$ and $θ_{f}$ at 360 MP. This means that 78% of specimen’s life remains that is equivalent to 643,500 cycles at 340 MPa.

Verification: The error between the predicted and experimental results in this approach is less than 10%.

These results show that the damage is continuous and accumulates during fatigue even if the fatigue stress is changed. It implies that for high-cycle fatigue wherein the values of temperature rise are small, the results of another fatigue stress can be used to determine the RUL. A fatigue test that could induce an adequate temperature rise can be applied to the pre-fatigued component to measure the steady-state temperature and determine the RUL.

Case IV. Variable loading from high-to-low stress (from 370 to 360 MPa)

In this case, after 30,000 cycles at 370 MPa, the fatigue stress is changed to 360 MPa. The circles in Figure 14 correspond to the stabilized temperature at the single stress of 360 MPa in Figure 4. The temperature stabilized after 5,000 cycles with a rise of 1.7 °C, which is equal to the damage of $ω_{E, N} = 0.66$ . This means that 34% of life is remaining. Considering the number of cycles needed to obtain temperature rise, the remaining useful life at the beginning of the second fatigue stress is 74,732 cycles.

Verification: Tests were continued at 360 MPa until fracture. The experimental number of cycles to fracture was 74,074. The estimated RUL is very close to the experimental result, with an error of less than 1%.

Case V. Variable loading from high-to-low stress (from 360 to 370 MPa)

A variable amplitude load is applied to the CS 1018 specimen from a low- to high-stress fatigue. The fatigue load is changed to 370 MPa after 74,074 load cycles at 360 MPa. The square points in Figure 15 are the same results presented in Figure 4 for changes in stabilized temperature at the single stress amplitude of 370 MPa. At the second amplitude load (370 MPa), the temperature change is measured to be 1.8 °C after 5,000 cycles. This is equal to the damage of $ω_{E, N} = 0.477,$ revealing that 52.3% of fatigue life is remaining. Taking into account the number of cycles expended to reach the stabilized temperature and considering the fatigue life at the single stress of 370 MPa (49,900 cycles), the predicted RUL is 31,908.

Verification: The experimental result for RUL is 34,144 cycles, which indicates the error between the prediction and experiment is less than 9%.

Case VI. Variable loading from low-to-high-to-low stress (from 360 to 380 to 370 MPa)

In this case, the specimen is subjected to 40,000 cycles at 360 MPa and 5,000 cycles at 380 MPa, before reducing the stress to 370 MPa. The circle points in Figure 16 are the results of temperature rise at the single fatigue stress of 370 MPa. The temperature rise after 5000 cycles at 370 MPa is measured to be 1.5 °C, which is equivalent to the damage of $ω_{E, N} = 0.392$ . Therefore, considering 5,000 cycles to reach the stabilized temperature, the estimated RUL at the beginning of the third fatigue stress is 35,339.

Verification: The experimental result for RUL is 38,121 cycles. The predicted results are conservative and the difference between the estimated and the experimental results is less than 8%.

Case VII. Variable loading from high-to-low-to-high stress (from 380 to 360 to 370 MPa)

A three-step variable load is applied in this case. The specimen is subjected to 4,000 fatigue cycles at 380 MPa, followed by 38,000 load cycles at 360 MPa before undergoing fatigue stress of 370 MPa. As shown in Figure 17, the temperature rise after 5,000 cycles at 370 MPa is 1.1 °C, equivalent to the damage value of $ω_{E, N} = 0.28$ . Therefore, considering the cycles needed to reach the steady state, the RUL at the beginning of the 3rd step of loading is predicted to be 40,934 cycles.

Verification: The experimental result for RUL is 44,121 cycles, less than 8% error compared to the predicted RUL.

The results of different variable amplitude cases summarized in Table 4 show that the estimated results are in an acceptable margin of error (less than 10%). The negative values for the margin of error belong to the conservative predictions. Contrary to Miner’s rule, which is unable to consider the loading sequence, in this method, the load sequence affects the rate of damage growth and, consequently, the temperature rise. Therefore, the predictions of this method are in good agreement with the experimental ones for both low-to-high and high-to-low-stress fatigue cases.

Conclusions

An experimentally-verified procedure is proposed to estimate the remaining useful life by measuring the temperature rise during fatigue. Two different damage parameters based on toughness reduction and entropy accumulation are developed using the temperature rise during the steady-state phase of fatigue. The comparison between damage parameters shows that the concept of fracture fatigue entropy (FFE) is more useful and more convenient for evaluating the damage state of the material. The toughness-based damage parameter is highly sensitive, and even a small 1 percent miscalculation in its estimation can result in significant inaccuracies when predicting the remaining useful life. Conversely, the entropy-based damage parameter offers greater stability, and its damage growth rate remains consistent regardless of the number of cycles until failure.

The CS1018 specimens are subjected to a repetitive Run-Stop-Cooldown (RSC) procedure to investigate the changes in the temperature rise of the steady-state phase. The results show that the stabilized temperature increases after each pause in the fatigue procedure, and it is a linear function of the number of fatigue cycles. This relation is used to estimate the remaining useful life of CS 1018 without knowing the loading history. The idea of gradual accumulating entropy until fracture is employed to estimate the damage value by leveraging the stabilized temperature. The accumulated entropy is harnessed to evaluate the material's capacity to generate entropy, thereby allowing the prediction of the remaining useful life for components experiencing fatigue.

The application of the method is examined in various loading conditions, including three different constant amplitude and four different variable amplitude cases. The test cases include high-to-low, low-to-high, low-to-high-to-low, and high-to-low-to-high stresses. First, the damage parameter is determined by measuring the temperature rise at a specific loading condition. Then the damage parameter is used to estimate the remaining life using the concept of FFE. The comparison between the estimated and experimental results shows that the method is applicable in both low- and high-cycle fatigue regimes. The method is capable of reliable prediction of remaining life in constant and variable amplitude cases with an error of less than 10%.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Center for Innovations in Structural Integrity Assurance (CISIA) under the US National Science Foundation award number 2052810.

ORCID iD

MM Khonsari

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Entropy-based damage model for assessing the remaining useful fatigue life

Abstract

Keywords

Introduction

Theory and formulation

Thermodynamics of fatigue

Damage parameters for remaining life assessment

Toughness-based damage parameter, ω T , N

Entropy-based damage parameter, ω E , N

Experiments and methods

Material and test equipment

Test procedure

Results and discussion

Evolution of Steady-State temperature

Evolution of damage parameters

Remaining life assessment

Case I. Constant loading at 370 MPa stress amplitude (low-cycle fatigue)

Case II. Constant loading at 360 MPa stress amplitude (mid-cycle fatigue)

Case III. Constant loading at 340 MPa stress amplitude (high-cycle fatigue)

Case IV. Variable loading from high-to-low stress (from 370 to 360 MPa)

Case V. Variable loading from high-to-low stress (from 360 to 370 MPa)

Case VI. Variable loading from low-to-high-to-low stress (from 360 to 380 to 370 MPa)

Case VII. Variable loading from high-to-low-to-high stress (from 380 to 360 to 370 MPa)

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Toughness-based damage parameter, $ω_{T, N}$

Entropy-based damage parameter, $ω_{E, N}$