# On the Thermodynamics of Friction and Wear―A Review

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Friction and Energy Dissipation

#### 2.1. Non-Sliding Contact

_{1}E

_{d}+ c

_{2}

_{1}is energy wear coefficient, E

_{d}is energy dissipation and c

_{2}is the residual volume. Figure 2 shows the wear volume as a function of the cumulated dissipated energy [18]. The results are presented for TiN and HSS (high-speed steels) versus alumina contacts, and both present a linear evolution.

**Figure 2.**Wear volume as a function of cumulative dissipated energy (reproduced from data in Fouvry et al. [18]).

_{d}/4Pδ

_{0}

_{0}is the sliding amplitude. The energy approach proposed by Fouvry and co-workers to characterize fretting contact is formulated for non-adhesive wear tribosystems, displaying a weak influence of third body. However, in many applications the presence of third body is a concern.

#### 2.2. Sliding Contact

**Figure 3.**Distribution of heat generated by deformation with the distance from the contact surface (reproduced from [32]).

**Figure 4.**Fretting wear against dissipated energy for 140 nm (Ti, Al)N:140nm TiN multilayer coating at different relative humidity. Operating Conditions: 1N load and 10Hz frequency (reproduced from Huq and Celis, [42]).

**Figure 5.**Wear volume against SN for three different materials (reproduced from Ramalho and Miranda, [43]).

## 3. Frictional Temperature Rise

_{1}(t) and f

_{2}(t) are boundary conditions, ${A}_{0},{A}_{0}^{\prime}$ are constants that depend on the material properties and ${T}_{0},{T}_{0}^{\prime}$ are the temperature limits in the moving range. They show that the bulk surface temperature corresponding to the transition from mild wear mechanism to the severe wear mechanism is about 200 °C for steel 52100.

**Figure 6.**Measured temperature profiles of a pin of steel 52100: (a) at about 2 mm from sliding surface; (b) 10 mm from the surface (reproduced from Wang et al. [73]).

_{2}O

_{3}-coated glass in a dry oscillatory sliding system to measure the surface temperature. Szolwinski et al. [76] use an infrared thermographic technique to assess the magnitude and distribution of near-surface temperature within a fretting contact between an aluminum alloy cylinder and flat.

_{1}, k

_{2}and Pe

_{1}, Pe

_{2}are the thermal conductivity and Peclet number of first and second bodies, respectively. The concept of heat partitioning factor is very useful in analysis of a tribosystem. By having the total heat generation at the interface of contacting bodies, one can estimate the amount of heat flux entering bodies.

_{1}is:

_{1}= ημuN

**Figure 7.**Temperature rise ∆T, against the heat entering body 1, Q

_{1}(reproduced from Amiri et al. [78]).

_{1}

_{1}from Equation (11) into Equation (10) gives a relation for the wear coefficient K as follows:

_{brass}= 4.3 × 10

^{−4}and for Bronze on Steel pair is K

_{bronze}= 2.02 × 10

^{−4}. The published value of wear coefficient for Brass on Steel (Rothbart, [80]) and Bronze on Steel (Rabinowicz, [81]) are:

_{brass}= 6 × 10

^{−4}

^{−3}< K

_{bronze}< 10

^{−4}

## 4. Entropy—Wear Relationship

_{n}= ∑

^{n}∆Q

^{(n)}/T

^{(n)}, where ∆Q

^{(n)}is the heat input to the rider during the nth time interval, and T

^{(n)}is the average absolute surface temperature of the rider during the nth time interval. Figure 9 shows their experimental results, plotted for normalized wear as a function of normalized entropy. Wear and entropy are normalized by the maximum value of the set in each test. Figure 9 demonstrates a strong relationship between normalized wear and normalized entropy flow, that is: Normalized wear = normalized entropy flow.

_{e}S + d

_{i}S

_{e}S is the entropy exchange with surroundings and d

_{i}S is the entropy produced internally by the system. The second law of thermodynamics states that the entropy production must be non negative, i.e.,:

_{i}S ≥ 0

_{e}S + d

_{i}S = 0; d

_{e}S = –d

_{i}S < 0

_{i}S in terms of experimentally measurable quantities, one must invoke the concept of thermodynamic forces and thermodynamic flows (Kondepudi and Prigogine, [107]).

_{j}, where each ${p}_{j}={p}_{j}({\zeta}_{j}^{k})$ depends on a set of time-dependent phenomenological variables ${\zeta}_{j}^{k}={\zeta}_{j}^{k}(t)$, k = 1, 2, …, m

_{j}. The entropy production can be expressed as the sum of products of the thermodynamic forces and the corresponding thermodynamic flows:

_{j}measures how entropy production and degradation interact on the level of dissipative processes p

_{j}. Bryant et al. [103] successfully applied their theorem to sliding systems involving wear and fretting. They conclude that the well-known Archard law for assessing wear in friction wear and energy-based models for fretting wear are the consequence of the laws of thermodynamics and the degradation-entropy generation theorem. It is worthwhile to mention that the degradation-entropy generation theorem provides a powerful tool in analyzing a system undergoing degradation within an irreversible thermodynamic framework. The theorem expresses the rate of degradation and entropy generation by applying the chain rule and makes no assumption on the thermodynamic state of the system (Bryant [108]). Therefore, it can be applied to the systems operating far from equilibrium. However, the applicability of the theorem in describing other type of degradation processes, i.e., wear and fretting, are yet to be experimentally investigated.

## 5. Entropy and Self-Organization during Friction

_{i}S/dt = 0 in Equation (18). Therefore, it is inferred that ${X}_{j}^{k}=0$, ${J}_{j}^{k}=0$ (Prigogine, [113]). As stated earlier, a tribosystem is an open, non-equilibrium system. Hence, its entropy is lower than the equilibrium state and entropy production is not zero. The lower entropy of the system asserts an increase in orderliness and self-organization. Generally, a system does not reach equilibrium because of the interference of external elements. In a tribosystem, external elements such as load, velocity, temperature, humidity, etc.,, push the system to operate far from equilibrium. Therefore, to prevent entropy growth, a highly ordered intermediate body forms on the interface. This is commonly referred to as tribo-film. Formation of tribo-films is the response of the system to external stimulus to reduce wear rate. If tribo-films are not formed during friction, tribosystem stops the friction by seizure or jamming (Gershman and Bushe [38]).

**Figure 11.**Crack healing by embedded capsules at the micro and meso scales (reproduced from Nosonovsky et al. [115]).

_{net}= S

_{macro}+ S

_{meso}+ S

_{nano}

^{2}S, can be expressed as:

_{i}and δJ

_{i}, are the deviations of X

_{i}and J

_{i}from their stationary values, respectively. Depending on the detailed form of the flows J

_{i}and forces X

_{i}, the state of the system could be stable or unstable. Therefore, the right-hand side of Equation (22) could be negative or positive (Coveney [118]). For a thermodynamically stable system, the right-hand side of Equation (22) should be non-negative. In what follows, it is shown how dependence of coefficient of friction and thermal conductivity on load, N and velocity, u could either retain system in stable condition or drive the system away from equilibrium.

^{2}and J = −k∇T = μNu, where μ and k are coefficient of friction and thermal conductivity, a phenomenological coefficient, L, can be defined as:

_{1}= −∇T/T

^{2}and J

_{1}= −k∇T = μNu, and the second term for current collection entropy production, X

_{2}= X

_{e}/T and J

_{2}= J

_{e}. X

_{e}and J

_{e}are voltage and electrical current, respectively. Therefore, the entropy production can be written as:

_{e}is:

_{o}is the coefficient of friction without electrical current. Equation (32) suggests that with application of electrical current, J

_{e}, in frictional contact the coefficient of friction decreases. Also, the higher the voltage, X

_{e}, the lower is the coefficient of friction. It is to be noted that in derivation of Equation (32), the entropy production, d

_{i}S/dt, is first differentiated with respect to J

_{e}and set to zero, then the coefficient of friction, μ, is integrated over J

_{e}.

^{*}, as follows:

_{o}is the thermal conductivity at reference temperature (e.g., ambient temperature), β is the temperature coefficient of conductivity, $\stackrel{.}{\epsilon}$ is the strain rate, K

_{b}and K

_{ε}are bulk modulus and coefficient of thermal expansion, respectively [52,53,54]. Abdel-Aal concludes that the variation in strain rate can cause anisotropy in thermal conduction, which in turn, may lead to blockage of heat flow in MAZ. The accumulated energy within MAZ may result in formation of protective layer, i.e., dissipative structure. Therefore, Abdel-Aal [54] postulates that formation of dissipative structures is the inherent response of the tribosystem to the external stimulus.

_{1}= −∇T/T

^{2}and J

_{1}= −k∇T = μNu and diffusion, with X

_{2}= −∇φ/T and J

_{2}= −γ

_{D}∇φ, Equation (18) results in the following expression for entropy production:

_{D}is the transport coefficient, p and A are pressure and nominal area of contact, p = N/A. Kozyrev and Sedakova assume that in non-equilibrium stationary state, the wear of the tribo-film is proportional to γ

_{D}and the product of pu is a characteristic of friction. Therefore, during stationary conditions, the analysis is performed for the conditions of minimum γ

_{D}depending on pu. Hence, the mathematical condition for that is:

_{D0}is the integration constant. Equation (36) indicates that under stationary conditions, as pu increases, γ

_{D}decreases. It is to be mentioned that γ

_{D}was assumed to be proportional to wear W. Therefore, under the stationary non-equilibrium conditions, Equation (36) offers a procedure for decreasing wear with increase in pu. Figure 12 shows the experimental wear results of the work of Kozyrev and Sedakova [121] for two different materials. Experimental wear tests pertain to a ring-plane configuration for two sets of contacting materials: F4K15M5 on 35KhM Steel and Sigma-3 on 35KhM Steel. Figure 12 shows that in some range of pu, the wear W does not grow with pu but even decreases. The length of this zone depends on material properties.

**Figure 12.**Wear as a function of pu for two different materials (reproduced from Kozyrev and Sedakova, [121]).

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Amiri, M.; Khonsari, M.M.
On the Thermodynamics of Friction and Wear―A Review. *Entropy* **2010**, *12*, 1021-1049.
https://doi.org/10.3390/e12051021

**AMA Style**

Amiri M, Khonsari MM.
On the Thermodynamics of Friction and Wear―A Review. *Entropy*. 2010; 12(5):1021-1049.
https://doi.org/10.3390/e12051021

**Chicago/Turabian Style**

Amiri, M., and Michael M. Khonsari.
2010. "On the Thermodynamics of Friction and Wear―A Review" *Entropy* 12, no. 5: 1021-1049.
https://doi.org/10.3390/e12051021