A thermodynamic entropy-based damage assessment with applications to prognostics and health management

Abstract

To advance the traditional structural integrity and life estimation approach, this article presents an explanation of damage and develops an alternative approach to damage assessment based on the fundamental laws of thermodynamics. This explanation provides a new way to perform prognostics and health management in terms of the irreversible thermodynamic framework. Specifically, it is proposed and demonstrated that entropy generated due to material degradation can be used as an index of damage. It is claimed that the use of an entropy-based damage index offers a unified and more comprehensive measure of damage. This is because energy dissipation associated with failure mechanisms that lead to material degradation involves irreversible processes, resulting in entropy generation, which is physically measurable as the materials degrade. We therefore conclude that entropy generation can be used to assess the degree of damage, the amount of the life of materials expended, and the extent of the life remaining. As an application of the proposed concept, this article discusses the entropic characterization of the corrosion-fatigue degradation mechanism, in which a standard entropy-based damage prognostics and health management approach is employed in integrity assessment and remaining useful life prediction of aluminum 7075-T651 specimens. Using entropy as a thermodynamic state function for damage characterization is shown to be effective in handling the endurance threshold uncertainties for prediction purposes.

Keywords

Structural integrity prognostics and health management entropy-based damage entropy generation corrosion fatigue remaining useful life

Introduction

The definition of damage has traditionally relied on the field variables chosen to describe the anticipated aging or degradation process. For example, for fatigue, damage variables used to describe the expected damage include the ratio of instantaneous to final crack lengths,¹ the linear summation of normalized number of cycles accumulated at different stresses (Miner’s² rule), the accumulation of normalized surface layer stress,³ the cumulative plastic strain ratio,⁴ the linear summation of persistent slip band length ratio,⁵ the accumulation of the endurance limit ratio,^6,7 and the ratio of defect surface area to total cross-sectional area.⁸ For wear damage, the local plastic deformation,⁹ the wear volume,¹⁰ and delamination of composites¹¹ have been used. Moreover, the definition of damage can vary at different geometric scales. For example, in the corrosion-fatigue degradation mechanism, transmission electron microscopy and dislocation modeling studies show that at the nano-scale, adsorbed hydrogen ions localize plastic deformation and lower the energy required for fracture, resulting in degradation.¹² At the micro-scale level, it has been shown that weakness of oxide film for extensive plastic deformation relative to the underlying metal may result in the localization of slip bands and a reduction in near-surface plasticity, ultimately leading to the formation of dislocation structures and voids.¹³ At the meso-scale level, damage constitutes the growth and coalescence of micro-cracks that initiate a crack. At a macro-scale level, damage is defined as the growth and propagation of macro-cracks that result in a final fracture.¹⁴

Difficulties in proposing a consistent definition of damage from the physical and mechanical points of view have compelled some researchers to look for a microscopically consistent definition in the context of the continuum damage mechanics (CDM).^8,15 For example, in CDM, damage, $D$ , as a thermodynamic internal variable is defined as the effective surface density of microdefects

D = \frac{A_{D}}{A_{i}}

(1)

where $A_{D}$ is the damaged surface area and $A_{i}$ is the initial (pre-damage) cross-sectional area. Due to the difficulty of directly measuring the density of defects on the surface or volume of materials, Lemaitre⁸ used the strain equivalence principle to correlate between other measurable properties of material (e.g. variation of elastic strain, module of elasticity, micro-hardness, density, and plastic strain) and damage. However, these different damage indices do not provide a consistent measure of damage that covers both observable and unobservable damages. In a thermodynamic framework, Lamaitre⁸ also utilized the thermodynamic relation between damage rate and dissipation of energy to drive the constitutive equations. He provided a comprehensive discussion on the general methods of deriving the constitutive equations for different degradation processes. The analytical expression of the dissipation potential (or equivalently, damage rate potential) is not of concern in this article. Rather, we propose a general way to quantify damage that is applicable to different degradation phenomena.

As the measurement of damage in conventional definitions is subjective and dependent on the choices of the observable variables that uniquely correlate with and describe the underlying physical processes leading to damage, entropy generation has been proposed by a number of researchers as a more inclusive and comprehensive measure to characterize damage. The underlying principle of this approach is that all damage processes and failure mechanisms share a common feature at a deeper level: energy dissipation. Energy dissipation is a fundamental measure for irreversibility, which in a thermodynamic treatment of non-equilibrium processes can be quantified by entropy generation. Energy dissipative processes can be directly related to thermodynamic entropy, for example, plasticity and dislocations,^16,17 erosion corrosion,¹⁸ wear and fracture, fretting corrosion,¹⁹ thermal degradation,²⁰ and associated failure of tribological components.^21,22

There are two distinct interpretations of entropy for damage characterization. In the statistical mechanics microscopic interpretation of entropy, introduced by Boltzmann,²³ the connection between damage and entropy may be established using equation (2), where the entropy, $S$ , as a macroscopic thermodynamic quantity can be connected with the statistical microscopic quantity, $W$ ,²⁴ describing “the probability that the system will exist in the state it is, relative to all the possible states it could be in”²⁵

S = K_{B} \ln W

(2)

where $K_{B} = 1.38 \times 10^{- 23} J K^{- 1}$ is the Boltzmann constant. The variation of the entropy, $d S$ , may be written as the sum of two terms, $d S = d^{e} S + d^{d} S$ , where $d^{e} S$ is the entropy supplied to the system through transfer of mass and heat and $d^{d} S$ is the entropy produced inside the system. The second law of thermodynamics states that $d^{d} S$ must be zero for reversible transformations and positive $(d^{d} S \geq 0)$ for irreversible transformations of the system with increase in disorder. According to Basaran and colleagues,^25–27 only the internal entropy production leads to degradation of the system. Using this approach and Boltzmann’s definition of entropy, Basaran and colleagues^25–28 and Temfack and Basaran²⁹ introduced a damage variable model as the ratio of probability of a microstructural state, $W$ , to the original reference state probability, $W_{0}$ . They calculated entropy generation and employed the Boltzmann notion of statistical entropy to describe the damage growth function for the Pb/Sn solder alloys. Tucker et al.³⁰ and Chan et al.³¹ also utilized the statistical mechanics³² interpretation of entropy for modeling damage and fatigue fracture initiations and growths in solder joints. In their model, damage is defined as the cumulative distribution function corresponding to the probability of a microstructure state, $W$ , occurring as a function of an entropic measure. However, in all of these models, a small-strain small-displacement assumption is used with application to electronic packaging under isothermal conditions. The macroscopic interpretation of entropy as the second law of thermodynamics offers an alternative definition of damage based on the interesting aspect of the principle of entropy increase. According to the second law of thermodynamics, natural processes evolve spontaneously in one direction only through the increase in entropy.³³ In accordance with the principle of entropy increase, manufactured organized components return to their natural conditions through degradation, which diminishes the reliability of material characteristics to the limit where the components are no longer functional.³⁴

Feinberg and Widom^35,36 made early efforts to describe a thermodynamic-based definition of damage. They developed a thermodynamic characterization of degradation dynamics, which employs the Gibbs free energy as a measure of material or component parameter degradation. According to their model, disorder/damage must increase entropy or reduce thermodynamic energies. The use of Gibbs free energy and its reduction to represent the degradation, however, is only valid when the pressure and temperature of the system remain constant,³⁷ which constrained their model of degradation to isothermal–isobaric conditions (e.g. Gibbs energy-based damage modeling cannot capture degradations due to thermal losses). Furthermore, they assumed that change in the system characteristic results from a log-time aging behavior versus time. This is true only if the damage accumulates at a fixed rate.³⁴ In fact, the emerging field of damage mechanics tracks entropy and thermodynamic energies to quantify the behavior of irreversible degradations including tribological processes such as friction and wear^38–42 and fretting and fatigue^{30,25,43–46} that incline to increase entropy generation of the components or structures. Following this trend, Bryant et al.⁴⁷ established a more general relationship between the damage of systems undergoing irreversible dissipative processes and the associated entropy generation called degradation-entropy generation (DEG) theorem. In their approach, Bryant et al. linked the traditional damage measure based on the observable markers of degradation (e.g. wear volume, crack size, and corrosion mass loss) to the associated entropy generation through a linear relation. Their actual results focused on tribological systems with a single dissipative process of wear.⁴⁷ They concluded that the famous Archard¹⁰ law for measuring wear in fretting wear is the consequence of the DEG theorem. Clearly, connection of the entropy generation to markers of damage requires that the entropy generated exclusively be generated for the specific degradation process that caused the observable marker. While this is theoretically valid, measurement of the entropy that can be exclusively attributed to a marker of damage would be nearly impossible. In another research performed by Sosnovskiy and Sherbakov,⁴⁸ it is mathematically confirmed that entropy generation represents damage metric. They used entropy generation to evaluate damage evolution of materials subjected to fatigue under variable temperature.

Imanian and Modarres⁴⁹ offered an alternative degradation assessment approach to formally describe the resulting damage. Their approach uses a theorem relating entropy generation to the dissipation process via the generalized thermodynamic forces and thermodynamic fluxes. In this theorem, entropy generation due to all degradation processes, including those that act synergistically, describes a normalized damage index. Failure occurs when the total entropy generation reaches a level called entropic endurance. At this level, the corresponding damage index reaches unity whereby cumulative damage reaches a level beyond which the specimen or component reaches a zone of concern (but not necessarily the point of failure that might be interpreted as an observable damage). Examples of the entropic endurance level include a point where it is judged that a small margin to failure might remain. The following attributes describe key properties of the entropic endurance:

Entropic endurance describes the capacity of a unit to withstand entropy.

Entropic endurance of the same units is equal.

Entropic endurance of different units is different.

Entropic endurance can be measured and involves stochastic variability.

Accordingly, if the entropy generation, $γ_{d}$ , monotonically increases starting at time zero from a theoretical value of zero or practically some initial entropy, $γ_{d_{0}}$ , and reaches the entropic endurance value, $γ_{d_{E}}$ , the corresponding damage index reaches unity. As such, damage due to all degradation processes damage can be expressed by the general index

D = \frac{γ_{d} - γ_{d_{0}}}{γ_{d_{E}} - γ_{d_{0}}}

(3)

In this definition, the evaluation of damage is relative to the initial damage, which is not necessarily zero. The initial entropic damage can be calculated using the correlation between the rate of damage and damage at different stages of degradation.⁵⁰

Using the entropy-based damage approach allows analysts to develop the corresponding structural integrity and assess the remaining useful life (RUL) of such structures. This recognition lets us to introduce an entropy-based prognostics and health management (PHM) framework which provides a strong foundation for failure prediction and RUL assessment of critical components and structures. The approach builds on the beneficial characteristics of the entropy-based damage model and conventional data-driven PHM frameworks.⁵¹

PHM approaches provide information about the performance and RUL of components by modeling degradation propagation. The techniques included in the PHM provide warnings before failures happen, reduce life cycle cost of inspection, and improve qualification tests assisted in design and manufacturing. Generally speaking, PHM models^51,52 are developed to combine the knowledge of the past trend and the current state of degradation with an endurance threshold to estimate the RUL.⁵³ Traditionally, these methods rely on the condition of the performance data, which are evaluated by empirically based physics-of-failure (PoF) models^54–58 or data-driven techniques such as machine learning algorithms.^59–61 The PoF models utilize the specific knowledge of products, such as failure mechanisms, material properties, loading profile, and geometry.^54–58 For example, in the PoF modeling framework, the corrosion-fatigue damage process is divided into four stages: (1) the pit nucleation, (2) growth of the nucleated pit, (3) surface crack initiation and its growth into a through-crack phase, and (4) through-crack growth to a prescribed critical length. For each of these stages, researchers have proposed empirical models for the life estimation.^62–69 In these models, the pit growth rate is widely followed by the Kondo⁶² model, whereby the pit remains hemispherical in shape and grows at a constant volumetric rate. The linear elastic fracture mechanism (LEFM) and the elastic–plastic fracture mechanism (EPFM) are often applied for analysis of the crack propagation stage and final instability.^57,65,70 However, such empirical methods are limited to simple failure mechanisms and are hard to model when multiple competing and common-cause failure mechanisms are involved.

The data-driven methods such as neural networks,⁵⁹ decision tree classifiers,⁶⁰ and Bayesian techniques^71,72 can obtain the complex relationship and the degradation trend in the data without the need for the particular product characteristics such as the degradation mechanism or material properties; however, they cannot identify the differences between various failure modes and mechanisms. In data-driven approaches, RUL is typically assessed by estimating damage trend, performing an appropriate extrapolation to the damage trend, and calculating RUL from the intersection of the extrapolated damage to a defined endurance threshold. In these methods, the threshold, such as the mean value of endurance thresholds, or a conservative lower bound of endurance thresholds, is commonly selected deterministically. However, the specification of the endurance threshold has a critical effect on the RUL prediction performance.⁷³ For example, setting a high threshold value can cause a component to fail before its scheduled time. However, a conservative low-threshold value can result in the premature abandonment of the component from the operation.

Using entropy generation as an index of damage can remedy some of the shortcomings of PoF and data-driven approaches in several ways. In comparison with PoF models, which traditionally rely on the dominant failure mechanism, entropy generation provides a consistent and far broader interpretation and measure of damage by incorporating multiple competing and common-cause degradation mechanisms. By its nature, entropy “as a measure of uncertainty”^28,31,74 comprises the variabilities in the microstructure states (e.g. chemical composition phases, point and planar defects, grain orientations, grain boundaries, dislocations, and configurations of micro-cracks and cavities). Therefore, it can provide a better estimate of life in the absence of detailed microstructural observations.³¹ Furthermore, the entropy-based PHM framework utilized in this article enables us to tackle the issue of endurance threshold determination by taking advantage of entropy as a thermodynamic state function independent of the path of the failure (which commonly depends on factors such as geometry, load, and frequency of load) from the initial state to the final failed state of the material, considering a known endurance limit.^34,43

The remainder of this article is organized as follows: in section “Entropy generation and damage,” the necessary thermodynamic background for assessment of entropic damage is presented. Section “Entropic assessment of damage in corrosion fatigue” describes the entropy generation function for evaluating corrosion-fatigue degradation. Section “Entropy-based prognostic framework” proposes an entropy-based PHM framework to predict the RUL of critical components and structures. Section “RUL prediction of AL samples subjected to corrosion fatigue” employs the entropy-based PHM method for RUL prediction of AL 7076-T651 samples subjected to corrosion fatigue, followed by conclusions in section “Conclusion.”

Entropy generation and damage

Using entropy as a broad index of damage, in order to assess damage induced in the system, every fundamental dissipative process active in the system should be identified. Using the conservation (balance laws) of energy, mass, and momentum, along with the Gibbs⁷⁵ relation and under the hypothesis of local equilibrium,^33,37,76,77 the entropy generation, $σ$ , can be written as the bilinear form of generalized thermodynamic forces $X_{i}$ and thermodynamic fluxes, $J_{i}$ , i = 1, …, n^37,49,76 as shown in equation (4). The concept of local equilibrium simply assumes that for a thermodynamic system, intensive variables, such as internal energy, specific mass, and volume, that define entropy and necessarily define the macroscopic state of the system can also be well defined locally, that is, over a small region.⁷⁶ Table 1 lists some of the thermodynamic forces and fluxes for some of the irreversible processes including fatigue, corrosion, creep, and wear

σ = \sum_{i = 1}^{m} X_{i} J_{i}

(4)

Table 1.

Examples of different thermodynamic forces and corresponding fluxes in some dissipative processes.

Primary mechanism	Thermodynamic force, $X$	Thermodynamic flow, $J$	Examples
Heat conduction	Temperature gradient, $\nabla (1 / T)$	Heat flux, $q$	Fatigue, creep, wear
Plastic deformation of solids	Stress, $τ / T$	Plastic strain, ${\overset{\cdot}{ϵ}}_{p}$	Fatigue, creep, wear
Chemical reaction	Reaction affinity, $A_{k} / T$	Reaction rate, $ν_{k}$	Corrosion, wear
Mass diffusion	Chemical potential, $- \nabla (μ_{k} / T)$	Diffusion flux, $j_{k}$	Wear, creep
Electrochemical reaction	Electrochemical potential, $\tilde{A} / T$	Corrosion rate density, $i_{corr}$	Corrosion

Qualification of the local equilibrium hypothesis for slow degradation mechanisms enables the use of equation (4) and allows us to express the evolutionary trend of the total volumetric entropy generation, $γ_{d}$ , representing damage, $D$ , by

D ~ γ_{d} | t = \int_{0}^{t} [σ | X_{i} (u), J_{i} (u)] d u

(5)

We must note that all degradation mechanisms leading to damage are in non-equilibrium states. Therefore, equilibrium thermodynamics, which is only concerned with equilibrium conditions, whereby a system achieves its maximum entropy with zero entropy generation rate, σ = 0, cannot describe the degradation processes which are commonly in non-equilibrium conditions. The local equilibrium hypothesis, however, enables us to deal with non-equilibrium changes from place to place and over time.^34,49 Two consequences of the local equilibrium hypothesis are that entropy “as a damage index” remains a valuable state function even in non-equilibrium situations and that it leads to an explicit expression for the entropy production from which dissipative processes in the bilinear form of thermodynamic forces and fluxes are inferred. However, the hypothesis is not appropriate for describing fast- or high-frequency phenomena such as ultrasound propagation, shock waves, and nuclear interactions.³³

Entropic assessment of damage in corrosion fatigue

Figure 1 shows the setup for an experimental process that involves a component subjected to the corrosion-fatigue degradation mechanism and being electrochemically monitored. The corrosion process occurs on the surface of the specimen under a cyclic load with the following oxidation and reduction reactions for a metallic electrode, $M$

M \leftrightarrow M^{z_{M}^{+}} + z_{M} e^{-}

(6)

O + z_{O} e^{-} \leftrightarrow R

(7)

where $O$ is a certain oxidant in solution, which results in the formation of reduction product, $R$ .⁷⁸

The entropy generation function for the corrosion-fatigue degradation mechanism in terms of the main dissipative processes may be expressed as follows.⁴⁹ Consider a specific control volume: the main dissipative processes during the corrosion-fatigue failure mechanism include heat losses (the first term in equation (8)), electrochemical losses induced by electronic current leakage resulting from activation overpotential (the second to fifth term in equation (8)) and Ohmic overpotential (the sixth term in equation (8)), diffusion losses (the seventh and eight terms in equation (8)), and mechanical losses (the ninth and tenth terms in equation (8)). Equation (8) expresses the contribution of these factors to the entropy generation in a specific control volume as

\begin{matrix} σ & = \frac{1}{T^{2}} J_{q} . \nabla T + \frac{1}{T} (J_{M, a} α_{M} {\tilde{A}}_{M} + J_{M, c} (1 - α_{M}) {\tilde{A}}_{M} + J_{O, a} α_{O} {\tilde{A}}_{O} + J_{O, c} (1 - α_{O}) {\tilde{A}}_{O}) + \frac{σ_{Ω}}{T} + \frac{1}{T} (J_{M, c} E_{M_{c o n c, c}} + J_{O, c} E_{O_{c o n c, c}}) + \frac{{\overset{\cdot}{ϵ}}_{p} : τ}{T} + \frac{{σ_{M}}_{e}}{T} \end{matrix}

(8)

where $T$ is the temperature; $J_{q}$ is the heat flux; $J_{M, a}$ and $J_{M, c}$ are the irreversible anodic and cathodic activation currents for oxidation reaction, respectively; $J_{O, a}$ and $J_{O, c}$ are the anodic and cathodic activation currents for reduction reaction, respectively; ${\tilde{A}}_{M}$ and ${\tilde{A}}_{O}$ are the electrochemical affinity for oxidation and reduction reaction, respectively; $α_{M}$ and $α_{O}$ are the charge-transport coefficients for oxidation and reduction reaction, respectively; $σ_{Ω}$ is the Ohmic dissipative energy; $E_{M_{c o n c, c}}$ and $E_{O_{c o n c, c}}$ are cathodic oxidation and reduction diffusion overpotentials, respectively; $τ$ is the stress tensor; ${\overset{\cdot}{ϵ}}_{p}$ is plastic strain rate; and ${σ_{M}}_{e}$ is the dissipative energy resulting from elastic deformation.

Using this experimental setup, corrosion-fatigue tests were performed in the high-cycle fatigue regime under the following assumptions for quantification of associated entropy generation during corrosion fatigue: (1) the corrosion process occurs close to the equilibrium state i.e. open circuit potential (OCP), and the classical theory of irreversible processes³³ does not deal with situations far from equilibrium; (2) entropy flow due to heat exchange is negligible in accordance with Naderi et al.⁴³ and Ontiveros,⁴⁴ whose results showed insignificant entropy generation due to heat conduction inside the solid under high-cycle fatigue. Although the high-cycle fatigue does not result in significant heat conduction, low-cycle fatigue can experience noticeable thermal heat generation and conduction;⁷⁹ (3) diffusion losses are eliminated since the solution species are assumed to be well mixed; (4) the effects of diffusivity and concentration of hydrogen at the crack surface are excluded from entropy calculation in accordance with Manson’s⁵⁴ study, indicating that above a critical frequency of load cycling (of 0.001 Hz), shorter cycle periods provide less time for diffusion and accumulation of hydrogen; (5) the Ohmic overpotential is ignored due to the placement of the Luggin capillary close to the working electrode; and (6) the corresponding current to OCP records are obtained based on the fit of the sum of two exponential functions to the part of the polarization curve where OCPs vary (i.e. around the second breakdown potential) in the corrosion-fatigue experiments.⁴⁹ According to these assumptions, the amount of entropy generated in each cycle of the corrosion-fatigue experiment was obtained in terms of the area of the hysteresis energy loops resulting from mechanical stress–strain⁸⁰ and the energy resulting from variation of corrosion potential–electrical charge in each cycle of loading and unloading.⁴⁹

Figure 2(a) represents the results of multiple entropic damage tests showing the evolution trend of corrosion-fatigue volumetric entropy for different loading conditions. Markers show the final value of fracture corrosion-fatigue entropy. The results show that the cumulative entropy values at the fracture points remain between 0.7 and 1.5 MJ m⁻³ K⁻¹. Figure 2(b) presents the box plot of the fracture entropies for each loading condition. The slim distribution of entropy to failure data points justifies the ability of entropy in treating the uncertainties associated with microstate variabilities.⁴⁹ Furthermore, it reveals the independence of entropy to the loading condition (i.e. failure path). Existence of the fracture entropy distribution can be interpreted as the presence of uncertainties such as instrumental measurement errors, legitimacy of assumptions considered in entropy evaluation, weak control of the experimental, operational and environmental conditions, and human error. Using $γ_{d_{E}} = 1.05 MJ m^{- 3} K^{- 1}$ as the mean of fracture entropies for training samples described in the next section and $γ_{d_{0}} = 0$ , the normalized cumulative volumetric entropy generation (i.e. damage metric) for AL specimens is shown in Figure 3.

Entropy-based prognostic framework

In this study, we propose an entropy-based PHM framework to estimate the RUL of critical components and structures. This framework, shown in Figure 4, benefits from conventional data-driven techniques and the superior advantages of the use of entropy as a damage index in comparison with common observable markers of damage.

According to this framework, the health assessment and RUL prediction can be implemented in four steps. First, the dissipative processes and associated data in the critical components under aging are determined. These processes and relevant parameters can be identified by the traditional failure modes, mechanisms, and effects analysis (FMMEA), which identifies the potential failure mechanisms for products under certain environmental and operating conditions. The volumetric entropy as a parameter of damage, which includes all the interactive failure mechanisms, is quantified then. The second step is the extraction of the features (e.g. root mean square, mean, variance, kurtosis, crest factor, peak to peak, auto correlation, and cross correlation) related to the monitored variables (e.g. temperature, electrical potential, corrosion current, mechanical stress, and solution concentration) used to measure dissipative processes and consequent entropy value, detection, and isolation of anomaly states by fault diagnosis approaches. The fault diagnosis approaches can include artificial neural network (ANN),^60,81,82 wavelet theory,^83,84 relevance vector machine,⁸⁵ support vector machine (SVM),⁸⁶ order tracking method,⁸⁷ and instantaneous power spectrum statistical analysis.⁸⁸ The third step includes the use of prognostic techniques (e.g. ANN,⁵⁹ fuzzy logic,⁸⁹ wavelet theory,⁸¹ relevance vector machine,⁸⁵ Bayesian methods (e.g. Kalman filter and particle filter (PF)^71,72), regression,⁹⁰ demodulation,⁹¹ time series analysis,⁹² and fusion prognostics techniques⁹³) for entropy-based damage model parameter estimation and prediction purposes.

The fourth and final step is RUL prediction. As discussed earlier, endurance threshold determination is critical in the performance of RUL prediction.^73,94 The entropy-based PHM framework suggested in this study handles the uncertainties related to endurance threshold by taking the advantage of entropy characteristics (e.g. providing the best estimate of degradation evolution and being independent of the failure path as a thermodynamic state function).

RUL prediction of AL samples subjected to corrosion fatigue

This article adapts the entropy-based damage evolution data from Imanian and Modarres’s⁴⁹ study to obtain the RUL of the AL7075-T651 coupons subjected to corrosion fatigue. Tests were carried out at the maximum cyclic stress chosen as 87%, 80%, 70%, 63%, and 57% of the corroded specimen yield stress, the stress ratio of 0.01, and the loading frequency of 0.04 Hz. The stop criterion was the failure of specimens. Figure 5 shows the implementation of the entropy-based PHM framework to the corrosion-fatigue degradation mechanism. The health assessment and RUL estimation of aluminum samples are demonstrated in two steps using two well-known methods in PHM. These steps are offline data analysis, including processing of training data such as extraction of features and classification of training data points to healthy and faulty labels, and online data analysis, including health assessment of test data points, estimation of online degradation model parameters, and RUL prediction. The details of each phase are discussed below.

Offline modeling

Assuming that the corrosion and fatigue are the main dissipative processes, features of strain and corrosion potential variations (i.e. maximum amplitude, root mean square, and skewness) along with energy loop values resulting from stress–strain hysteresis and corrosion potential–electrical charge hysteresis were processed for training samples. Three specimens were selected as training samples from each test condition having a group of five specimens. To evaluate the state of the component’s health, the k-nearest-neighbor (k-NN)⁹⁵ approach was employed.

Selection of k-NN among other classification methods such as ANNs,⁹⁶ SVMs,⁹⁷ and fuzzy classifiers (FCs)⁹⁸ is based on the fact that k-NN needs the least number of assumptions regarding the data. For example, ANN and SVM require to be parametrized, especially if certain assumptions are needed to be made (e.g. if the data is linearly separable in SVM). Nevertheless, in a separate research, authors considered to evaluate the performance of k-NN compared with other classifiers mentioned above, as they can outperform each other based on the data set characteristics.^99,100 The k-NN approach is a non-parametric classification method in which a new data point is classified based on its vicinity to the k-neighboring data points of known classes. In this method, the Euclidean distance is calculated from a new data point to the centroid of the nearest neighbors from each class, and the new data is classified to the shortest centroid distance. The healthy and faulty classes were selected based on the portion of the specimen life. Data from the initial 10%–30% and 70% lifetime defined the healthy and faulty class, respectively.

In k-NN, the selection of k is subject to the type of data. To obtain a proper value of k at which the classification output is stabilized, sensitivity analysis should be performed. In this study, the impact of the presence of noise in the classification was examined in altering the location of the nearest neighbors’ centroid. In the sensitivity analysis, the change in the distance of a test point to the centroid of the nearest neighbors was calculated as the value of k was increased by one at each step, as shown in Figure 6. k was selected such that enough neighbors were selected to produce a stable neighbor’s centroid.

Fault level (FL) is determined by taking a moving average of the k-NN binary output over time with a window of 100 data points. This enables the definition of anomaly detection threshold. An anomaly was acknowledged when the FL > 0.7. Based on the faulty portion of the lifetime, this threshold is defined as a minimum operating requirement level for an item (the item may still be operational, but not satisfactory). An example of a classification output plot is depicted in Figure 7.

Online modeling

By choosing the normalized volumetric entropy parameter as the index of damage to be monitored, the Bayesian updating method can be used to estimate the parameters of the degradation model and to predict age. The Bayesian updating approach provides a general, rigorous method for dynamic state estimation problems. The idea is to build a probability density function (PDF) of the system model states (i.e. the degradation model parameters in this article) based on all available information. PF is a method for implementing a recursive Bayesian filter using Monte Carlo simulations, where in comparison with other approaches (extended Kalman filter, Kalman filter, and unscented Kalman filter), PF provides a better performance when either the system dynamics or noises are nonlinear. Herein, it approximates the damage evolution trend (Figure 3) parameters’ PDF by a set of particles sampled from the normal PDF and a set of associated weights denoting probability masses.⁷¹

As the PF requires models developed from known system behavior, the particles in the PF method are generated and recursively updated by the system states evolution model shown in equation (9), where the random walk model is considered for evolution of degradation trend parameters $a_{0_{n}}$ and $a_{1_{n}}$ , a measurement model depicted in equation (10), and an a priori estimate of the system model states PDF

\begin{matrix} a_{0_{n}} = a_{0_{n - 1}} + ω_{a_{0}} \\ a_{1_{n}} = a_{1_{n - 1}} + ω_{a_{1}} \end{matrix}

(9)

D_{n} = a_{1_{n}} n + a_{0_{n}} + θ

(10)

where $x_{n} = [a_{0_{n}}, a_{1_{n}}]$ is the degradation trend parameter vector at cycle n while its elements are subject to normal distribution error, $ω_{a_{i}} = N o r m a l (0, β_{a_{i}}), i = 1, 2$ , with zero mean and standard deviation, $β_{a_{i}}$ . $D_{n}$ is the entropic damage measurement at cycle n which is subjected to a Gaussian noise, $θ = N o r m a l (0, β_{D})$ , and its progress model is obtained from the linear regression of damage trends depicted in Figure 3, with the average of coefficient of determination, $R^{2}$ is equal to 0.98, estimated from the training samples’ degradation evolution trend regression. Although in this article, both system dynamics and noises are considered linear and Gaussian, respectively, authors consider PF as a general approach that can be used for the future investigation of the effect of different distributions of noises on the RUL prediction performances.

The goal of PF implementation is then to estimate the probability distribution of the degradation model parameters, $a_{0_{n}}$ and $a_{1_{n}}$ , given a series of entropic damage measurements, $D_{n} = [D_{1}, D_{2}, D_{3}, \dots, D_{n}]$ . The PF is implemented by initiating the degradation model parameters by a set of particles, $x_{0}^{i}$ , where $i = 1, 2, \dots, N_{s}$ .

Within the Bayesian framework, the posterior PDF of the system model states, $P_{r} (x_{0 : n} | D_{1 : n})$ , at cycle $n$ can be approximated by

P_{r} (x_{0 : n} | D_{1 : n}) \approx \sum_{i = 1}^{N_{s}} w_{n}^{i} δ (x_{0 : n} - x_{0 : n}^{i})

(11)

where $δ (\cdot)$ is the Dirac delta function; $x_{0 : n}$ and $D_{1 : n}$ are the set of all states and damage measurements up to cycle $n$ ; and $x_{n}^{i}, i = 1, \dots, N_{s}$ , is a set of random samples drawn from $P_{r} (x_{0 : n} | D_{1 : n})$ with associated weights $w_{n}^{i}, i = 0, \dots, N_{s}$ , normalized such that $\sum_{i = 1}^{N_{s}} w_{n}^{i} = 1$ .

Sampling/importance resampling is a commonly used algorithm to attribute importance weight, $w_{n}^{i}$ , to each particle, $i$ , as

w_{n}^{i} = \frac{P_{r} (D_{1 : n} | x_{n}^{i}) P_{r} (x_{n}^{i})}{π (x_{n}^{i} | D_{1 : n})}

(12)

A recursive formulation in order to update the weights can be obtained as

w_{n}^{i} = w_{n - 1}^{i} \frac{P_{r} (D_{n} | x_{n}^{i}) P_{r} (x_{n}^{i} | x_{n - 1}^{i})}{π (x_{n}^{i} | x_{n}^{i}, D_{1 : n})}

(13)

where the importance distribution $π (x_{n}^{i} | x_{n}^{i}, D_{1 : n})$ can be approximated by $P_{r} (x_{n}^{i} | x_{n - 1}^{i})$ , which is an arbitrarily chosen distribution.⁷¹

RUL prediction

As discussed earlier, given a series of measured entropic damage values, $D_{n}$ , the PF technique enables the estimation of the degradation model parameters $(a_{0_{n}} and a_{1_{n}})$ , where in the updating process, $N_{s}$ , samples are used to approximate the posterior PDF of them. Each sample denotes a candidate for the system model state vector, $x_{n}^{i} = [a_{0_{n}}, a_{1_{n}}]$ , $i = 1, 2, \dots, N_{s}$ , so the prediction of damage, $D$ , would have $N_{s}$ possible trajectories with the corresponding importance weight $w_{n}^{i}$ . The lth step ahead of the prediction of each trajectory at cycle $n$ is calculated by

D_{n + l}^{i} = a_{1_{n}}^{i} (n + l) + a_{0_{n}}^{i}

(14)

The estimated PDF of the damage prediction, $h (D)$ , at lth step ahead can be estimated by

h (D) = P_{r} (D_{n + l} | D_{0 : n}) \approx \sum_{i = 1}^{N_{s}} w_{n}^{i} δ (D_{n + l} - D_{n + l}^{i})

(15)

Defining a deterministic entropic endurance which corresponds to endurance threshold level as $D_{f} = 1$ , the RUL probability estimation, $R_{n}^{i}$ , of the ith trajectory at cycle $n$ can be obtained by solving the following equation as

1 = a_{1_{n}}^{i} (n + R_{n}^{i}) + a_{0_{n}}^{i}

(16)

The PDF of the RULs at cycle $n$ can be approximated by

P_{r} (R_{n} | D_{0 : n}) \approx \sum_{i = 1}^{N_{s}} w_{n}^{i} δ (R_{n} - R_{n}^{i})

(17)

However, if the variability in the endurance threshold is considered independent of the deterioration distribution, Abdel-Hameed¹⁰¹ showed that the RUL cumulative density function at cycle $n$ can be derived from the convolution law. Herein, the distribution of the predicted total life is obtained based on the similar approach proposed by Abdel-Hameed¹⁰¹ and Nystad et al.,¹⁰² whereby the variabilities in the endurance thresholds are obtained from the fracture entropy gains of training samples.

Prognostic results

Using the k-NN classification method, anomalies were identified when the test data FL crosses the anomaly detection threshold. Once the anomaly was detected, the PF algorithm was initiated to predict the RULs. The initial values of the degradation evolution trend parameters $(a_{0_{n}} and a_{1_{n}})$ were obtained from the least square regression of degradation trend, using the healthy interval of the data for each test specimen. These parameters were updated by the PF algorithm in each cycle step. Figure 8(a) shows an example of the prediction results. The left-truncated endurance threshold and the PDF of the predicted total life are shown in the red contour plot and blue line, respectively. Figure 8(b) shows the distribution of the degradation trend at the anomaly criteria, resulting from the degradation model parameter’s uncertainties.

The same procedure was applied to the remaining specimens. The percentage of the mean of the predicted RULs and actual RULs and the minimum and maximum absolute value of error between the predicted RULs at different endurance thresholds and actual RULs are shown in Table 2. The error interval falls in the range of 0.1%–25.5%. This implies that using entropy as a measure of damage can handle the uncertainties related to the endurance threshold to a good extent. The performance of predictions, however, can be enhanced by reducing the uncertainties discussed earlier. Other studies such as Nystad⁷³ and Fantoni and Nordlund¹⁰³ suggest statistical solutions to manipulate the endurance threshold uncertainty in predicting the RUL, or using the opinion of experts with expertise within the relevant field to provide useful information about the threshold probability distribution. With the use of entropy as a damage index, however, this study tried to provide a practically scientific approach to control the endurance threshold uncertainty effects on RUL prediction.

Table 2.

RUL prediction results.

Stress (MPa)	Actual RUL (% of life)	Mean of estimated RULs (% of life)	Error between actual RUL and mean of estimated RULs (% of life)	Minimum error between actual RUL and estimated RULs (% of life)	Maximum error between actual RUL and estimated RULs (% of life)
405	32.3	37.1	4.8	1.2	21.5
405	32.8	28.3	4.4	1.9	12.8
365	11.0	11.7	0.7	0.1	23.9
365	23.3	20.2	3.1	0.3	23.3
330	26.3	17.8	8.5	0.2	13.3
330	30.8	30.3	0.5	0.1	11.5
330	23.2	8.15	15.4	1.4	23.2
295	31.6	42.5	10.9	2.5	25.5
295	30.0	15.4	14.6	0.1	24.1
265	28.9	33.6	4.7	1.2	24.2
265	31.5	35.7	4.2	0.2	15.7

RUL: remaining useful life.

Conclusion

This article discusses an entropy-based definition of damage and its uses in a PHM approach to estimate RUL prediction of critical components and structures within the irreversible thermodynamic framework. The suggested approach not only offers a science-based foundation for PHM methods but also benefits from the superior advantages of the use of entropy as a damage index, in contrast with the common observable markers of degradation. Entropy as a measure of damage provides a unified non-empirical model of degradation evolution that can capture the effect of multiple competing and common-cause failure mechanisms. Entropy as a measure of uncertainty can provide the best estimate of the degradation evolution trend in the absence of microstructural observation. Furthermore, entropy as a thermodynamic state function is independent of the failure path and is capable of handling the uncertainties related to endurance thresholds.

The proposed prognostic approach was demonstrated by modeling the corrosion-fatigue degradation mechanism and tested in a laboratory environment. The thermodynamic entropy generated in the corrosion-fatigue degradation mechanism was then used to predict the RUL of AL 7075-T651 samples. Combining entropy-based damage theory with real experimental data and the standard PHM framework, we showed that the structural integrity of complex failure mechanisms such as corrosion fatigue can be evaluated scientifically, experimentally, and statistically. Furthermore, using the entropy as a degradation measurement enables us to handle the uncertainties associated with the endurance thresholds and to efficiently predict RUL. Although the RUL prediction results support the adequate performance of the entropy-based damage model, the advantages of this method remain to be explored for different materials, various operational conditions (e.g. different loadings, loading frequencies, loading ratios, and activation overpotentials), different geometries, various environmental conditions (e.g. different temperatures, corrosive solution pHs, solution flow rates, and radiations), and other failure mechanisms.

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 was a part of ongoing research through grant number N000141410005 from the Office of Naval Research (ONR).

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