## Abstract

The science base that underlies modelling and analysis of machine reliability has remained substantially unchanged for decades. Therefore, it is not surprising that a significant gap exists between available machinery technology and science to capture degradation dynamics for prediction of failure. Further, there is a lack of a systematic technique for the development of accelerated failure testing of machinery components. This article develops a thermodynamic characterization of degradation dynamics, which employs entropy, a measure of thermodynamic disorder, as the fundamental measure of degradation; this relates entropy generation to irreversible degradation and shows that components of material degradation can be related to the production of corresponding thermodynamic entropy by the irreversible dissipative processes that characterize the degradation. A theorem that relates entropy generation to irreversible degradation, via generalized thermodynamic forces and degradation forces, is constructed. This theorem provides the basis of a structured method for formulating degradation models consistent with the laws of thermodynamics. Applications of the theorem to problems involving sliding wear and fretting wear, caused by effects of friction and associated with tribological components, are presented.

## 1. Introduction

Manufacturing transforms nature's raw materials into highly organized finished components, reduces entropy and increases thermodynamic energies. Ageing or degradation, which tends to return these components back to their natural state, must increase entropy and reduce thermodynamic energies (Feinberg & Widom 1995, 1996, 2000), to be consistent with the laws of thermodynamics. This recognition led to the creation of a thermodynamic degradation paradigm, wherein experimentally it was shown that degradation in the form of wear correlated with ‘entropy flow’ (Doelling *et al*. 2000) produced by accompanying irreversible processes occurring at the wearing surface.

At present, most degradation models are collections of heuristic equations, with little in common. As noted by Ludema (1995), tribology literature contains a preponderance of wear mechanisms each based on many different physical principles, often expressed in terms of many different variables. These phenomenological models are limited primarily to the specific system being examined. Additionally, friction and wear—manifestations of the same interfacial physics—are often treated as separate phenomena. Proper assessment of these phenomena, leading to a fruitful generalization, requires a fundamental unifying principle.

In general, degradation processes involve different mechanisms with distinctive features, types, rates and sequences of dissipative processes. Nevertheless, to elicit permanent and irreversible changes to bodies, at least one of the processes *must* be dissipative. The crux of this article is that all types of permanent degradation are irreversible processes, which disorder a system and generate irreversible entropy to agree with the second law of thermodynamics. Therefore, quite naturally, entropy can be used in a fundamental way to quantify the behaviour of irreversible degradation, including tribological processes such as friction and wear (Klamecki 1980*a*,*b*, 1982, 1984; Zmitrowicz 1987*a*–,*c*). For sliding wear, this irreversible entropy is generated by irreversible dissipation associated with non-conservative friction forces. Similarly, other forms of degradation such as fretting (Dai *et al*. 2000) and fatigue (Bhattacharya & Ellingwood 1998) of machine components are consequences of irreversible processes that tend to add disorder to the system. Disorder increases with time until a critical stage is reached whereby failure occurs. Simultaneous with the rise in disorder, entropy monotonically increases. Indeed, the emerging field of damage mechanics tracks entropy and thermodynamic energies (Lemaitre 1985; Bhattacharya & Ellingwood 1998, 1999; Voyiadjis 1999). Thus, entropy and thermodynamic energies offer a natural measure of component degradation. Dissipative processes can be directly linked to thermodynamic entropy, or associated thermodynamic energies, e.g. plasticity, dislocations (Weertman & Weertman 1964), wear, fracture (Knott 1973; Maugis 2000), fatigue (Izumi *et al*. 1981; Mura 1987), fretting, corrosion (Uhlig & Revie 1985), thermal degradation and associated failure of tribological components. We note that Feinberg & Widom (1995, 1996) related material or component parameter degradation to Gibbs free energy and predicted changes in the system characteristic results with a log-time ageing behaviour versus time.

Examples of dissipative irreversible processes (Van Wylen & Sonntag 1968; De Grote & Mazur 1984; Bejan 1988) include viscous dissipation in fluids, unrestrained expansion of gases, mixing of different chemical species, fracture and plastic flow in solids (Green 1955; Collins 1980; Fahrenthold & Venkataraman 1993, 1994) and thermally or chemically induced changes of materials. The change in irreversible entropy δ*S*′ can be calculated via the theory equations (2.9*a*)–(2.9*d*) with the first law equation (3.1), if the work, heat or energy dissipated is known. For fluids, this is the dissipation function (Keller 1976; Bejan 1988). For plastic dissipation, this is the plastic work (De Grote & Mazur 1984). For dry or poorly lubricated friction and wear involving metals, work dissipated by rubbing involves work of plastic deformation (Van Wylen & Sonntag 1968; Klamecki 1984) and other dissipative effects (Robbins & Krim 1998). In fact, Tsuya (1976) observed for copper sliding against steel, intense plastic deformations that occur in a 10–50 μm thick skin layer on the softer copper, and attributed plastic dissipation in this ‘severely deformed region’ as a principal mechanism of friction force. Rigney & Hirth (1979), Heilmann & Rigney (1981), Rigney *et al*. (1984) and Rigney & Hammerberg (1998) equated the friction energy to the plastic work dissipated in this region, and obtained good agreement with Tsuya's measured friction force. Kennedy (1989) measured the near-surface deformations due to sliding using microscopic observations of the contact region and compared these values with values predicted by his analytical model. In the analytical model, finite-element viscoplasticity techniques were developed to model high rate plastic strains in the vicinity of a moving contact. Kennedy (1989) suggested that the thickness of the severely deformed region for the loading conditions in his experiment was less than 300 μm. The thickness, however, is dependent on the load and speed in the experiment. Suh (1986) estimated the friction force for sliding of metals via mechanics of asperity adhesion, shearing and ploughing, and adhesive wear and abrasive wear by ploughing. These mechanisms assumed irreversible plastic deformations coupled friction forces and energy dissipation. On this, Klamecki's (1980*a*,*b*, 1982, 1984) thermodynamic analysis of friction and wear applied conservation of energy, mass and entropy for bodies in sliding contact. Dai *et al*. (2000) analysed the entropy production associated with fretting wear. Zmitrowicz (1987*a*–,*c*) balanced mass, momentum, energy and entropy between bodies in contact, to predict friction and wear.

This paper develops a thermodynamic characterization of degradation dynamics, which employs entropy, a measure of thermodynamic disorder, as the fundamental measure of degradation. This assumption was implicit in Feinberg & Widom's (1996, 2000) analyses with thermodynamic energies, and explicit in Doelling *et al*. (2000) and Ling *et al*. (2002). In this paper, we establish a general relationship between the degradation of systems undergoing irreversible dissipative processes, and the concomitant entropy produced. Specifically, we link components of material degradation and production of thermodynamic entropy to the irreversible dissipative processes that drive the degradation. Through the dissipative processes, we relate the rate of degradation to production of irreversible entropy, encapsulate the formulations into an equation, and finally summarize the results into a degradation-entropy theorem. An example is presented that illustrates the application of the theorem to a problem involving degradation by wear caused by rubbing friction during sliding.

## 2. Formulations

### (a) Definitions

A *dissipative degradation process* *p*_{i} is the minimum group of physical sub-processes for which the entropy production produced by the group is non-negative, i.e. .

A *degradation mechanism* is a sequence of one or more dissipative degradation processes *p*_{i} (*i*=1, 2, …, *m*) that degrade or impair the functionality of a material or material body.

A *degradation measure* *w*=*w*{*p*_{1}, *p*_{2}, …, *p*_{i}, …, *p*_{m}} is a non-negative, non-decreasing continuous function, differentiable in one or more variables *p*_{i}, associated with a degradation mechanism.

### (b) Thermodynamics of degradation

Suppose a degradation mechanism consists of *i*=1, 2, …, *n* dissipative processes *p*_{i}, where each describes an energy, work or heat characteristic of the process, and depends on a set of time-dependent phenomenological variables , *j*=1, 2, …, *m*_{i}. To accumulate the effects of the processes on overall degradation or ageing, let a degradation measure(2.1)which depends on phenomenological variables via the *n* processes *p*_{i}. Any dissipative process *p*_{i} *must* produce an irreversible entropy , characterized by the same set of variables . Here, the prime denotes irreversible entropy, subscript *i* references specific process *p*_{i} and superscript *j* indicates the phenomenological variable of process *p*_{i}.

In accordance with the second law, the degradation mechanism must generate total irreversible entropy(2.2)The second law mandates non-negative entropy generation *and* the sum over the dissipative processes. The rate of degradation d*w*/d*t* can be determined by applying the chain rule to equation (2.1),(2.3)The rate of entropy d*S*′/d*t*, which is entropy generation, via equation (2.2) and the chain rule is(2.4)In equations (2.3) and (2.4), and denote entropy generation and degradation rate contributions arising from *specific* process *p*_{i}. For stationary systems or systems near equilibrium, irreversible thermodynamics (Prigogine 1967; Bejan 1988; Kondepudi & Prigogine 1998) expresses entropy generation as the product of generalized forces and generalized rates or flows . We note in equation (2.3), . Comparing sums in equations (2.4) and (2.3),(2.5)Since the second law mandates non-negative entropy generation, signs in equation (2.4) of multiplying factors must be identical, i.e. . Equations (2.3) and (2.4) share rate factors . Irreversible thermodynamics considers forces as drivers of flows . Each can depend on all forces (De Grote & Mazur 1984) and intensive quantities (e.g. temperature *T*) associated with the dissipative process via (Bejan 1988)(2.6)where subscript *i* was dropped in equation (2.6) for clarity. For systems near equilibrium or stationary, relation (2.6) is invertible (Hillert & Agren 2006) and usually assumed linear (Bejan 1988). Non-negative entropy production demands symmetric, positive definite *L*_{qj}. Applications of equations (2.4) and (2.5) have explained diverse phenomena—thermoelectric effect, diffusion (Bejan 1988), reactions (Kondepudi & Prigogine 1998) and phase changes, amongst others—and have given an alternate derivation of Kirchhoff's voltage law for resistive networks (Županović *et al*. 2004).

### (c) Degradation force and degradation coefficient

Equation (2.4) can be constructed via principles of thermodynamics (Prigogine 1967; Bejan 1988; Kondepudi & Prigogine 1998). Since equations (2.3) and (2.4) share, an entropy production analysis (obtained by constructing equation (2.4)) can elucidate the rates associated with degradation equation (2.3). In analogy to generalized thermodynamic force , we call the ‘generalized *degradation force*’ and define the *degradation coefficient*(2.7)which exists, since *except when the system is not degrading*. Equation (2.5) was used in equation (2.7). The last term in equation (2.7) means ∂*w*/∂*S*′ with process *p*_{i} active, which suggests *B*_{i} measures how entropy generation and degradation interact on the level of processes *p*_{i}, rather than process variables .

Since increments of degradation are non-negative, coefficients *B*_{i}≥0. Finally, if equations (2.5) and (2.7) are applied to equations (2.3) and (2.4), then(2.8)

### (d) Degradation-entropy generation theorem

The preceding formulations can be summarized into the following theorem:

*Degradation-entropy generation theorem*. Given an irreversible material degradation consisting of *i*=1, 2, …, *n* dissipative processes characterized by a set of time-dependent variables , *j*=1, 2, …, *m*_{i}. Assume the degradation can be described by a degradation measure, equation (2.1). Then

the degradation rate is a

*linear combination*of the components of entropy production of the dissipative processes*p*_{i},the degradation components proceed at rates determined by the entropy production of the dissipative processes

*p*_{i},the generalized ‘degradation forces’ are

*linear functions*of the generalized ‘thermodynamic forces’ , andthe proportionality factors

*B*_{i}are the degradation coefficients given by equation (2.7).

Integration of item (i) of the previous theorem yields a corollary to the theorem: the total amount of degradation is a linear combination of the total components of entropy produced by the dissipative processes *p*_{i}.

### (e) Conservation of energy

All systems must obey conservation of energy, stated by the first law of thermodynamics(2.9a)where *E* is internal energy; *Q* and *W* are heat flow and work across the boundary of the relevant control volume; and *η*_{k} and *N*_{k} are chemical potential and number of moles of species *k*. A change in entropy,(2.9b)consists of a reversible change d*S*_{e} from entropy flow, and an irreversible change from entropy generation d*S*′. We note that entropy flow arises from heat transfer via heat flow d*Q* and matter flow ,(2.9c)A change in the number of moles,(2.9d)consists of a chemical reaction term d′*N*_{k} and a matter transport term d_{e}*N*_{k}. At equilibrium, a system's entropy is maximum and entropy production ceases: d*S*′/d*t*=0 (Kondepudi & Prigogine 1998). A system produces entropy (d*S*′/d*t*>0) until equilibrium. Stationary systems have d*E*=0 and d*S*=0.

Our interest is the irreversible effects of dissipation caused by work of non-conservative forces. This dissipated work must eventually diffuse through heat flow d*Q* and/or mass flow , as an entropy flow d*S*_{e}/d*t*. Via equation (2.9*a*)–(2.9*d*), the irreversible entropy produced can be linked to the work dissipated (Frederick & Chang 1965; Klamecki 1984; Bejan 1988). Open systems demand balancing flows of entropy, heat, work, energy and mass over a control volume about the degrading body or system, to construct open system counterparts of entropy generation equation (2.4), and possibly degradation equation (2.3).

### (f) Degradation analysis procedure

The preceding formulations suggest an approach for degradation analysis.

From knowledge of the degradation mechanism, list the irreversible processes and variables . Often, the process

*p*_{i}can be energy dissipated, or may be posed in terms of lost work, heat transferred or a thermodynamic energy such as internal energy or Gibbs free energy.Obtain entropy generation d

*S*′/d*t*of equation (2.4) via irreversible thermodynamics. Historically, this has involved applying laws of thermodynamics to a control volume about the body in question. Klamecki (1980*a*,*b*, 1982, 1984) presents examples relevant to tribology.Using equation (2.3), obtain an expression for the degradation rate d

*w*/d*t*.Via equation (2.4), obtain process rates , found in both equations (2.3) and (2.4).

Via equations (2.3), (2.4) and (2.7), get thermodynamic forces and degradation forces . By measuring coefficients

*B*_{i}of equation (2.7), can be related to . For and*B*_{i}, if process*p*_{i}is an energy dissipated, then definition of entropy suggests ∂*S*′/∂*p*_{i}=1/*T*_{i}, where*T*_{i}is a temperature.

If needed, via equation (2.6), relate rates to thermodynamic forces .

## 3. Applications

### (a) Single variable systems

In some systems, degradation *w*=*w*{*p*(*ζ*)} involves principally a single process *p*(*ζ*) with but one phenomenological variable *ζ*. Equations (2.3) and (2.4) yieldwhere *J*=d*ζ*/d*t*, *X*=d*S*′/d*p*(d*p*/d*ζ*) and *Y*=d*w*/d*p*(d*p*/d*ζ*).

The degradation coefficient (see equation (2.7)) can be obtained from a ratio of equations (2.3) and (2.4) to relate *Y* to *X*. If *w* and *S*′ are measured at common times *t*, thenThen equation (2.3) gives(3.1)Besides *ζ*, irreversible thermodynamics suggest that *B* can depend on intensive variables such as temperature *T* (Bejan 1988). For single variable systems, therefore, equation (3.1) suggests a direct relation between degradation rate and irreversible entropy production.

### (b) Degradation wear due to friction rubbing

Wear in bodies in sliding contact is measured as volume *w*_{v} of material lost. A counter body moves beneath and rubs against a fixed wearing body. The counter body applies friction force *F*_{μ} directed along the counter-body motion onto the wearing body. Consider a ‘wear control volume’ enclosing the sliding surface and near interior regions of the wearing body. The friction force *F*_{μ} through distance d*x* does work d*W* on the control volume, i.e.where *μ* is the friction coefficient and *N* is the normal load. The minus sign on d*W* indicates work done on the system from outside the control volume.

This work is dissipated within the control volume. For sliding of ductile metals, Rigney & Hirth (1979) identified the dominant dissipative process *p* to be work of plastic deformation. Let us assume that (i) rubbing is at steady speed and force, and is a stationary process. In other words, for steady-state sliding, changes to the internal energy d*E* and entropy d*S* contained within the small wear control volume that surrounds the sliding surface and near interior regions of the wearing body are small. (ii) Energy transport effects of material loss are small, is negligible compared with other terms in equation (2.9*c*), thus permitting us to treat the system as closed. (iii) There are no chemical reactions occurring. (iv) All friction work is dissipated within the control volume, i.e. d*p*=−d*W*. Applying assumptions (i)–(iv) to equation (2.9*a*)–(9*c*) yieldsThe rate of irreversible entropy generated due to friction can be obtained as follows:(3.2a)where contact temperature *T* arose in equation (3.2*a*) from ∂*S*′/∂*p*=(d*S*′/d*W*)(d*W*/d*p*)=1/*T*. For d*S*′/d*t*≥0, since *T*≥0, *F*_{μ} and d*x*/d*t* must have the same signs in the relevant free body diagram for the slider; this is consistent with the friction force always opposing the direction of the sliding velocity.

Comparing equation (3.2*a*) with equation (2.4),(3.2b)*ζ*=*x*, d*p*/d*x*=*F*_{μ} and thus(3.2c)Also, *J*=*J*(*X*) by equation (2.6) is consistent with the observed friction force dependency on sliding speed and temperature.

Applying equations (3.1) or (2.3) yields(3.2d)where *w*_{v} represents wear volume and *Y*=*BμN*/*T*.

### (c) Archard's wear law

One of the most commonly used relationships for assessing wear in a friction pair is due to the work of Archard (1953, 1980). It states: , which relates *w*_{v} to *N* and *x*, via wear coefficient *k* and hardness *H* of the softer of the material pair.

With load and temperature constant, the time rate of wear can be obtained by differentiation of Archard's law as follows:(3.2e)Comparing equation (3.2*e*) with equation (3.2*d*),(3.2f)Constant *B* can be measured via equation (2.7), i.e. *B*=d*w*/d*S*′, using d*S*′=−d*S*_{e} for a stationary process.

Doelling *et al*. (2000) conducted extensive experimental measurements of wear in an apparatus where a stationary copper specimen was in boundary lubricated contact with a rotating steel cylinder. They reported a series of measurements to establish relationship between normalized wear as a function of normalized entropy flow. The entropy flow, the negative of entropy production for a stationary process, was calculated using , where Δ*Q*^{(l)} is the heat input to the slider during the *l*th time interval and *T*^{(l)} is the corresponding average absolute surface temperature of the stationary rider on the rotating cylinder. Both temperature and wear rates were measured and the slope of normalized wear plotted versus the normalized entropy flow was evaluated. This slope is precisely the degradation ratio *B* defined in §2*c*, equation (2.7).

Having determined *B*, equation (3.2*f*) was used to estimate the wear coefficient, *k*. The average of a series of repeatable tests yielded *k*=1.01×10^{−4}. For same metals under poor lubrication, Rabinowicz (1980) gives *k*=1.0×10^{−4}. Note that the *k* estimated by equation (3.2*f*) arose from measured wear, temperatures and forces; Rabinowicz's *k*, calculated via Archard's wear law (1953), arose from measured wear, forces and distance. Additional tests show similar results with remarkable accuracy when compared to published wear coefficients. These experiments reveal that wear (a measure of material degradation) and entropy are intimately related, and the relationship between entropy and temperature can be put to use for prediction of material degradation.

### (d) Application to fretting wear

Fretting involves rubbing between bodies with small, cyclic motions. Fretting degrades components through surface wear and structural fatigue. Fretting wear of metals involves intense plastic deformations near the surface of the wearing body, sometimes accompanied with corrosion or oxidation of material. Rise of bulk temperatures are usually moderate, of the order of tens of degrees Kelvin. The fretting process is gradual, with the fretted component in a state of equilibrium or quasi-equilibrium.

During the last two decades, Mohrbacher *et al*. (1994), Huq & Celis (2002), and later Fouvry *et al*. (1997, 2003, 2007) and Fouvry & Kapsa (2001) developed an energy-based model for fretting wear. Exhaustive measurements by both groups showed the Archard wear law to be an inappropriate descriptor of their data. Mohrbacher *et al*. (1994) found that wear volume *w*_{v} (measured via geometry of a wear scar), when plotted versus total friction energy dissipated *E*_{μ}, produced a straight line. Fouvry *et al*. (1997, 2003, 2007), Fouvry & Kapsa (2001), Huq & Celis (2002) and Lee *et al*. (2005) confirmed this linear relation for fretting wear of various coatings on substrates.

Dai *et al*. (2000) treated fretting wear as an irreversible thermodynamic process, in a critically stable state near equilibrium, but in the process of transitioning to a new equilibrium state. At equilibrium, entropy maximizes and entropy production ceases. Perturbations of the system about the equilibrium state reactivate the dissipative processes and entropy production. For perturbations, thermodynamic quantities including entropy production are expressed as variations about the equilibrium state (Kondepudi & Prigogine 1998). For stable systems, the first variation of entropy production about the equilibrium state vanishes, which demands that the intensive thermodynamic states, including temperature, assume equilibrium values. For a critically stable system, the rate of the second variation of entropy vanishes. With this condition, Dai *et al*. (2000) equated the perturbed entropy flow to the perturbed entropy production, and solved for wear as the mass flux component of entropy flow. The entropy flow component due to heat conduction was ignored in this equation, which limits the use of the formulation of Dai *et al*. (2000).

Our theorem suggests that the rate of fretting wear can be written as . Following Mohrbacher *et al*. (1994), Fouvry *et al*. (1997, 2003, 2007), Fouvry & Kapsa (2001) and Huq & Celis (2002), we assume work of plastic deformation, driven by friction force *F*_{μ}, to be the dominant dissipative process *p*. Letting *p* be the dissipation from the friction force, an analysis similar to that which led to equation (3.2*d*) suggests(3.3)The delta notation on *X* and *J* indicate perturbations from the current equilibrium state, and the two arises from the second variation of entropy (Dai *et al*. 2000). Near equilibrium, temperature *T* is very near the equilibrium temperature, as shown by Szolwinski *et al*. (1999). The first and last terms of equation (3.3) suggest that the volume of material lost *w*_{v} *should be proportional to the friction energy E*_{μ} *dissipated*, with coefficient *B*_{f}*=*2*B*/*T*. This shows that the energy-based models for fretting wear are a consequence of the laws of thermodynamics and the degradation-entropy generation theorem, when viewed through the formalism of equations (2.1)–(2.9*d*).

## 4. Discussion

Coefficients *B*_{i}, defined by the formulations of equation (2.7), are material constants relating generalized degradation forces to generalized thermodynamic forces. Material properties such as thermal conductivity, magnetic permeability, fluid viscosity, elastic modulus or fracture toughness are defined by formulations of physics, but of necessity, the constitutive relations of these formulations must involve heuristic measurements. In §3*c*, coefficients *B*_{i} were measured, used and compared with the results of Rabinowicz (1980) based on fundamentally different measurements. Coefficients *B*_{i} can be geometrically interpreted as ratios of components of gradients and , for two hypersurfaces, *w*=*w*{*p*_{i}} and *S*′=*S*′{*p*_{i}}, in a space {*p*_{i}, *i*=1, 2, …, *m*} spanned by the processes *p*_{i} (which serve as coordinates in the space). For degradation by wear, the surface defines a wear curve (Bayer 1994) or wear map (Bhushan 2002). Each *B*_{i} gives the ratio between the local slopes of the surfaces, along the direction *p*_{i}.

A possible application of this procedure is in the development of a methodology for *accelerated testing of degradation*. It deals with an enabling technology for predicting behaviour of a system undergoing degradation and ultimate failure after a long period of time, based on short-term laboratory experiments. As mentioned in §2*b*, equations (2.3) and (2.4) have common rate factors that depend on system parameters . If the flow rates are judiciously chosen, then the rate of degradation d*w*/d*t* in equation (2.3) can be observed without waiting for long times. Equations (2.3)–(2.7) suggest that one can conduct an accelerated failure testing scheme by increasing process rates while maintaining ‘equivalent’ forces and to obtain the same sequence of physical processes, in identical proportions, but with a higher rate. Simply increasing rates may alter the physical processes (Ling *et al*. 1997). For example, moderate heat hatches an egg; if heating is accelerated without maintaining equivalent forces, the egg cooks. Via equation (2.5), altering any could change every .

As an example, to accelerate wear testing, equations (3.2*a*) and (3.2*b*) suggest increasing sliding (rubbing or slip) speed *J*=d*x*/d*t* while maintaining equivalent thermodynamic force *X* and degradation force *Y*. Higher d*x*/d*t* will accelerate wear, but higher temperatures from friction heat will affect constancy of *X*=*F*_{μ}/*T* via contact temperature *T*, and *Y*=*BμN*/*T* through dependencies of *B* and possibly *μ*, on *T*. This suggests adjusting the normal force *N* and temperature *T*, to compensate and maintain equivalent *Y*.

## 5. Concluding remarks

The science that captures degradation dynamics in a modelling paradigm suitable for early failure prediction, structured development of accelerated failure testing, and control of machine and structure maintenance is of particular interest in many disciplines. However, the modelling and analysis of machine and structure reliability has remained substantially unchanged for decades. In this paper, we presented a thermodynamic characterization of degradation dynamics, which employed entropy (a measure of thermodynamic disorder) as the fundamental measure of degradation, and suggested methods of measurement of state variables that are functionally related to entropy, which is, necessarily, an implicit system variable. Degradation of machinery components is a time-dependent phenomenon that arises because effects of irreversible processes accumulate to disorder the component. Therefore, it follows that thermodynamic entropy can be used to properly characterize the extent of disorder, the rate at which it progresses, and measure the component degradation. Hence, the second law of thermodynamics provided the framework for predicting failure, based on the rate of production and accumulation of entropy. The theory presented lays a thermodynamically consistent foundation for applying this concept to a variety of failure problems, including the tribological applications (adhesive wear and fretting wear) presented here as case examples. Application to tribology systems involves placing a tribological control volume about the region where the dissipative processes occur. Godet (1984, 1990) identified this region as the ‘third body’. The work can be extended to determine the critical damage parameter in processes involving fretting fatigue and to predict fretting fatigue life of machine element (cf. Quraishi *et al*. 2005).

## Acknowledgments

The authors would like to thank the Accenture Endowed Professorship in Manufacturing Systems Engineering, University of Texas at Austin and the Dow Chemical Endowed Chair in Rotating Machinery at Louisiana State University for support to conduct this work.

## Footnotes

- Received December 17, 2007.
- Accepted March 13, 2008.

- © 2008 The Royal Society