Degradation: Data Analysis and Analytical Modeling

Abstract

A central element in life-cycle modeling of engineered systems is the appropriate understanding, evaluation, and modeling of degradation. In this chapter we first provide a formal definition and a conceptual framework for characterizing system degradation over time. Afterward, we discuss the importance of actual field data analysis and, in particular, we present a conceptual discussion on data collection. We also present briefly the basic concepts of regression analysis, which might be considered the first and simplest approach to constructing degradation models. Regression analysis will be used later to obtain estimates of the parameters of degradation models. As an example, the special case of estimating the parameters of the gamma process (see Chap. 5) is presented. This chapter is not intended as a comprehensive discussion on degradation data analysis, as this topic has been widely studied in a variety of different research fields, and many tools and procedures are available for modeling degradation data.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

4.1 Introduction

A central element in life-cycle modeling of engineered systems is the appropriate understanding, evaluation, and modeling of degradation. In this chapter we first provide a formal definition and a conceptual framework for characterizing system degradation over time. Afterward, we discuss the importance of actual field data analysis and, in particular, we present a conceptual discussion on data collection. We also present briefly the basic concepts of regression analysis, which might be considered the first and simplest approach to constructing degradation models. Regression analysis will be used later to obtain estimates of the parameters of degradation models. As an example, the special case of estimating the parameters of the gamma process (see Chap. 5) is presented. This chapter is not intended as a comprehensive discussion on degradation data analysis, as this topic has been widely studied in a variety of different research fields, and many tools and procedures are available for modeling degradation data. If the reader is interested, some of the most relevant references with respect to failure data in engineering problems are [1, 2].

Finally, the discussion presented in Chaps. 1–3, which has provided motivation for the study of engineered systems subject to failure, as well as an overview of the mathematical background in stochastic processes, will serve as the foundation for modeling degradation analytically. In the last part of the chapter, and as an introduction to the rest of the book, we provide a conceptual framework for characterizing system degradation over time and define the appropriate random variables that will be used later. We discuss the general properties of progressive and shock degradation mechanisms, which are illustrated with several examples of physical degradation in various engineering fields. This chapter is intended as a conceptual and general discussion of degradation before we present specific analytical degradation models in detail in Chaps. 5 through 7.

4.2 What Is Degradation?

When an engineered system is put into use, physical changes to the system occur over time. These changes may be the result of internal processes, for instance, natural changes in material properties, or external processes, such as environmental conditions and operating stresses. Regardless of the cause, these changes may result, over time, in a reduced capacity of the system to perform its intended function.

We measure the capacity of a system by one or more physical quantities that serve as performance measures, such as the inter-story drift of a building, the vibrational signature of a bridge, or the tread depth of a tire. By the term degradation (or equivalently, deterioration), we mean

the decrease in capacity of an engineered system over time, as measured by one or more performance indicators.

Thus degradation is a process that describes the loss of system capacity over time. We make a distinction in this book between the definition of degradation given above and the actual physical processes that result in the decline in capacity. As noted in [3], what we define as degradation above is in reality only the observable damage produced by a number of different physical processes that may, themselves, be unobservable. For example, in the case of concrete bridge decks, physical changes due to corrosion, cracking and spalling, load related fatigue, and so on [4] occur over time as a result of exposure and system use; the processes related to these phenomena are typically not directly observable. However, these processes all manifest themselves through changes in performance measures , and the latter is what we refer to as degradation. In this sense, theoretical and empirical models of the physical processes that result in system damage are quite valuable (and in some cases, critical) in developing effective models of degradation. Ben-Akiva and Ramaswamy [3] pioneered an approach to this problem using latent variables or processes , a concept that was first introduced in social sciences to model those characteristics that are not easily measurable or directly observable in a population [5]. While several attempts have been made to link the physical changes observed in the system to the system’s capacity to perform its function [3, 6–8], these procedures are generally quite data intensive and suffer from computational limitations; nevertheless, this remains an open and very important problem in all aspects of engineering. However, we will not address this issue directly, and our main concern will be with the characterization of degradation as the reduction of the system capacity over time.

In engineering practice, system capacity is often characterized by an index or rating that is intended to combine a number of performance indicators into a single measure that represents the system state. Examples of such indices include the Present Serviceability Index (PSI) in pavement management, the Utah Bridge Deck Index (UBDI) for concrete bridge deck management, [9–13]. While these indices do serve as a guide for determining whether the system performance at a given time is acceptable, they have little predictive value [14], which is crucial to supporting operational and maintenance decisions. In this book, we will study predictive models for degradation that incorporate inherent randomness due to such factors as material variability, changes in operating conditions, and variable environmental factors.

4.3 Degradation: Basic Formulation

We desire a formulation of system performance over time that explicitly incorporates randomness in the design, manufacture, and operation of the system. To that end, let us assume that a new system (device or component) is placed in operation at time 0, and let $V_0$ be a positive random variable that measures the initial capacity of the system (also referred to as the nominal life). The nominal life of a system is generally determined by the system’s design and manufacturing, and is independent of the operating conditions once the system is placed in service. Let D(t) be a random variable that measures the accumulated degradation by time t, and let V(t) be the remaining capacity of the system at time t. The remaining capacity at time t will simply be the nominal life decreased by the accumulated degradation up to time t, provided the system remains operational at time t; that is,

$$\begin{aligned} V(t)=V_0-D(t), \end{aligned}$$

(4.1)

Conceptually, failure occurs when the remaining life declines to zero; however, for our purposes, it will be useful to define performance states characterized by remaining life falling below a prespecified critical value [15] known as a limit state . Many maintenance and intervention models are based on control-limit policies that call for a particular action once a limit state is entered. A particularly important limit state, which will be widely used in this book, corresponds to a minimum performance level (here designated by $k^{*}$). Once this limit state is reached, the system will be removed from service (see Fig. 4.1), or replaced. We refer to this state as the failure limit state ; even though a structure may still be minimally operational past this state, its continued use will pose unacceptable risks, and for all intents and purposes, it will be considered to have “failed” and will require complete replacement. The selection of $k^{*}$ is usually obtained based on experience; frequently, $k^*=0$ but in some cases it is reasonably to assume that $k^*>0$.

Fig. 4.1

Basic formulation of degradation

Once the limit state $k^*$ has been defined, we can revise our expression for remaining life as follows:

$$\begin{aligned} V(t)=\max (V_0-D(t),k^*). \end{aligned}$$

(4.2)

The system lifetime can then be defined as

$$\begin{aligned} L=\inf \{t\ge 0:V(t)\le k^*\}, \end{aligned}$$

(4.3)

or equivalently,

$$\begin{aligned} L=\inf \{t\ge 0:D(t)\ge V_0-k^*\}. \end{aligned}$$

(4.4)

Note that we can interpret the device lifetime L as the first passage time for the total degradation process $\{D(t), t\ge 0\}$ to reach $V_0-k^*$.

Other limit states may similarly be defined that correspond to acceptable performance levels determined, for instance, by a regulatory agency; i.e., a serviceability limit state . These states may indicate the need for a preventive intervention or maintenance but might not require complete replacement of the system, and again, the intervention times will be determined as first passage times to a limit state.

If the system is systematically maintained (repaired preventively and/or at times of failure), we can define system availability at time t as

$$\begin{aligned} A(t)=P(V(t)\ge k^*), \quad t\ge 0. \end{aligned}$$

(4.5)

Based on models developed to describe nominal life and degradation over time, we are interested in estimating such quantities as:

• the probability distribution of capacity of the system at time t and, if it exists, in the limit as $t\rightarrow \infty$;

• the first passage time distribution for the capacity to fall below a prespecified threshold level; and

• the system availability at time t and, if it exists, the limiting system availability (this is of particular importance in cases where the system is systematically reconstructed—see Chap. 8).

4.4 Degradation Data

This book is concerned with models that characterize system degradation; that is, models that describe the deterioration in system performance over time. To “calibrate” these models, to estimate model parameters and to validate model performance requires the collection of data on actual system behavior. Data collection involves the structured gathering of empirical observations of systems, either under controlled, experimental conditions, or under uncontrolled operating conditions. Because it is often difficult to observe the physical changes that accompany degradation directly and continuously, we often monitor surrogates for these physical changes, or alternately, we may monitor some system performance indicator over time. While our main focus in this book is on model development, in this section we present an overview of the nature, problems, and challenges of collecting and analyzing data to characterize degradation.

4.4.1 Purpose of Data Collection

One of the primary objectives of data collection in modeling degradation is to predict the time until the system reaches particular operational states. The types of data gathered for infrastructure degradation and reliability studies, and the methods used for their analysis, can be generally aggregated into two main directions. The first involves the direct study of the time at which system failure occurs. The vast majority of reliability studies are related to failure time estimation for systems that are replaced upon failure (so-called non-repairable systems ). Common experimental techniques involve placing statistically identical items on test under operating conditions (or accelerated operating conditions) and observing the time of failure of each item. Because not all items may have failed by the end of the study period, failure time studies typically involve censored data. In these cases, precise failure times are not known, but the censored observations provide a lower bound on the actual failure time. This is particularly true in the case of infrastructure components that are designed to be highly reliable and have life spans of several decades. Statistical methods for dealing with censored data have a long history in the field of survival analysis and life testing; some further reading on this topic can be found in [16–18].

When modeling failure times, data is used to estimate parameters of a (positive) lifetime random variable. Common distributions used in modeling time to failure include the exponential, gamma, Weibull, lognormal, and several other less common distributions (e.g., inverse Gaussian , Birnbaum–Saunders, Gompertz–Makeham). A number of references are available for the statistical properties of these distributions, including [1, 2, 19]. For reliability prediction, moment-based parameters, such as the mean and variance of lifetime, are often not of primary interest. Rather, engineers may be more interested in estimating quantiles of the lifetime or (similarly) failure probabilities for given (fixed) mission lengths. The choice of distribution to fit often involves the phase of life that is of interest, as determined by the shape of the hazard function , and many techniques have been developed that address modeling the hazard rate directly as a linear or polynomial function; cf. [20].

A second direction for data collection and analysis in degradation modeling involves situations where actual physical changes that lead to deterioration of system performance can be measured. Examples include material fatigue induced by crack formation and propagation, material removal due to wear or thermal cycling, corrosion, and fracture. If direct measurements of these processes can be made over time, the analyst often has more information available that may allow modeling of the actual failure mechanism. In cases where actual degradation processes are not observable, it may still be possible to observe a performance measure that acts as a surrogate for degradation, for instance, decreasing power output of an electronic device over time. Techniques for modeling degradation paths over time are quite complex, and necessarily employ analytical models of specific physical failure mechanisms. These models generally involve the effects of stressors such as temperature, duty cycle, vibration, humidity on the material properties of a system. In contrast to direct measurement of failure times, these degradation models are often used to predict when the measured degradation (or its performance surrogate) reach a threshold that results in failure. Variability due to the initial material properties (manufacturing process) as well as actual operating conditions leads to the attainment of the failure threshold, and hence this approach can also lead to estimation of the lifetime distribution; some additional information on this approach can be found in [21].

Whether working with failure time observations or with observations of degradation or performance, highly reliable systems and those that are designed for long mission lengths may require accelerated testing . In accelerated testing, the level or intensity of stressors are magnified beyond what normal operating conditions would dictate in order to induce premature degradation or failure. There is a great body of work related to accelerated testing; suffice it to say that the design and analysis of accelerated tests for failure prediction is quite complicated and involves a great deal of engineering judgement.

4.4.2 Data Collection Challenges

As technology evolves toward more precise and less expensive data acquisition systems, modeling the degradation process of engineered systems should become a common practice. Today it is possible to install sensors and smart chips to measure and record data about the system performance over the life of an engineered device. In some areas, this practice belongs to the area of system health monitoring and materials state information. This information is used to carry out real-time monitoring and for prognostic purposes. Thus, the next generation of reliability field data will be richer in information and as the cost of technology drops, cost/benefit ratios decrease and applications spread to different practical problems [22].

Future data will also come from the development of better accelerated tests. These will require new lab techniques and methods to incorporate the main sources of uncertainty that are found in the field like load demands, temperature, humidity, material oxidation, etc., [1]. In this field, scale models and testing facilities such as the geotechnical centrifuge [23] have been used extensively.

Furthermore, the development of analytical tools to replicate actual experimental data is an area of research that is gaining a lot of attention. Frequently, simulations are used in situations where experiments are not feasible for practical or ethical reasons . The main questions associated to this issue are related to the assumptions, the validity and the conditions required for a simulation so that it can serve as a surrogate for an experiment. Thus, simulation techniques should guarantee that the results are as reliable as the results of an analogous experiment [24]. Further discussions on this topic can be found in [25–28].

4.5 Construction of Models from Field Data

The selection of the best degradation model is guided by both field data and some understanding of the mechanical laws that describe the system performance. If there is information about the physics that drive the behavior of the system, the mechanical performance can be expressed in the form of a differential equation, or a system of differential equations with some randomness that can be associated to, for instance, the model parameters (e.g., rate, material properties) [1]. A classic example is the case of fatigue of materials expressed in terms of the crack growth rate; thus, degradation can be described as:

$$\frac{da(t)}{dt}=C\times [\Delta K(a)]^m$$

where C and m are constants, and a(t) is the crack size; and $\Delta {K}$ is the range of the stress intensity factor, i.e., the difference between the stress intensity factor at maximum and minimum loading $\Delta {K} = K_{max}-K_{min}$, where $K_{max}$ and $K_{min}$ are the maximum and minimum stress intensity factor respectively [29]. Another example is the automobile tire wear (wear rate), which is modeled as: $D(t)/dt=C$; where C is a constant. The selection of the best mechanical model depends upon the physics of the problem at hand and it is a topic that is not in the scope of this book.

Sometimes the complexity of the degradation problem makes it hard to find a unique mathematical formulation and the only information available is field data. In these cases, the only option is to make inferences from failure time observations or from data about the system condition at different points in time. The former provides information about the lifetime distribution, while the latter can be used to model and understand the system performance over time; information that can be used later to build a mechanistic model of the degradation process (see Chaps. 5–7). In this section, we will briefly mention the basic concepts of regression analysis, which can be interpreted as the most basic degradation model; literature about regression analysis is abundant, but some useful information can be found in [30, 31].

4.6 General Regression Model

Let us assume that the degradation path of the system consists of a vector of field measurements $\{y_1, y_2, \ldots ,y_m \}$ made at discrete points in time $\{t_1, t_2, \ldots , t_m\}$, which reveal the actual condition of the system. Let us also assume that the system performance is characterized by a model denoted by $D(t)=y'(t)$; e.g., target degradation model (see Fig. 4.2).

Fig. 4.2

Description of the general degradation model

Then, the relationship between actual data and the model at time $t_i$ can be written as:

$$\begin{aligned} y(t_i)=y'(t_i)+\varepsilon (t_i);\quad i=1\cdots m \end{aligned}$$

(4.6)

where $\varepsilon (t_i)=y(t_i)-y'(t_i)$ is a measure of the error (residual) at time $t_i$ and is usually modeled as a random variable normally distributed; i.e., $N(0, \sigma _{\varepsilon })$. The form of $y'(t)$ is obtained from a mechanical model or can be selected arbitrarily. For example, several commonly used models for degradation are shown in Table 4.1; where $\mathbf{B}=\{\beta _0,\beta _1, \ldots \beta _k\}$ is a set of parameters that fully characterize the model. For example, if it has a linear form, $y(t_i, \mathbf{B})=\beta _0+{\beta _1t_i}+\varepsilon (t_i)$. In practice, it is usually assumed that the set of parameters B are independent of $\varepsilon$, and that $\sigma _{\varepsilon }$ is constant [1]. It is important to stress that although frequently a predefined model for $y'(t)$ is selected, occasionally, the form of degradation is unknown and, therefore, nonparametric regression techniques are required to analyze the data.

Table 4.1 Common regression models useful to describe degradation

Due to the inherent variability of the problem, the set of parameters B are uncertain, which leads to possible different degradation paths with the same general trend. For example, Fig. 4.3 shows the measurements of the crack size in a fatigue test of an Alloy-A [32], which is a standard degradation process in materials subjected to repeated loads. In this figure every curve represents the result of a specimen built and tested under the same conditions. It can be observed that there is some important variability in the results.

Fig. 4.3

Fatigue crack data of Alloy-A (Data reported in Lu and Meeker, 1993 [32])

In more complex structures this degradation process is more difficult to evaluate and the uncertainties more difficult to quantify. For example, in the area of asphalt pavements the surface of the top asphalt course is permanently exposed to the combined action of traffic loading and climatic effects. Among the different phenomena affecting the functionality and durability of these materials, asphalt oxidation is recognized in as one of the most relevant weather-related deterioration processes. Oxidative hardening is defined as the process by which the asphalt binder present in the mixture becomes stiffer as a consequence of its chemical reaction with the oxygen present in the air. The main consequence of this chemical process is that the mixture becomes more fragile, which in turn makes it more susceptible to undergo fracture. This is of particular concern during low temperature seasons where this condition can promote the appearance of cracks at the surface, affecting the overall serviceability, functionality, and durability of the pavement structure. Figure 4.4 presents the increase in the expected normalized dynamic modulus (i.e., the increase in modulus with respect to the modulus at the moment of opening the pavement to traffic) during the initial 5 years of a pavement. Note that there is a significant variability, depending upon the construction process, i.e., different compaction levels leading to different air void contents, and on other aspects such as the chemical kinetics that describes the coupled effect of oxidative hardening and the mechanical viscoelastic response of the material [33].

Fig. 4.4

Increase in modulus of asphalt mixtures with different air void content as a consequence of oxidative hardening (modified after Caro et al. [33])

These examples show the importance of quantifying the randomness of B, which is clearly problem-related and can be described by a multivariate normal distribution with mean vector $\mu _\mathbf{B}$ and covariance matrix $\Sigma _\mathbf{B}$ (see Meeker and Escobar [1]). Finally, and for completeness, the analysis should also take into account the set of parameters p that are important to describe the process but are not necessarily random; for instance, the geometry. Then, Eq. 4.6 can be rewritten as:

$$\begin{aligned} y(t_i)=y'(t_i, \mathbf{B}, \mathbf{p})+\varepsilon (t_i);\quad i=1\cdots m; \quad j=1\cdots k \end{aligned}$$

(4.7)

4.7 Regression Analysis

Finding the best degradation model requires identifying the function $y'(t)$ (we drop $\mathbf{p}$ for now) and the parameters $\mu _\mathbf{B}$ and $\Sigma _\mathbf{B}$. Thus, a regression has the following form:

$$\begin{aligned} \mathbb {E}[Y|t]=y'(t,\hat{\mathbf{B}}) \end{aligned}$$

(4.8)

where $\hat{\mathbf{B}}$ is the best estimator of the vector parameter $\mathbf{B}$. For example, for the case of a linear regression: $y'(t, \hat{\mathbf{B}})=\hat{\beta }_0+\hat{\beta }_1 t$. The function $y'(t)$ is obtained by evaluating various models (e.g., see Table 4.1) and selecting the one with the least cumulative error; this error is evaluated as:

$$\begin{aligned} \Delta ^2=\sum _{i=1}^n(y_i-y'_i)^2;\quad i=1,2, \ldots ,n, \end{aligned}$$

(4.9)

where $y'_i$ is the value of the proposed model and $y_i$ the value of the actual data point at time $t_i$ ($i=1, \ldots ,m$ data points). Frequently, the error is also evaluated in terms of what is called the mean square error (MSE) of the regression:

$$\begin{aligned} \text {MSE}=\frac{1}{n}\sum _{i=1}^n(y_i-y'_i)^2;\quad i=1,2, \ldots ,n, \end{aligned}$$

(4.10)

The error term in Eq. 4.7 it is usually assumed to have a constant variance, i.e., $\varepsilon \approx N(0,\sigma _{\varepsilon }^2\,=\,{\text {constant}}$). However, if there is significant variation in the degrees of scatter of the control variable (i.e., data value at an inspection time), the conditional variance of the regression equation will not be constant and $\varepsilon \approx N(0,\sigma _{\varepsilon }^2=q(t))$. In these cases, Eq. 4.9 needs to be evaluated as [31]:

$$\begin{aligned} \Delta ^2=\sum _{i=1}^nw_i(y_i-y'_i)^2;\quad i=1,2, \ldots ,n, \end{aligned}$$

(4.11)

where $w_i$ is a weight assigned to the data such that data points in regions of small conditional variance (i.e., small $\sigma _{\varepsilon }^2$) should carry higher weights than those in regions with larger conditional variance. These weights are assigned inversely proportional to the conditional variance [31]; i.e.,

$$\begin{aligned} w_i=\frac{1}{\sigma ^2_{\varepsilon (t_i)}(y'(t_i))^2} \end{aligned}$$

(4.12)

The estimation of the parameters of the regression, i.e., $\mathbf{B}$ (Eq. 4.7), can be obtained by minimizing $\Delta ^2$ in Eq. 4.9 or 4.11; i.e.,

$$\begin{aligned} \min _\mathbf{B}\sum _{i=1}^n(y_i-y'(t_i, \mathbf{B}))^2;\quad \quad \min _\mathbf{B}\sum _{i=1}^nw_i(y_i-y'(t_i, \mathbf{B}))^2 \end{aligned}$$

(4.13)

This method is usually referred to as the method of least squares . It is important to mention that Eqs. 4.9 and 4.10 should be modified if there is some correlation between the observation times and the data values [34]; however, this is not usually the case in degradation problems.

There is a vast amount of literature available about regression analysis; conceptual discussions on particular aspects as well as specific examples in Civil engineering problems and calculation details can be found in, for example, [31, 34, 35]. In the following two subsections we will briefly summarize some important aspects of linear and nonlinear regression models. The case of multivariate regression models will not be discussed here but the details can be found in [31, 35].

4.7.1 Linear Regression

The case of linear regression $y'(t,\mathbf{B})=\beta _0+\beta _1t$ has been widely studied and the derivation of the estimative for the parameters $\beta _1$ and $\beta _2$ can be obtained using the method of least squares. Let’s consider a sample of observed data pairs of size n, i.e., $\{(t_1,y_2), (t_2,y_2),$ $\ldots , (t_n,y_n)\}$, where, for example, $t_i$ is the time at which the system is inspected and $y_i$ the result of the inspection in terms of a given performance measure. Then, the parameters of the regression equation can be obtained analytically by solving Eq. 4.13 where $y'(t)$ has a linear form:

$$\begin{aligned} \min _\mathbf{B}\sum _{i=1}^n(y_i-y'(t_i, \mathbf{B}))^2=\min _{\{\beta _0,\beta _1\}}\sum _{i=1}^n(y_i-\beta _0-\beta _1t_i)^2 \end{aligned}$$

(4.14)

Then, computing the derivative of Eq. 4.14 with respect to the parameters and equating to 0, leads to (for the case of constant variance) [31]:

$$\begin{aligned} \hat{\beta }_0=\frac{1}{n}\sum _{i=1}^ny_i-\frac{\hat{\beta }_1}{n}\sum _{i=1}^nt_i=\bar{y}-\hat{\beta }_1\bar{t}\end{aligned}$$

(4.15)

$$\begin{aligned} \hat{\beta }_1=\frac{\sum _{i=1}^ny_it_i-n\bar{y}\bar{t}}{\sum _{i=1}^nt_i^2-n\bar{t}^2}=\frac{\sum _{i=1}^n(t_i-\bar{t})(y_i-\bar{y})}{\sum _{i=1}^n(t_i-\bar{t})^2}, \end{aligned}$$

(4.16)

where $\bar{y}$ and $\bar{t}$ are the corresponding sample means, and n is the sample size. Therefore, the least-squares regression equation is:

$$\begin{aligned} \mathbb {E}[y|t, \hat{\mathbf{B}}]=\hat{\beta }_0+\hat{\beta }_1t \end{aligned}$$

(4.17)

4.7.2 Nonlinear Regression

In most degradation problems the functional regression among variables (e.g., time and performance measure) is not always linear; on the contrary, frequently it shows nonlinear trends. The basic idea of nonlinear regression is the same as that of linear regression; the main difference is that the prediction equation $y'(t)$ (Eq. 4.7) depends nonlinearly on one or more unknown parameters. For instance, $y'(t)=\beta _0+t/(1+\beta _1)^2$; also some typical examples are shown in Table 4.1. It is important to stress that the definition of nonlinearity actually relates to the unknown parameters and not to the relationship between the covariates and the response. A comprehensive review of nonlinear regression models and many practical examples can be found in [30, 36, 37].

Frequently, nonlinear regression models are constructed from expressions linear in the parameters. For example (dropping $\mathbf{B}$ for now);

$$\begin{aligned} y'(t)=\beta _0+\beta _1g(t) \end{aligned}$$

(4.18)

where g(t) is a nonlinear function of t. A common model that follows this approximation is the polynomial regression, which can be written as follows:

$$\begin{aligned} y'(t)=\beta _0+\beta _1t+\beta _2t^2+\beta _3t^3+\cdots +\beta _nt^{n} \end{aligned}$$

(4.19)

whose parameters can be computed using the least-squares method described above. Another important example of transforming a nonlinear function into a linear expression is the following: consider the nonlinear function $y'(t)=\beta _0\exp (\beta _1t)$; then, by taken logarithm in both sides we get that $\ln y'(t)=\ln \beta _0+\beta _1t$ and the regression equation can be computed as:

$$\begin{aligned} \mathbb {E}[\ln y'|t]=\ln \beta _0+\beta _1t \end{aligned}$$

(4.20)

Example 4.14

In asphalt pavements, fatigue is a critical failure mechanism. Consider two asphalt mixtures subjected to a standard fatigue test¹ [38] and whose results are shown in Table 4.2.

Table 4.2 Fatigue data of two asphalt mixtures

Based on this information we can construct a degradation model via regression analysis. The fatigue curve can be described by the following equation:

$$\begin{aligned} \log (N)=C-{ m}{\log }(S) \end{aligned}$$

(4.21)

where N is the number of cycles to failure at an stress/strain amplitude S; and C and m are constants to be determined. Note that rearranging Eq. 4.21 we get

$$\begin{aligned} { {NS}}^m=C \end{aligned}$$

(4.22)

which is usually referred to as the $S-N$ relationship. Equation 4.21, is a nonlinear regression, which can be expressed as a linear regression. Note that Eq. 4.21 can be expressed also as: $\log (S)=\alpha -\beta \log (N)$. Then, using the least-squares method, the estimates of the regression coefficients for the first asphalt mixture are $\hat{\alpha }_1=-2.5291$ and $\hat{\beta }_1=-0.2620$; and for the second asphalt mixture: $\hat{\alpha }_2=-2.1199$ and $\hat{\beta }_2= -0.3406$. This leads to the regression degradation model shown in Fig. 4.5. Furthermore, the fatigue formulation, Eq. 4.22, for both mixes becomes:

$$\begin{aligned} NS^{\hat{m}_1}&=\hat{C}_1=NS^{-3.817}=9.653\end{aligned}$$

(4.23)

$$\begin{aligned} NS^{\hat{m}_2}&=\hat{C}_2=NS^{-2.936}=6.224 \end{aligned}$$

(4.24)

where $m=1/\beta$ and $C=\alpha /\beta$.

Fig. 4.5

Asphalt fatigue degradation model based on experimental data

4.7.3 Special Case: Parameter Estimation for the Gamma Process

Data analysis is essential to build any model, and degradation is not an exception. In Chap. 3 we presented the basics of the most important models that we will later develop in more detail in Chaps. 5–7. Among them, there is one particular case that is particularly important, i.e., the gamma process . It is used mostly to model progressive degradation since it is somewhat an improvement over rate-based models (see Sect. 4.9.2). The gamma process will be discussed in more detail in Sect. 5.5.1. In this section, we will present an approximation, described in [39], to find the parameters of the gamma process (i.e., the scale u, and shape v, parameters) from empirical data. For this task, we would present the results obtained by using two main methods: Moment Matching (MM) and Maximum Likelihood (ML) .

The MM and the ML methods can be used also in other models described later such as when obtaining the parameters for phase-type distributions (Chap. 6). Some references will be given when necessary.

4.7.4 Moment Matching Method

Let us define the target degradation model as $D(t)=y'(t)$ (Sect. 4.6). Furthermore, consider that the underlying degradation process is represented by a gamma process (see Eq. 5.50 in Sect. 5.5.1) with scale parameter u and shape parameter v(t). Then we can use the MM method to define the parameters of the gamma process that describe D(t).

The expected value and variance of the accumulated deterioration at time t (i.e., calendar time), D(t), with $t\ge 0$ are:

$$\begin{aligned} \mathbb {E}[D(t)] = \frac{v(t)}{u}\quad \text {and}\quad Var[D(t)]=\frac{v(t)}{u^2}. \end{aligned}$$

(4.25)

The expected deterioration function can take any form depending of the problem at hand; however, as discussed later in Sect. 4.9.2, it is reasonable to assume a power law for the expected deterioration at time t, v(t), [39]; i.e., $v(t)=c t^b$, for some constants $c > 0$ and $b>0$. This kind of relationship is often present in many practical applications [9, 13].

For the particular case in which the exponent b of the power law is known, the nonstationary gamma process can be transformed into a stationary gamma process by making the following time transformation. Since $z=t^b$ then $t=z^{1/b}$ [39], and therefore the expected value and the variance in Eq. 4.25 become:

$$\begin{aligned} \mathbb {E}[D(t)] = \frac{cz}{u}\quad \text {and}\quad Var[D(t)]=\frac{cz}{u^2}. \end{aligned}$$

(4.26)

which result in a stationary gamma process with respect to the transformed time z.

Suppose now that the set $\{y_0,y_1,\ldots ,y_n\}$ are the results from inspections taken at times $\{t_0,t_1,\ldots ,t_n\}$. Then, the transformed inspection times can be computed as: $z_i=t_i^{b}\;\text {with}\; i=0,1,2, \ldots ,n$; and the transformed times between inspections can be defined as $w_i=t_i^b-t_{i-1}^b=z_i-z_{i-1}$. This means that the deterioration increment, $\Delta _i=D(t_i)-D(t_{i-1})$, has a gamma distribution with shape parameter $cw_i$ and scale parameter u for all i. The corresponding observations of $\Delta _i$ are given by: $\delta _i = y_{i}-y_{i-1}$. Then, the estimators $\hat{c}$ and $\hat{u}$ from the method of moments are given by [13]:

$$\begin{aligned}&\frac{\hat{c}}{\hat{u}} = \frac{\sum _{i=1}^{n} \delta _i}{\sum _{i=1}^{n} w_i} = \frac{y_n}{z_n} = \frac{y_n}{t_n^b} \end{aligned}$$

(4.27)

$$\begin{aligned}&\frac{\hat{c}}{\hat{u}^2} \left( t_n^b - \frac{\sum _{i=1}^{n} w_i^2}{t_n^b} \right) = \sum _{i=1}^{n} \left( \delta _i-\frac{y_n}{t_n^b} w_i\right) ^2, \end{aligned}$$

(4.28)

Note that the first equation involves the sum of the observed damage increments, which leads to the total damage observed, i.e., $y_n$, which occurs at time $t_n$ (i.e., total time). In other words, the last observation is enough to fit the first moment, as it contains the information from all the previous damage increments.

4.7.4.1 Maximum Likelihood

The method of maximum likelihood estimates c and u by maximizing the log-likelihood function of the observed damage increments $\delta _i = y_{i}-y_{i-1}$. As these are independent, their joint density can be defined as $f_{\Delta _1,\ldots ,\Delta _{n}}(\delta _1,\ldots ,\delta _{n})$, which is simply the product of the individual gamma densities,

$$\begin{aligned} f_{\Delta _i}(\delta _i)=\frac{u^{\Delta v_i}\delta _i^{\Delta v_i-1}}{\Gamma (\Delta v_i)}\exp (-u\delta _i) \end{aligned}$$

(4.29)

where $\Delta v_i = v(t_i)-v(t_{i-1})=c(t_i^b-t_{i-1}^b)$, for $i=1,\ldots ,n$.

Then, the likelihood of the observed degradation increments takes the form:

$$\begin{aligned} l(\delta _1,\ldots ,\delta _{n} | c,u)&= \prod _{i=1}^{n} f_{\Delta _i}(\delta _i)\nonumber \\&= \prod _{i=1}^{n} \frac{u^{c(t_{i}^b-t_{i-1}^b)}}{\Gamma (c(t_{i}^b-t_{i-1}^b))} \delta _i^{c(t_{i}^b-t_{i-1}^b)-1} \exp {(-u\delta _i)}. \end{aligned}$$

(4.30)

A system of equations is obtained by evaluating the partial derivatives of the log-likelihood function of the degradation increments with respect to c and u. Then, the estimatives $\hat{c}$ and $\hat{u}$ can be solved from [13]:

$$\begin{aligned} \hat{u}&= \frac{\hat{c}t_n^b}{y_n} ,\end{aligned}$$

(4.31)

$$\begin{aligned} t_n^b \log {\left( \frac{\hat{c}t_n^b}{y_n} \right) }&= \sum _{i=1}^{n-1} (t_{i+1}^b-t_{i}^b)\{\psi (\hat{c}(t_{i+1}^b-t_{i}^b))-\log \delta _i\} , \end{aligned}$$

(4.32)

where $\psi (x)$ is the digamma function, defined as the derivative of the logarithm of the gamma function: $\psi (x) = \frac{d \log \Gamma (x)}{dx}=\frac{\Gamma '(x)}{\Gamma (x)}$, and can be computed with a standard software, e.g., MATLAB®. Observe that Eq. (4.31) is the same as the Eq. (4.27) corresponding to the first moment fitting in the MM method.

Note that for the maximum likelihood estimator of u obtained from Eqs. 4.31 and 4.32, the expected deterioration at time t can be written as [39]:

$$\begin{aligned} \mathbb {E}[D(t)]=y_n\left( \frac{t}{t_n} \right) ^b \end{aligned}$$

(4.33)

Example 4.15

The objective of this example is to estimate the parameters of a gamma process using the two fitting methods described above (i.e., MM and ML). In this illustrative example, degradation data are obtained from simulation of a gamma process with shape parameter $v(t)=c t^2$ ($c=0.005$), for $0\le t\le 120$; and scale parameter $u=1.5$. The results are used as if they were actual field data observations, for which the parameters of the gamma process will be obtained.

Fig. 4.6

Observations of the system state of various artifacts taken at times intervals of $\Delta t = 2.5$ years

Thirty sets of data were obtained numerically; this information is assumed to correspond to field data for different artifacts. The thirty degradation data sets were divided in three groups of 10 artifacts each; in each group, data was collected at a specific and fix time interval; i.e., there were three different inspection strategies. The time intervals selected for each strategy are: $\Delta t=\{0.5,1,2.5\}$ years, thus obtaining $n=\{240,120,48\}$ measurements of an artifact condition in each set, respectively. The observed data of five artifacts of the set with $\Delta t=2.5$, are shown in Fig. 4.6.

Based on the previous discussion (Sects. 4.7.4 and 4.7.4.1), and given the form of the shape parameter (i.e., $v(t)=c t^2$), the value of $\hat{c}$ and $\hat{\beta }$ of the gamma process for each artifact data are calculated using both the MM and ML methods. Afterwards, the difference (i.e., error) of the estimative of the parameters for each artifact with respect to the parameters of the actual process, from which experimental data was generated, is calculated as: $\varepsilon _i=(\hat{z}_i-z)\times 100/z$, where z can be either c or $\beta$. Then, the mean relative error was computed for each group, j, of ten artifacts (with observations at the same time interval) as: $\bar{\varepsilon }_j=0.1\cdot \sum _i^{10}\varepsilon _{i,j}$; with $j=1,2,3$ and i the artifact number. The results are shown in Table 4.3.

Note first that, in this particular case, the ML method performs better than the MM method, for all data sets (i.e., smallest $\bar{\varepsilon }$). Although for the first set the errors are quite similar (around $18\,\%$ for $\hat{c}$ and $23\,\%$ for $\hat{\beta }$), they become further apart as the number of data points increase. For instance, for the third data set, the error for $\hat{c}$ in the MM method is $15\,\%$ while in the ML method is $5\,\%$, and the error for $\hat{\beta }$ is $19\,\%$ and $11\,\%$ for the MM and ML method, respectively. In summary, the error diminishes in both methods as more data points are available, but decreases faster for the ML method compared with the MM method. This is expected, as the ML method takes into account the entire density function.

In Figs. 4.7a, b we show various sample paths constructed with the parameters given by the estimators shown in Table 4.4; which correspond to specific artifacts. Besides, the mean deterioration $\mathbb {E}[D(t)]$ from the fitted gamma processes and the mean deterioration of the actual gamma process are plotted. Note that $\mathbb {E}[D(t)]$ of the fitted gamma processes are the same, for both algorithms. This is so, because $\mathbb {E}[D(t)]$ is proportional to the ratio $\hat{c}/\hat{\beta }$, which depends only on the last data point $(t_n, y_n)$ for both algorithms, according to Eqs. (4.27) and (4.31). Note also that for this particular data set, the estimated mean deterioration is greater than the actual mean deterioration.

Fig. 4.7

Degradation sample paths evaluated using the parameters evaluated by (a) MM method; and (b) ML method

Table 4.3 Mean relative error $\bar{\varepsilon }$ (in $\%$) for each data set

Table 4.4 Parameters of the gamma process used to build the sample paths shown in Figs. 4.7a, b

4.8 Analytical Degradation Models

In Sects. 4.4–4.7 we briefly discussed the importance of field data in modeling degradation and presented a first approximation using regression analysis. However, most of this book is concerned with analytical models. Then, in this and the following sections, we will provide a conceptual framework for characterizing system degradation over time and define appropriate random variables that will be used in the subsequent chapters.

4.8.1 A Brief Literature Review

Degradation modeling is challenging because it involves the interaction of environmental conditions with material and other physical properties of the system. There are many approaches available in the literature for modeling physical changes that can result in a reduction of system capacity. These approaches vary depending upon the problem at hand and the scope of the analysis. Physical changes such as crack initiation and growth, material corrosion, material removal, etc., and physical models of these phenomena may be quite detailed. However, it is not always an easy task to identify how these physical changes lead to a reduction in system capacity, which is how we define degradation. Then, in this book degradation models will focus not specifically on physical changes but rather on a more general model of reduction in capacity over time.

In the literature, many models assume that degradation is defined by a functional class with a set of parameters to be determined [13, 40, 41]. There are also models based mainly on the theory of stochastic processes; some examples can be found in [40, 42–44]. Markov processes have been used extensively, see for instance, [45–50]. Recently, a significant amount of research has been carried out based on models that use information obtained at different points in time to reevaluate the predictions about the system performance. Most of these methods include Bayesian probability; see, for example, [51–53]. A review of common probabilistic models for life-cycle performance of deteriorating structures can be found in [11]. Some additional references that may be of interest are [10, 11, 40, 51, 54–58].

To summarize, the literature on degradation modeling spans the spectrum from physical modeling of mechanical and chemical processes through life-cycle modeling of an idealized system state over time. What is clear is that degradation is a general response to the interaction of many different ongoing physical processes within the system. Each of these processes causes physical changes that lead to deterioration in performance. Moreover, some of these processes may be generally independent, while others may have complicated interactions. The reality is that actual physical changes in complex systems are often very difficult to observe and monitor in situ, leading us to embrace a more conceptual notion of degradation that allows modeling of a variety of physical mechanisms.

4.8.2 Basic Degradation Paradigms

Because of the challenges in modeling a variety of physical changes that cause system performance to degrade over time, most degradation modeling asserts two primary degradation classes, namely

• continuous (progressive or graceful) degradation; and

• degradation due to discrete occurrences (shocks).

Conceptually, it is convenient for a variety of reasons to classify degradation in this way. From an observational viewpoint, certain mechanisms, such as corrosion or continuous material removal due to friction or heat, fit naturally within the progressive deterioration category. These mechanisms generally involve very small changes in physical properties that occur continuously over a long timescale. Other changes, such as loss of material due to a sudden collision and disruptions due to failure of a component that may not cause immediate system failure, are more appropriately viewed as shock degradation. Mathematically, the stochastic models suitable for modeling continuous degradation are quite different from those suitable for modeling shock degradation. Because the drivers of progressive deterioration and shocks are typically different (and may be relatively independent), a general mathematical model of degradation can be constructed that consists of a superposition of models for each degradation class (see Chap. 7). In what follows, we provide practical examples and discuss models for both graceful and shock-based degradation separately before presenting a general model that incorporates both classes of degradation.

4.9 Progressive Degradation

4.9.1 Definition and Examples

Progressive degradation, also called graceful degradation, is the result of the system’s capacity/ resistance (life) being continuously depleted at a rate that may change over time. As an example, three realizations of progressive degradation are shown in Fig. 4.8. Note that progressive deterioration may actually consist of a series of discrete damage occurrences, but if the actual damage at any point in time is very small, say

$$\begin{aligned} D(t)-D(t^-)<\varepsilon , \end{aligned}$$

(4.34)

for some arbitrarily small $\varepsilon$ and the timescale is long, we model it as continuous degradation.

Fig. 4.8

Realizations of progressive (graceful) degradation of a system or component

Progressive degradation is generally the result of a mechanical process that may be driven by internal or external system conditions. Some examples of well known, and widely studied, progressive mechanical degradation processes are:

• Wearout of engineered devices is observed in most mechanical devices that have been used for a time period close to their service life (e.g., tire treads or a piston continuously contacting a cylinder). This phenomenon is also observed in pavements of roadways and runways and bridge structures.

• Material fatigue is a degradation process that occurs in devices or structures subjected to repeated loading and unloading cycles. Fatigue leads to microscopic cracks, which frequently form at the boundary (e.g., surface) of the element. Eventually a crack will reach a critical size, and the structure will fracture [59]. Fatigue problems have been widely studied in, for example, aeronautical engineering [60, 61]; and in pavement structures [62, 63].

• Corrosion is the gradual loss of material (primarily in metals) that reduces the component strength or deteriorate its appearance as a result of the chemical reaction with its environment, and it is frequently favored by the presence of chlorides or bacteria. Corrosion may concentrate on specific points forming “pits”, which lead to crack initiation and propagation, or it can extend across a wide area corroding the surface uniformly. Deterioration models of steel structures have been widely discussed. Two cases in point are corrosion in marine environments (offshore structures); e.g., [64–66]; and corrosion in pipelines in [67].

• Degradation of reinforced concrete structures results from a reduction of the structural capacity caused mainly by chloride ingress, which leads to steel corrosion, loss of effective cross section of steel reinforcement, concrete cracking, loss of bond and spalling [68–70].

• Concrete biodeterioration is a consequence of the activity of bacteria that uses the sulfur found within the concrete microstructure, weakening it and increasing porosity; which, in turn, reduces the resistance and favors chloride ingress [71, 72].

• Pavement deterioration may be caused by three main processes: (1) fatigue cracking in asphaltic layers (or other stabilized layers), caused by the repetition of traffic loads, (2) permanent deformation or rutting in unbounded layers (mainly in the natural soil layer or subgrade), and (3) low temperature cracking in the asphalt course layer. Most pavement damage models are empirical and based on experimental data; however, some analytical models have been proposed recently. More information about these mechanisms can be found in [73, 74].

• Moisture damage refers to the effects that moisture causes on the structural integrity of any material. For example, it has been recognized as one of the main causes for early deterioration of adhesives and asphalt pavements. In the particular case of pavements, this phenomenon includes chemical, mechanical, thermodynamical and physical processes, each of them occurring at different magnitudes and rates [75, 76].

4.9.2 Models of Progressive Degradation

Progressive degradation is characterized by a continuous process; that is, loss of system capacity that has the form:

$$\begin{aligned} D(t)=\int _0^t\delta (\tau )d\tau , \end{aligned}$$

(4.35)

where $\delta (t)$ is a degradation rate at time t, measured in capacity units per time unit; for example, the loss of material due to corrosion per year, or the annual increase of concrete porosity due to bacterial activity. The degradation rate over time $\{\delta (t), t\ge 0\}$ may itself be a stochastic process, or the parameters associated with an empirical deterioration law may be assumed to be unknown to reflect the variability observed in a sample of deterioration data [51].

In some cases it may be reasonable to assume a particular mathematical form for the degradation process based on experimental data or physical models, so that degradation may take the following general form:

$$\begin{aligned} D(t)=h(t-t_e) \text { for } t > t_e, \end{aligned}$$

(4.36)

where $t_e$ is usually known as the time to deterioration initiation (e.g., time to corrosion initiation; see, for example, [69, 70]). The function h may take a linear, nonlinear, or any other form based on the problem at hand. It is important to note that the specific form chosen for the function h depends heavily on the physical properties of the specific system at hand (e.g., material characteristics, geometry, environmental conditions). Three examples of these type of models are presented in Fig. 4.9.

Fig. 4.9

Examples of progressive deterioration models; data: $u_0=100$, $\alpha _1=1.25$, $\alpha _2=0.2$, $\alpha _3=0.057,$ and $p=1.5$

In many cases there are abundant data available to justify the form of Eq. 4.36 for specific deterioration processes. For example, [40] reports that many studies use degradation trends following a power form $h(t)=t^b$. For instance, for the expected degradation of concrete due to corrosion of reinforcement $b=1$; for sulfate attack to concrete $b=2$; for the diffusion-controlled aging $b=0.5$ [9]; creep $b=1/8$ [13]; and for scour-hole depth $b=0.4$ [41].

4.9.3 Performance Evaluation

Let us assume that the system starts operating at time $t=0$, and that the initial capacity has a known deterministic value $V(t=0)=V_0=v_0$. Then, the capacity of the system at time t can be expressed in terms of a deterioration rate as:

$$\begin{aligned} V(t)=v_0-\int _0^{t}\delta (u)du \end{aligned}$$

(4.37)

for $t\ge 0$. Note that the rate does not necessarily need to be constant over time. Some examples of degradation based on deterministic time-dependent rates are shown in Fig. 4.10.

Fig. 4.10

Examples of rate-based deterioration models

An overview of random deterioration rate-based models can be found in [11]. If we assume that the minimum acceptable performance threshold is deterministic; i.e., $k^*$, the life of the system, i.e., L, or the time to failure, can be obtained as follows:

$$\begin{aligned} L=\inf \{t>0:\int _0^{t}\delta (u)du=v_0-k^*\}. \end{aligned}$$

(4.38)

Equation 4.38 basically states that the system fails once the capacity available, i.e., $v_0-k^*$, is fully used.

4.10 Degradation Caused by Shocks

4.10.1 Definition and Examples

Shock-based degradation occurs when discrete amounts of the system’s capacity are removed at distinct points in time. Shocks are events that cause a significant change in a system’s performance indicator over a very small time interval. By significant we mean (Fig. 4.11)

$$\begin{aligned} D(t)-D(t-\Delta )>\xi , \end{aligned}$$

(4.39)

where $\xi$ is some arbitrary, positive, “large enough” value and $\Delta$ is some arbitrary, positive, “small enough” value, and we typically compress the time of occurrence of the damage to a single point. Generally, we use shock degradation when the damage that occurs at a particularly point in time is meaningful or observable. The size of the shock that occurs at time t is defined as the discontinuity in the degradation function $D(t)-D(t^-)$. Practically speaking, we may classify deterioration as shock degradation if significant damage occurs continuously but over a very short time interval (as shown in Fig. 4.11).

Fig. 4.11

Realization of a sudden event (i.e., shock)

Shocks are assumed to occur randomly over time according to some physical mechanism, with each shock causing measurable damage to the system. We will denote the occurrence time of the ith shock as $T_i$ and the size of the ith shock as $Y_i$; where,

$$\begin{aligned} Y_i=D(T_i)-D({T_i}^-) \end{aligned}$$

(4.40)

Between the occurrence of shocks, the system state may or may not change continuously. For ease of exposition, in this section and in most of the book we will assume that the system degrades only at times where shocks occur.

Some examples of shock degradation include electrical, mechanical, or infrastructure systems subjected to, usually, unexpected extremely large demands; for example,

• Overcurrent in electronic devices occurs when a conductor experiences a spike in electric current, leading to excessive generation of heat. Possible causes for overcurrent include short circuits, excessive load, and incorrect design. In general overcurrent problems can be considered as shocks. However, in this case, if the failure does not occur (damage to equipment or electrical components of the circuit), the system remains in a condition “as good as new.”

• Earthquake damage occurs when civil infrastructure (e.g., bridges, buildings) is subjected to a sudden acceleration which causes large inertial forces resulting in structural damage. This damage may result in the failure of one of various structural elements leading to the collapse of the structure. Mid-size earthquakes may not cause a collapse, but may cause damage (e.g., loss of stiffness) that accumulates with time reducing the structure’s ability to withstand future events.

4.10.2 Models of Shock Degradation

Shock-based degradation has been used extensively in the literature (c.f. [77]), and several common assumptions are made that lead to different models.

The simplest models assume that the system will be unaffected by any disturbances below a specific threshold. Effectively, a system failure will occur only if the size of a shock exceeds a pre-specified threshold $k^*$ (see Fig. 4.12) [78].

Fig. 4.12

Independent shock-based damage models

If damage does not accumulate, the system will be in one of two states: “as good as new,” $V(t)=V_0$, or in a failed state, $V(t)\le k^*$. Then, the system will fail at the ith shock if

$$\begin{aligned} Y_i>V_0-k^*. \end{aligned}$$

(4.41)

Furthermore, the life of the system L, which is the same as the time to first failure, is given by:

$$\begin{aligned} L=\inf \{t_n: Y_n>V_0-k^*, n=1,2,\ldots \}, \end{aligned}$$

(4.42)

This type of models have been used in modeling the fracture of brittle materials such as glass [79] and the failure of bridges due to overloads. Additional details can be found in [78], and a discussion on the applicability of this model will be presented in Chaps. 5–9.

The independent shock-based failure model given above is too simplistic to incorporate actual physical damage caused by successive shocks, therefore, models in which damage accumulates are generally more realistic. In cumulative damage models, the system is subjected to randomly occurring shocks, and each shock adds a random amount of damage to the damage already accumulated. Here the total degradation D(t) by time t is given by:

$$\begin{aligned} D(t)=\sum _{i=1}^{N(t)}Y_{i} \end{aligned}$$

(4.43)

where N(t) is the number of shocks that have occurred by time t. Note that in many practical applications the time between shocks is also random; therefore, {$N(t), t~\ge ~0\}$ is a random process (a counting process as discussed in Chap. 3). A sample path of this type of process is given in Fig. 4.13 and described in [80, 81].

Fig. 4.13

Damage accumulation as a result of random shocks

In this model, the remaining capacity of the system at time t is given by:

$$\begin{aligned} V(t)=V_0-\sum _{i=1}^{N(t)}Y_{i} \end{aligned}$$

(4.44)

and, as in Eq. 4.38, for a given failure or maintenance threshold $k^*$, the life, L, of the system is obtained by,

$$\begin{aligned} L=\inf \left\{ t>0: \sum \nolimits _{i=1}^{N(t)}Y_{i} \ge V_0-k^*\right\} \end{aligned}$$

(4.45)

Extensive research has been carried out on mathematical models for shock degradation; see for instance [77, 82–93].

4.10.3 Increasing Damage With Time

Increasing damage with time: in this type of model, shocks are independent but not necessarily identically distributed. Thus, the statistical properties of the shock size distribution may increase or decrease with time. This model is very convenient when dealing with the performance of systems where damage accumulates according to the previous state of the system. For instance, in the case of building structures located in seismic regions [95, 96]. Then, every earthquake causes some damage and the effect of the following event depends on the system state at the time of the event.

Two modeling alternatives are available for this type of problems. In the first, the shock size distribution parameters are not stationary; i.e., $Y_i\sim F(\mu (t), \eta (t), \ldots )$. The second option is that damage accumulates according to a function g(Y, V), which should be continuous, nondecreasing in Y (shock size) and nonincreasing in V (system state). Then, if shock sizes, i.e., $Y_i$, are iid and occur at times $t_1,t_2, \ldots$. The degradation caused by shock $Y_i$ is $g(Y_i,V(t_i-))$. Then, the accumulated damage at a given time t can be computed as:

$$\begin{aligned} D(t)=\sum _{i=1}^{N(t)}g(Y_i,V(t_i-)). \end{aligned}$$

(4.46)

where, for instance,

$$\begin{aligned} g(y,v(t_i-))=\beta \frac{y}{v(t_i-)} \end{aligned}$$

(4.47)

Note that in this case, shocks are dependent on the system state [97].

4.11 Combined Degradation Models

Finally, in practice, there are problems that require some variations of progressive and shock models as described in previous sections. Here, we will describe some interesting cases.

4.11.1 Progressive and Shock Degradation

General life-cycle models describe the performance (i.e., degradation) of a system or a component throughout its lifetime. Then, once the system is put in service, damage starts accumulating as a result of progressive degradation or sudden events (i.e., shocks) until it fails. A sample path describing the performance of structural system throughout its lifetime is depicted in Fig. 4.14.

Fig. 4.14

Loss of remaining life as a result of both progressive degradation and random shocks

If the initial capacity of the system is $v_{0}$ and if D(t) describes the degradation function, the capacity of the component by time t can be expressed as:

$$\begin{aligned} V(t)=v_{0}-D(t) \end{aligned}$$

(4.48)

Furthermore, based on the assumption that the structure is subjected to both continuous and sudden damaging events, and that they are independent, the degradation by time t can be computed as:

$$\begin{aligned} D(t)=\int _0^{t}\delta _{p}(u,\mathbf {p}(u))du+\sum _{i=1}^{N(t)}Y_{i} \end{aligned}$$

(4.49)

where N(t) is the number of shocks by time t, $Y_{i}$ is the loss of capacity caused by shock i; $\delta _{p}(t,\mathbf {p}(t))>0$ describes the rate of some continuous progressive degradation process; and $\mathbf {p}(t)$ is a vector parameter that includes all random variables that influence the process. Then, combining Eqs. 4.48 and 4.49, the condition of the system by time t can be computed as:

$$\begin{aligned} V(t)=v_{0}-\left[ \int _0^{t}\delta _{p}(u,\mathbf {p}(u))du+\sum _{i=1}^{N(t)}Y_{i}\right] \end{aligned}$$

(4.50)

and the life of the system requires solving,

$$\begin{aligned} \int _0^{L}\delta _{p}(u,\mathbf {p}(u))du+\sum _{i=1}^{N(L)}Y_{i}=v_0-k^* \end{aligned}$$

(4.51)

for L, if it exists.

4.11.2 Damage With Anealing

Damage with annealing. In some cases the system may recover a certain amount of capacity, $\Delta Y$, after the ith shock and before the shock $i+1$ (see Fig. 4.15). Then, if the system recovers with a function A(Y, t) after a shock of size Y, the accumulated damage (degradation) at any time t within the time interval between the ith and the $(i+1)$th shock is:

$$\begin{aligned} Y_i-A(Y_i,t) \quad \text {for}\quad T_i\le t\le T_{i+1} \end{aligned}$$

(4.52)

where $Y_i$ is the shock size at time i. Therefore, the condition of the system at any time t would be

$$\begin{aligned} D(t)=\left[ \sum \nolimits _{i=1}^{N(t)-1}Y_i-A(Y_i,(T_{i+1}-T_{i}))\right] +[Y_{N(t)}-A(Y_{N(t)},(t-T_{N(t)}))] \end{aligned}$$

(4.53)

where $T_{N(t)}$ is the time at which the N(t) event occurs. Note that the time between shocks is a random variable and therefore N(t) is also a random variable. In an application of this model, Takacs [94] considered the following recovery model: $A(Y_j,(t-T_j))=Y_j\exp (-\alpha (t-T_j))$, where $0<\alpha <\infty$. This type of behavior is common in some materials such as rubber, fiber reinforced plastics, asphalt, steel, and in general in most polymers [94]. Note that this type of behavior is a combined form of progressive and shock-based deterioration. The life of the system in this case can be computed similarly as in Eq. 4.45.

Fig. 4.15

Shock damage accumulation with annealing

4.12 Summary and Conclusions

This chapter presents the fundamentals of degradation modeling. Thus, we first discuss important conceptual issues about the meaning of degradation and the way in which it affects the system’s performance over time. Afterwards, we address the problem of data collection and analysis. It is argued that degradation models should be built based on actual data obtained from field observations of the physical performance of the system. This, however, is not an easy task, specially in the case of systems with expected long lifetimes such as civil infrastructure. Nevertheless, the most basic degradation model can be constructed using regression analysis. Although, this is a natural and common approximation, regression analysis by itself lacks completeness in the estimation of the physical nature of degradation and the uncertainties associated to the process.

We believe that understanding and modeling analytically the uncertain nature of the process is central to built useful degradation models. Then, in this chapter we have also presented the fundamentals of analytical degradation models. In particular we have focused on the formulation behind the two main degradation mechanisms: progressive and shock-based. In every case, we have briefly mentioned some examples of their manifestation in practice and outlined the mathematical formulation. In particular we have focused on explicitly define three aspects: (1) the degradation function, D(t); (2) the condition state of the system at a given time t, V(t); and (3) the life (time to failure) of the system, L. Also a general degradation model was outlined. In all cases various references were provided for the reader to find more detailed applications. The concepts treated in this chapter will be used extensively in the rest of the book.