kinetics of friction and thermal energy

8/8/2019 Kinetics of Friction and Thermal Energy

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Wear 254 (2003) 884900

On the interdependence between kinetics of friction-releasedthermal energy and the transition in wear mechanisms

during sliding of metallic pairs

Hisham A. Abdel-AalDepartment of Mechanical Engineering Technology, York Technical College, 5426 Gwynne Avenue, Charlotte, NC 28205, USA

Abstract

Sliding of complying solids is often associated with the release of thermal energy. This energy accumulates within the mechanicallyaffected zone (MAZ) of the rubbing pair. The accumulation of thermal energy within the MAZ tends to maximize the potential energyat the interface. Now, since a maximized potential energy renders the sliding system unstable, one (or both) materials will respond in amanner that consumes (dissipates) part or all of the accumulated energy in order to re-establish system stability or at least equilibrium. Thematerial response may be in many forms: oxidation, crack initiation, wear debris generation, transition in wear mechanism, etc. As such,one may consider that these processes are intrinsic responses by the material to dissipate energy. Moreover, many of these responses aretriggered at different stages of rubbing according to the balance between the rate of external thermal energy release (which is a factor ofthe nominal operation parameters) and the rate of thermal energy accumulationRTEA (which is mainly a function of thermal transportproperties of the rubbing pair). An interesting feature of this view is that the later quantityRTEAis directly related to the ability of theparticular solid to dissipate thermal loads. This quantity, which is termed here as the heat dissipation capacity (HDC), is directly relatedto the state of blockage of energy dissipation paths within the rubbing solid. The objective of this paper is therefore to study the relationbetween the change in the HDC of a sliding solid and the transition in the mechanism of wear. It is shown that there exists an inversecorrelation between the change in the HDC and the transition in the mechanism of wear. Moreover, it is also shown that a so-called ratioof residual heat (RRH, representing the ratio between the actual thermal load and the part of that load that is not dissipated by the solid)

is a significant parameter that influences the magnitude and mechanism of wear. The findings are applied to explain the wear behavior oftwo tribo systems: a titanium (Ti6Al4V) sliding on itself and sliding on a steel (AISI M2) counterpart. 2003 Published by Elsevier Science B.V.

Keywords: Wear transition; Oxidational wear; Wear of titanium

1. Introduction

When two complying materials rub against each other, atangential force representing the frictional traction will beexperienced at the interface. This force will perform workon each of the rubbing surfaces. A portion of that workwill be consumed in: plastically deforming a sub-contactlayer termed here as the mechanically affected zone (MAZ)in both rubbing materials and; initiating some secondaryprocesses (e.g. acoustic and luminescence emission, plasmaemission). The major portion of that work, however, termed

Abbreviations: CL, center line; HDC, heat dissipation capacity; LE,leading edge; MAZ, mechanically affected zone; NCI, nominal contactinterface; RRH, ratio of residual heat; TE, trailing edge; TFA, thermalflux accumulation

Tel.: +1-704-563-2658. E-mail address: [email protected] (H.A. Abdel-Aal).

as residual heat (RH) will be transformed into thermalenergy.

There are several processes that take place during sliding.In order for one process to take place, a particular amountof work needs to be supplied. This work is provided by thework of the friction tractions. In general, each process hasits own distinct energy requirement. Also, each process ex-pends a particular amount of energy. This situation is sum-marized in Fig. 1, where a quantitative comparison betweenthe energy requirements of the process taking place in slid-ing is presented. It is seen that most of the work of frictionforce is transferred into thermal energy. This does not meanthat the various associative processes are of less importanceto a friction process.

The availability of that energy at the interface, as wellas, within the MAZ violates the thermodynamic stability ofthe sliding pair (at least instantaneously). So that, if thisnewly available energy is not consumed by some mechanism

0043-1648/03/$ see front matter 2003 Published by Elsevier Science B.V.doi:10.1016/S0043-1648(03)00243-6


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Nomenclature

B thermal effusivity (W m2/KS1/2)Cv specific heat at constant volume (kJ/(kg K))Ck specific heat measuring isothermal volumetric

change (W/(m K))

Cp specific heat at constant pressure (J/m3 K)D thermal diffusivity (M2/S)e strain (m/m)e strain rate (S1)

E modulus of elasticity (GPa)h specific enthalpyh rate of change of enthalpy

H hardness expressed as pressure (GPa)K thermal conductivity (W/(mK))K apparent (effective) thermal conductivity

(W/(m K))Kb bulk modulus (GPa)

Km mechanical dilatation contribution tothermal conductivity (W/(mK))K coefficient of thermal expansion

(m/(m K))K coefficient of thermal expansion in

stress (J/m3 K)Pe Peclet numberQa actual amount of heat conducted through a

contact spot (W/m2)Qgen heat generated due to friction (W/m2)T temperature (K)Tmax maximum temperature attained in a single

contact cycle (K)T rate of temperature rise of the surface

(K/S)Uslid sliding speed (m/S)V volume (m3)

Greek symbols

temperature coefficient of conductivity(K1)

Poissons ratio temperature coefficient of diffusivity

(K1) thermo-mechanical coupling factor

coefficient of friction heat partition factor

Subscripts

max maximumxx in the plane of slidingzz normal to the plane of sliding

Superscripts

m moving solids stationary solid

(intrinsic or otherwise) one can envision that the MAZ ofboth materials will undergo some damage. That is, releaseof the thermal energy will maximize the potential energyof the interface as well as that of the MAZ. Since a maxi-mized potential energy of a system is inherently an unstablestate, one can postulate that particular surface activities will

be triggered to dissipate the excess energy, and re-establishthermal stability where relative motion between the slidingsurfaces is maintained at a minimal energy expenditure. Inthis sense, one can envision that particular responses aretriggered at particular progressive stages of sliding in accor-dance to the rate and level of energy accumulation within thecontacting layers of the rubbing pair. One of such responsesis the formation of protective oxide layers.

The ability of a sliding metal to form an enduring pro-tective oxide layer is influential to the resistance of wear athigh temperatures [13]. The formation of an effective pro-tective layer is a multifaceted process, in which the ability ofthe rubbing metal to diffuse (dissipate) thermal loads is cru-

cial [47]. Indeed, it was found [8] that the degradation ofthat ability during rubbing leads to the accumulation of ther-mal energy within the contacting layers. Such accumulationmakes available the energy required to surpass the energybarriers necessary to form a protective (glaze) oxide layer.

To understand the mechanism by which this energy issupplied or acquired by the reactive gases, it is essentialto study the origins of thermal flux accumulation (TFA)within the MAZ of the sliding solid. This understanding,however, hinges on envisioning the thermal scenarios takingplace within the MAZ within a non-orthodox prospective.In particular, it is necessary to view TFA as a protective

response that is intrinsic to the sliding solid. This responseaims at re-establishing a state of thermal equilibrium afterthat state has been violated due to the work performed bythe friction force on both solids.

The objective of this paper is to provide an alternativeview of the oxidationwear transition process. This view isbased on the consideration that oxidation, transition in wearregimes and, probably, wear particle generation itself areforms of intrinsic responses on part of the particular slidingmaterial. Such responses are triggered by the presence ofresidual energy introduced to the sliding surfaces by way ofthe work of the frictional forces. The goal of such responsesis to dissipate (consume) this excessive energy, thus lower-ing the potential energy of the surfaces to an instantaneousminimum that re-establishes thermodynamic equilibrium be-tween the sliding surfaces. As will be shown this view linksthe external influences on sliding (nominal load, speed, etc.)to the intrinsic properties of the particular solid in sliding.

2. Aspects of thermal transport in sliding

It can be postulated that any sliding solid has a so-calledheat dissipation capacity (HDC). This quantity determinesthe portion of the externally applied thermal load that the


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Fig. 1. Qualitative comparison between the energy requirements of the process taking place in sliding. The work of the friction force initiates manyprocesses at the atomic level. These, in turn, dissipate into heat. The amount of heat resulting is around 8090% of the work of the friction force.

particular solid can dissipate under a given set of sliding pa-rameters (mechanical load and sliding speed, etc.). The termdissipation is used here to merely indicate the transport ofthermal energy from the interface to the bulk of the rub-bing solid. Fundamental to this postulate is the distinction

between two heat quantities: the amount of heat generatedat the surface due to friction (Qgen) and the amount of heatthat is actually transported through the contact spot betweenthe sliding solids (Qa, say). The quantity Qgen is a functionof the mechanical properties of the sliding pair, their relativespeed and, surface finish. It represents the actual externalthermal loading of the sliding pair. The second quantity, Qa,is intrinsic and represents the loading limit of the particu-lar rubbing material. That is, the maximum thermal load arubbing material can dissipate. This quantity is limited bythe HDC which may be higher than or less than the actualapplied thermal load. If the HDC is greater than Qgen thenthermal accumulation will not take place. However, whenQgen is greater than the HDC, thermal accumulation takesplace and the immediate surfaces would be subject to ther-mal instability effects.

The distinction between the two quantities, Qgen and Qa,raises an interesting question regarding the residual portionof the friction force work (RH). In particular, what are theorigins of that RH, and by what mechanism does this heataccumulate within the MAZ?

Fig. 2 illustrates the mechanistic steps by which the RHmay accumulate within the MAZ. Fig. 2a depicts the idealsituation, where the rate of heat generation at the surfaceQgen is less than or equal to the total rate of heat dissipation

(conduction) through both contacting surfaces (the movingand the stationary). That is when,

Qgen Qma + Qsa (1)where the superscripts m and s denote the moving and the

stationary solids, respectively.In an ideal situation, thermal flux accumulation cannot

take place as the rate of heat production due to friction is atleast equal to the total rate of heat removal (through) con-duction by both surfaces. That is, the sliding system is inthermal equilibrium (stability) at least instantaneously. Theideal situation is not necessarily a dominant one. In mostcases, the total rate of heat removal will not remain con-stant throughout the duration of sliding (especially duringthe run-in period). The rate of heat removal may be affectedby many factors, mainly the dependence of the thermal trans-port properties on both temperature [9] and pressure [10].

The rate of heat production, as well, will not remain constant.This is due to the dependence of the mechanical propertieson temperature. This condition is particularly pronounced inthe run-in period where the sliding system is in an unstablecondition. This state (stabilityinstability fluctuation) beingtransient in nature, emphasizes the role of the ratio of therate of heat generation at the surface to the rate by whichthe surface temperature is elevated. At any rate, however,in most cases the rate of heat generation at the surface willovertake the total rate of heat removal by both mating sur-faces (Fig. 2b), so that,

Qgen > qma + qsa (2)


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Fig. 2. Mechanistic steps through which the residual heat (RH) mayaccumulate within the mechanically affected zone (MAZ). (a) Rate of heatgeneration at the surface (Qgen) is less than or equal to the total rate of heatdissipation (conduction) through both rubbing surfaces. (b) Initiation of theeffusionaccumulation process rate of heat generation at the surface ( Qgen)is greater than the total rate of heat dissipation (conduction) through bothrubbing surfaces. (c) Continuation of the effusionaccumulation process:the effused heat seeks the path of least resistance.

Now the difference between the two sides ofEq. (2) wouldyield the RH. That is,

Qgen

qma +

qsa

=

qRH = QRH (3)

Now, since the MAZ of each surface would be instanta-neously blocked (congested), the residual heat will try to

dissipate through the mating surface. That is, the residualheat will tend to effuse out of one surface toward the matingsurface. This effusion process (Fig. 2c) will continue as longas Qgen > Qa. The end effect, however, will be the exis-tence of a super saturated, super heated thermal layer withinthe contacting layers of both mating solids. That is, the ef-

fusion process will lead to the creation of a layer dominatedby TFA (i.e. accumulated thermal energy).To consume the newly introduced thermal energy, oxi-

dation may take place thus forming protective layers. Foran oxidative reaction to be sustained, however, a minimumlevel of thermal flux intensity has to be established at leastduring the initial stages of the reaction. Such thermal inten-sity condition may be formulated in terms of the duration ofthermal energy injection through the surface, compared tothe time needed by the material to redistribute this injectedenergy. That is, if the rate of energy injection through thesurface is greater than the rate at which the material redis-tributes this energy, a sufficient intensity level, in essence,

may take place and vice versa.The rate of energy injection to the surface is represented

by the thermal conductivity of the material. Whereas, therate of redistribution of the injected energy is representedby the diffusivity. As such, sustainment of a sufficient levelof intensity will depend mainly on the balance, or the ratio,between the thermal conductivity and the thermal diffusiv-ity at all operation temperatures. This ratio is representedby the so-called thermal effusivity B. The effusivity is a de-rived quantity that controls the penetration of heat within agiven material given by B = KD1/2, where K is the con-ductivity and D the diffusivity (alternatively this quantity is

termed as the coefficient of heat penetration). Therefore, inorder to maintain the critical intensity level, the diffusionof heat in the direction of sliding has to be considerablyminimal compared to the conduction of heat normal to thedirection of sliding; and that lateral conduction should alsobe smaller compared to normal conduction. This argument,in essence, simplifies to the necessity of a directional effect(or an anisotropic thermal behavior) of all metals in slidingcontact regardless of their nominal thermal properties.

2.1. Limitation of conventional thermal view

A central problem concerning the foregoing postulateis that thermal anisotropy is not reflected in conventionalthermo-tribological analysis. This is due to the linear na-ture of the heat conduction theory adopted early on intribo-analysis. This theory, which was formed by the pow-erful ideas of Rosenthal [11], Jaeger [12], Blok [13], Holm[14] and Archard [15] among others, rests on a fundamentalpremise that uncouples the thermal and mechanical statesof the rubbing material. This is mainly due to the dominantinfluence that the theory of moving heat sources (originallyintroduced by Rosenthal [11]) have had on the thermalanalysis in friction. This theory was originally developedfor welding, where the solid is allowed to expand freely


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upon heating. The same theory (without adjustment to thenature of sliding surfaces) was adopted to analyze frictionalheat problems. This lead to a situation where the effect ofmechanical dilatation on thermal conduction in sliding wasoften disregarded. Thus, within the realm of conventionaltheory, thermal conduction is at most influenced by temper-

ature rise and not by a combination of temperature elevationand mechanical effects. The nature of sliding, however, sug-gests that coupling both thermal and mechanical influenceson conduction is more representative of the physical picture.

Due to the topography of surfaces, free expansion of thecontact spots does not take place. Instead, upon attemptingto expand, the hot spots will be resisted by neighboring rel-atively cold regions that will tend to restrict the expansion.So, the work done by the solid in attempt to expand will bealso released as thermal energy. This situation is generallyrepresented by the so-called generalized heat equation [16](originally driven by Jeffrey [17]). This equation incorpo-rates the effect of mechanical loading on thermal conduction

through the inclusion of mathematical terms that representthe work done on the solid to prevent free expansion. Thisis achieved by incorporating the effect of mechanical dilata-tion on energy transport. Naturally, this equation is highlynon-linear and presents mathematical difficulties upon at-tempting an analytical solution. However, despite such math-ematical hindrance, the consideration of the coupled statereveals several aspects of the thermal kinetics in sliding, thatare otherwise not reflected within conventional linear the-ory. In particular, within a non-linear frame a so-called ap-parent thermal conductivity [18] is defined. This parameterreflects the strong dependence of thermal conduction on the

strain rate and the speed of sliding. This dependence, as willbe shown, may indeed cause anisotropic thermal behavior,even for nominally isotropic solids. Such an anisotropy may,under suitable loading and sliding conditions, dominate theheat transfer process and affect wear particle generation andtransitions.

3. Heat conduction in a strained solid

The flux transport in an elastic medium may be expressedin the form [19],

h dV =

sq n ds (4)

where q is the heat flux vector flowing into the elasticmedium. Note that the energy production rate within thebody (source term) has been neglected in Eq. (4). The sur-face integral (right-hand side ofEq. (4) is related to the vol-ume integral by,

s

q n ds =

V

q dV (5)

Substituting Eq. (5) in Eq. (4) yields:

q

=h (6)

For a deformable body, the specific enthalpy is a function ofboth temperature and volume increase of the body. Thatis,

h = h(T, e) (7)where

e = e(11) + e(22) + e(33) = VV

Note that e(11), e(22), and e(33) are the so-called LagrangianPrincipal Strain values given by the roots of the equation,

e3 ILe2 +IILe IIIL = 0with

IL = eii, IIL = 12 (eiiejj eijeij), IIIL = |eij|The time rate of change of the specific enthalpy is thuswritten as,

h =

h

T

e

T +

h

e

T

e (8)

The first term (h/T)e is recognized as the volumetricspecific heat, Cp, under constant pressure. The second term(h/e)TCk can be defined as another type of specificheat that measures the isothermal change in volume perunit volume of the continuum body. Since the volume changein a continuum body is a result of mechanical straining, theheat capacity Ck serves as a bridge to thermo-mechanicaldeformation. The specific heat Ck is, in general, a functionof temperature. Expanding Ck in a Taylor series with respect

to the temperature gives,

h

e

T

Ck(T) = Ck(T0) +

Ck

T

T0

(T T0) + (9)

The heat capacity Ck at the reference temperature, Ck(T0)can be assumed zero [20], implying that the value of Ck ismeasured with regard to its value at the reference tempera-ture (or is a function of the temperature rise). As such, thetemperature dependence ofCk(T) may be expressed in termsof a temperature rise above the reference temperature, thatis,

h

e

T

=

Ck

T

T0

T (10)

Substituting from Eq. (10) in Eq. (8), the time rate changeof the specific enthalpy per unit volume is obtained as:

h = CpT +

Ck

T

0

Te (11)

where

Ck

T

0

Ck

T

T0


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Substituting from Eq. (11) in Eq. (6), we may write

q = CpT + K(T)Te, K

Ck

T

0

(12)

The quantity, K, retrieves the coefficient of thermal expan-sion in stress defined in linear thermo-elasticity. This is re-

lated to the coefficient of linear thermal expansion (whichmeasures the change in strain per degree rise in temperature)by,

K = 3Kb, 3Kb = 3 + 2 = E1 2V

Note that Eq. (12) reflects the dependence of K on thetemperature rise and involves a non-linear coupling betweenthe thermal and the mechanical fields.

3.1. Thermo-mechanical coupling

Eq. (12) may be rewritten as:

q = CpT

1 + KTeCpT

= CvT

1 +

3KbTK2Cp

e

KT

(13)

Eq. (13) may be rearranged in the form:

q = CvT

1 +

e

KT

(14)

where is a non-dimensional quantity representing thethermo-mechanical coupling factor given by

= 3KbK2 T

Cp

The ratio e/(KT ), also non-dimensional, measures the me-chanical strain rate e relative to the thermal strain rate KT.Note that the thermo-mechanical coupling factor is com-posed essentially of material properties. As such, in a slidingsituation, will depend on the material. On the other hand,the ratio e/(KT ) depends mainly on the sliding conditions(nominal loading and sliding speed). So that, the last termin Eq. (14), (e/(KT )), may be taken to represent the in-teraction between the material properties and operation con-

ditions. Alternatively, this same quantity may be taken toindicate the response of a particular rubbing pair to externaloperational conditions.

3.2. Thermal conduction in sliding solids

If different portions of a mass of matter are at differenttemperatures, thermal energy is transported from the highertemperature region to the lower temperature region. Thethermal conductivity is the property of a material which pro-vides a quantitative measure of the rate at which thermalenergy is transported along the thermal gradient.

Heat transport can be related to microscopic behavior.In a semi-classical picture of the insulator, input of heatexcites vibrations of the nearby atoms. Because atoms areconnected by chemical bonds, the vibrational energy is dis-sipated through excitation of vibrations of adjacent atoms.Realistic modeling of thermal conduction accounts for quan-

tization of these lattice vibrations, called phonons. Heat isthus transferred through phonons colliding with each otherand possibly with defects or grain boundaries. Because rais-ing pressure (or lowering temperature) raises vibrationalfrequencies and densification increases the chances of col-lision, the conductivity (K) increases as the pressure P in-creases (or as the temperature T decreases). In addition totransport by conduction, a hot material produces black bodyradiation which travels as an electromagnetic wave. Both ra-diation and phonon collision take place simultaneously. So,the total conductivity at any temperature is the sum of twocontributions: the lattice contribution Klatt and the radiationcontribution Krad.

It is immaterial whether thermal conduction involves elec-tronic or vibrational excitations, both can be described asdamped harmonic oscillators that are, in general, affectedby pressure and temperature [2123]. The variations in theconductivity with pressure and temperature do influence thetemperature rise in a sliding system and thereby influencethe mechanism of wear (or the material resistance to wear).While this is the complete physical picture, the mathemati-cal description is not as comprehensive. This is, mainly, dueto the high order of non-linearity that would be introducedin the mathematical equations when a complete descriptionis attempted. Moreover, characterization of the conductiv-

ity of materials under a combination of high pressures andtemperatures is plagued with experimental difficulties thatrender few reliable data available. In this article, only thedependence of the thermal conduction on temperature willbe considered.

3.2.1. Modeling of the thermal conductivity

In general, the variation in the thermal conductivity formost engineering materials with temperature falls into threebasic categories. These are summarized in Table 1. Math-ematically, the variation of the thermal conductivity with

Table 1Classification of materials according to the variation in the conductivitywith temperature

Materialclass

Conductivity behavior with temperature

Class a Conductivity drops with temperature elevation (e.g.carbon steels, sapphire, and zirconium)

Class b Conductivity increases with temperature elevation (e.g.stainless steels, duralumin, and cast iron)

Class c The temperatureconductivity curve includes an inflationpoint. That is, the conductivity of the material increases(or drops) with temperature, reaches a maximum (or aminimum), then drops (or increases); (e.g. titanium, zincand vanadium)


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temperature may be represented in several forms (i.e. ex-ponential, linear, biharmonic, etc.). The choice of a partic-ular mathematical form depends, mainly, on the allowableerror margin introduced when fitting experimental measure-ments. To obtain a closed form solution of the heat equation,however, it is necessary to adopt a form that would allow

the application of the currently available mathematical lin-earization tools. In particular, it is necessary to adopt a linearrepresentation of the thermal conductivitytemperature rela-tionship. This is because a linear form directly lends itself toefficient application of a straightforward quasi linearizationtechnique known as the Kirchoff transformation [24]. Thechoice of a linear form of variation in the conductivity is notan ad hoc approach. In fact, a linear relation between theconductivity and the temperature has physical origins thatstem from conventional models of the electronic behaviorof metals.

In seeking an analytical expression that reflects the depen-dence of the conductivity on temperature it is to be remem-

bered that the thermal conductivity is of electronic origins.Within conventional considerations, the thermal conductiv-ity can be divided into two components. The first is due tothe crystal itself and is independent of any free electrons orelectron holes. Whereas, the second is a result of the mo-tion of the electrons. In the case of good conductors (suchas most metals), the second component is of more impor-tance than lattice conduction. It is to be noted that withinthis model, the radiation contribution to conductivity is to-tally neglected. This is justified within the limits of mostengineering applications that involve metals. The analysis,however, is not quite accurate when ceramics are involved

[25]. At any rate, for many practical applications, neglectingthe lattice contribution is sufficient. So, the thermal conduc-tivity due to heat current carried by the electrons in a metalmay be written as:

K = 2B2nf

3JmvL (15)

Now, the mean free path of the electrons, L, may be consid-ered (as a first approximation) independent of the tempera-ture. Whence, the variation in the conductivity with temper-ature may be expressed as:

KK0 dK =

0

Md (16)

where Mis a constant representing the temperature indepen-dent values in Eq. (15). Integrating Eq. (16), we may expressthe thermal conductivity at any temperature as:

K() = K0(1 + ) (17)where K() is the conductivity of the material at the partic-ular temperature , and is a constant that represents theso-called temperature coefficient of conductivity. The termK0 in Eq. (17) represents the thermal conductivity at a ref-erence temperature that is not necessarily equal to zero.

3.3. Apparent thermal conductivity

Introducing the Fourier law of heat conduction

q = KTin Eq. (14) results in,

K2T = CvT (18)where

K = K(T) CpeK(2T) (19)

is the apparent thermal conductivity. The apparent thermalconductivity is the true conductivity of the medium under theparticular state of loading. Eq. (19) indicates that K is a su-perposition of two components: the temperature rise-inducedeffect, K(T), and the dilatational effect which is representedby the coupling factor . This later component, however,mainly varies in proportion to the strain rate

e.

Upon rearranging Eq. (19) in the form ofEq. (17) aidedwith the definition of the thermo-mechanical factor , theapparent thermal conductivity is written as:

K = K0

1 +

KmK0

T

(20)

where

Km = 3eKbK2TEq. (20) represents a working formula that allows the es-timation of the effective thermal conductivity for a given

set of sliding conditions (provided that the strain rates arepredetermined, experimentally or otherwise). The term 2Tmay also be estimated, as a first approximation, from the un-coupled heat conduction equation and through an iterativeprocedure the actual conductivity may be calculated.

The main implication ofEq. (20), however, is that thermalconduction in rubbing depends on the direction and mag-nitude of the strain rate. That is, whether the rubbing ma-terial is strained in tension or in compression will have aneffect on the local effective conductivity of the sliding ma-terial. It follows that if the material is undergoing tensilestrain (positive strain rate) the local effective conductivity isreduced, whereas if the material is undergoing compressivestrains (negative strain rate) the local conductivity will beaugmented. Now, since the different layers of a rubbing solidare strained in different rates, the different contacting layerswill conduct heat according to the local magnitude and di-rection of the strain rate. Moreover, since a sliding contactspot undergoes different strain modes and rates (the lead-ing edge (LE) of the sliding contact will experience com-pressive strains, while the trailing edge (TE) will experiencetensile strains [26]) combined with the compressive strainrates normal to the direction of sliding, a directional effectwith respect to the local thermal conductivities will takeplace. Thus, regardless of the nominal thermal properties


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of the sliding solid an anisotropic heat conduction behav-ior will be exhibited by any material element located withinthe MAZ. The magnitude of that anisotropy will depend onthe local strain rates and their respective orientation withrespect to the plane of sliding. In all, depending on the ma-terial and the sliding parameters (nominal load and speed) a

solid that is nominally thermally isotropic may very well ex-hibit anisotropic thermal conduction, and thereby anisotropicthermal effusion. The local intensity of thermal accumula-tion will follow that anisotropy (note that the effusivity is thesquare root of the product of the thermal conductivity andthe heat capacity). Now, since the density of the oxide andthe oxidation rate follow the local intensity of thermal accu-mulation, different parts of the slider will attain the criticalthickness necessary for wear protection at different times.This will cause the slider to exhibit different wear rates atdifferent locations. That is, in general, the leading edge willtend to build up the oxide layer slower than the trailing edge.Whence, wear is expected to be higher at the leading edge

and relatively lower at the trailing edge. The details of this ef-fect, however, are considered beyond the scope of this paper.

4. Results and discussion

4.1. Behavior of the thermal conductivity in sliding

To illustrate the behavior of the thermal conductivity insliding, Eq. (20) was applied to the experimental measure-ments of Seif and coworkers [2731]. Table 2 provides asummary for the experimental conditions and main proper-

ties used in the work of these authors.Strain rates at different points of the nominal contact sur-

face, both in the plane of sliding and the normal plane, werereconstructed from the point-wise velocity vectors obtainedoriginally by Seif et al. [27]. The temperature rise at eachof these points, published elsewhere [3234], are shown inFig. 3. These were used to calculate the contribution of thethermal perturbation to the thermal conductivity of the AISI1018 specimen at the respective points (i.e. the term K0T).The temperature coefficient of conductivity, , meanwhilewas obtained by linear regression of the conductivity dataextracted from the work of Touloukian et al. [35].

Table 2Summary of the conditions of sliding for the work of Seif et al. [2731]

Tribo system Stationary rectangular pin-rotatingcircular disk

MaterialPin Steel AISI 1020Disk Steel AISI 1018

Sliding speed (m/s) 0.45Nominal load (N) 80Nominal contact stress (MPa) 13.333Nominal contact area (m2) 6 106

Fig. 3. Temperature rise along the width of the nominal contact surface forthe AISI 1020AISI 1018 rubbing pair studied by Seif. T1 is the constantconductivity temperature, T2 represents the temperatures corrected forthe variation in the conductivity with temperature only, and T3 depictsthe temperatures corrected for both temperature variation and mechanicaldilatation effects.

Fig. 3 depicts the distribution of the temperature rise alongthe width of the nominal contact surface. The temperaturesplotted in the figure are corrected for non-linear effects (i.e.for the variation of the thermal conductivity with temperature(T2) and the thermo-mechanical coupling effect) (T3) bymeans of the Kirchoff transformation [23].

Fig. 4 depicts the distribution of the total strain rates alongthe same points for which the surface temperatures were

evaluated. Fig. 4a is a plot of the strain rates in the planeof sliding exx, whereas Fig. 4b is a plot of the strain ratesin the normal plane ezz. It is noted that the leading edgeof the slider experiences high strain rates compared to thetrailing edge. Moreover, the strain rate in the sliding planechanges from being compressive at the leading quadrant tobeing tensile at the trailing half of the specimen. The strainrate in the normal plane (Fig. 4b) also displays some uniquefeatures. Note the difference between the strain rate at theleading edge and those at the trailing edge. Note also thesteep gradients of the strain rate experienced by the leadingquadrant of the specimen.

Strain rates from Fig. 4a and b, along with an estimate ofthe term 2T(obtained from linear theory) were substitutedin Eq. (20) to compute an initial estimate of the point-wisevalues of the thermal conductivity in the plane of slidingand the normal plane. These initial values were subsequentlyused to refine the estimate of 2T. The refined value, inturn was re-substituted in Eq. (20) to obtain a correctedvalue of the conductivity. Computations were carried outuntil two successive values of the conductivity for the samepoint fall within a predetermined accuracy. The results ofthat procedure are provided as Fig. 4c, which depicts thepoint-wise distribution of the thermal conductivity along thenominal contact interface (NCI) of the steel specimen.


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Fig. 4. Distribution of the total strain rates along the width of the nominalcontact surface of the steel rubbing pair shown in Fig. 1. (a) Strain rates inthe plane of sliding (exx). (b) Strain rates in the plane normal to the planeof sliding (ezz). (c) Point-wise distribution of the thermal conductivityalong the nominal contact interface, K(T) represent the values of theconductivity as a function of the temperature rise only.

To establish a qualitative comparison between the thermaland the mechanical effects on the conductivity, for this par-ticular specimen, a third curve was computed. This curve,which is labeled K(T), represents the values of the con-ductivity as a function of the temperature rise only. Thesewere computed by neglecting the strain rate contribution inEq. (20). That is, by assuming that the conductivity is give

by the equation:

K(T) = K0(1 + T)

where the temperature T is extracted from the curve labeledT3 in Fig. 3.

An interesting feature of the figure is that the effectiveconductivities Kxx and Kzz are considerably lower than thetemperature based conductivity K(T) (on average the differ-ence amounts to 20%). It is to be noted also that the ef-fective thermal conductivity of the specimen in the normalplane Kzz does not vary significantly from K(T) (averagevariation of Kzz/K(T) 0.2). This indicates that for thisparticular loading condition, the mechanical effects tend toslightly counteract the temperature-induced degradation ofthe conductivity.

Thermal conductivity in the plane of sliding (lateral ther-mal conductivity), Kxx, exhibits an interesting trend. At theleading quadrant, the conductivity in the plane of sliding is

considerably higher than that in the normal plane. Whereas,throughout the trailing half of the specimen, the thermal con-ductivity in the normal direction is considerably higher thanthat in the plane of sliding. This trend, while in general isconsistent with the temperature distribution given in Fig. 3,lends a new interpretation to the kinetics of heat within theNCI.

Conventional analysis explains the trend of the temper-ature distribution (Fig. 3) in terms of the theory of Jaeger[12]. That is, due to the motion of the specimen convectiveheat fluxes will take place and heat will tend to diffuse to-ward the trailing half of the specimen. Fig. 4c, however, of-

fers a complementary explanation: due to the difference inthe effective conductivities, the leading quadrant will havean HDC that is higher in the plane of sliding than that inthe normal plane. Thus, at the leading quadrant, resistanceto thermal flow will be higher in the direction normal to theplane of sliding. Since heat is primarily conducted throughthe path of the least thermal resistance, the heat generatedwithin the first quadrant will tend to flow laterally paral-lel to the plane of sliding. Such a tendency will continue,point-wise, until the values of the effective conductivity inboth directions are locally equal. The location of the pointof equal conductivity will depend on the distribution of thestrain rate and its local magnitude in addition to the localtemperature rise. Starting at the point of equal conductivi-ties (approximately at X/W = 0.5 in Fig. 3) heat will againfollow the passage of the least resistance. Only, this pas-sage will likely be normal to the plane of sliding due tothe effect of the strong temperature gradient. Now, due tothe differences in the effective conductivities, a correspond-ing difference in the thermal effusivities of the leading andtrailing portions of the specimen will take place. This willlead to the tendency of the portions of the highest effusivity,trailing portions of the specimen, to retain heat in the nor-mal direction and allow thermal flow in the lateral direction(also toward the trailing half of the slider).


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Laterally, however, the situation is reversed. That is, thelateral effusivity in the leading half is higher than that at thetrailing half. As such, the leading portions of the slider willtend to accumulate heat parallel to the sliding plane. Thisheat will tend to gravitate toward the trailing half where thelateral resistance is lower. This will cause the bulk of the

thermal load to shift toward the trailing half of the slider.As a consequence the temperature at the trailing half will,in general, be higher than those at the leading half. The dif-ference in the effusivity will increase relative to the strainrate (which may be assumed to vary with the speed of slid-ing). Thus, at higher speeds the difference will increase andthe shift in thermal loading will be more significant. Conse-quently, the point of maximum temperature will shift towardthe trailing edge.

High effusivity indicates a resistance to an abrupt changein the thermal state [4]. This resistance is exhibited as lo-cal retention of heat within the layers of the MAZ due tohigh resistance to heat penetration. This situation may lead

to two different scenarios, each of which depends on theamount of gain or drop in the directional effective conduc-tivity. The first scenario will take place if the mechanicaleffects are not as severe. Due to the local difference in theeffusivity, heat will accumulate mainly at the trailing halfof the specimen. This accumulation will provide more en-ergy, to be acquired by the reactive gases, at the trailinghalf than the leading half of the specimen. Now, with in-crease in the nominal load and the speed, the generatedheat (thermal load at the NCI) will increase. Depending onthe difference in the HDC and that of the effusivity, theHDC normal to the plane of sliding may become signifi-

cantly constricted. This will lead to a critical situation inwhich the heat flux will be forced to flow parallel to theplane of sliding due to a lower lateral thermal resistance anda congested normal plane. This tendency will continue aslong as the HDC in the lateral direction is not impaired. Inthis case, the lateral flow of heat may be viewed as an in-trinsic relief mechanism that is invoked by the material tocounteract the destructive effects of thermal energy accumu-lation.

While the lateral flow of heat relieves the normal planefrom the destructive effects of heat accumulation, it causeslateral thermal accumulation. Depending on the lateral HDC,lateral heat accumulation will cause a corresponding degra-dation in the material resistance to mechanical shear. Thus,under a suitable combination of nominal loading and slid-ing speed shear stability may take place. This will lead tothe delamination of the oxide layer and mark the transitionin the wear mechanism from oxidation to delamination. Thepostulated indication for such transition is the growth of theso-called ratio of residual heat (RRH) [5,8] in associationwith the degradation of the HDC normal to the direction ofsliding. The RRH of a material represents the percentage ofthe applied thermal load which cannot be dissipated instan-taneously by the material. The RRH may also be viewed asa qualitative measure of the amount of thermal energy that

is instantaneously available for fueling oxidative reactionsor secondary processes within the MAZ of the material.

4.2. Correlation to wear regime transition

To examine the lateral heat flow postulate, wear data of

a titanium (Ti6Al4V)tool steel (AISI M2) rubbing pairwere analyzed. The data were acquired by Straffelini andMolinari [36], who carried wear experiments at a rangeof speeds (0.30.8 m/s) and loads (50, 100, and 200 N).Tables 35 summarize the experimental conditions of Straf-felini and Molinari and the thermo-physical properties of therubbing pair, respectively. Fig. 5 depicts the variation in thethermal properties of a stress free sample of the materialsused in the test.

The experiments identified two wear mechanisms thattook place irrespective of the counterface and applied load:oxidative wear at the lowest sliding velocities (0.30.5m/s)

and delamination wear at the highest (0.60.8 m/s). This be-havior was attributed to the different, bulk, temperatures thatwere reached on the contacting surfaces during sliding thushighlighting the importance of thermal softening effects. Theauthors reasoned that as the temperatures increased (due toan increase in load or decrease in the thermal conductivityof the counterface), the plastic strain rate at the contactingasperities also increased, thus leading to a corresponding in-crease in the wear rate. They also predicted that the mecha-nism responsible for the increase in the plastic strain rate isreversible dislocation motion. However, no explanation wasgiven to how would the dislocation motion acquire the ki-netic energy necessary to move, especially that the thermal

conductivity of the particular titanium alloy increases withtemperature (see Fig. 5). The increase in the conductivitywill lead to faster heat dissipation from the counterface thusminimizing the amount of thermal energy that can mobilizethe dislocations. An alternative view that avoids the contra-dictions of Straffelini and Molinaris arguments originatesfrom the lateral heating hypotheses.

The starting point is to compute the flash, and not the bulk,temperatures reached in the sliding of TiM2 pair using theclassical solution for a moving anisotropic medium [37], i.e.

Table 3

Experimental conditions for the work of Straffelini and Molinari [36]

Test configuration Disk-on-disk

Materials Ti6Al4V stationaryAISI M2 moving

Disk dimensionDiameter (mm) 40Thickness (mm) 10

Sliding speed (m/s) 0.30.4Nominal normal loads (N) 50, 100, 200Depth of plastic zone (m) 40Coefficient of friction, 0.30.4Total sliding distance before steady state (m) 1770


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Table 4Experimentally determined wear rates for the Ti6Al4VAISI M2 sliding pair

Sliding speed (m/s) 0.3 0.4 0.5 0.6 0.7 0.8Wear rate (mm3 /m) 0.01 0.0098 0.0086 0.0078 0.0072 0.007Maximum temperature (C) 458.55 511.17 555.35 593.78 627.96 658.87Normal strain rate, ezz (s1) 75 98 107.5 117 126 140Lateral strain rate,

exx (s1) 15 20 25 30 35 40

Table 5Properties used in the calculation of the HDC and RRH for the titanium(Ti6AL4V)tool steel (AISI M2) rubbing pair

Material Ti6Al4V AISI M2

(kg/m3) 4420 7600K0 (W/m K) 5.8 48.6

D0 (m2/s 106) 2.15 0.148B0 (W/mS1/2 K 103) 3.955 12.642 (K1) 0.0001983 0.0006146 (K1) 0.001608 0.0010258Cp (kJ/kg K) 610 452Tmelt (K) 1933 1760Kb (GPa) 112 140K (m/m K) 9.7 11.9y (GPa) 0.970 1.517

Fig. 5. Variation of the thermal properties of titanium with temperature.

T = q(Cp)1/2

8(3K1K2)3/2

t0

d

(t )3/2

A

exp

Cp

4(t )

[X1 X1 U1(t )]2K1

+ X22

K2

dX1 dX2 (21)

where K1 and K2 represent the effective (apparent) ther-mal conductivities in the plane of sliding and the plane nor-mal to it. Strain rates were estimated from wear data (seeAppendix A), then substituted in Eq. (20) to obtain the effec-tive conductivities. The heat partition factor was estimatedfrom the expression [12],

Fig. 6. Evolution of the flash temperatures with time for a titanium(Ti6AL4V)tool steel (AISI M2) sliding pair. Nominal test load, 50 N.(a) Evolution of temperature with time for the different speeds. (b) Max-imum temperature attained for each speed.


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= Ks

zz

0.795 Kszz + Kmzz

Pe

where the superscripts s and m denote the stationary andthe moving specimens, respectively, and Pe is the Pecletnumber.

Fig. 6a depicts the evolution of the flash temperatures asa function of the contact time at each of the speeds usedby Straffelini and Molinari under a nominal load of 50 N.Fig. 6b depicts the maximum temperatures reached at eachof the sliding speeds used to evaluate Fig. 6a. For each of thetemperatures plotted in Fig. 3a, the effective thermal con-ductivity in the direction of sliding (Kxx) and the normal(Kzz) were calculated. These values were subsequently usedto evaluate two ratios that represent the change in the ther-

Fig. 7. Variation of the conductivity components at the different sliding speeds (load = 50N). (a) The ratio Kxx/Kzz for the different sliding speedsplotted against a non-dimensional temperature T/Tmax, where Tmax is the maximum temperature reached in a loading cycle at a particular sliding speed.(b) Variation of the ratio Kxx(U)/Kxx (0.3), at different sliding speeds.

mal conductivity as a function of sliding speeds and load-ing. These being the ratio between the lateral conductivity(conductivity in the plane of sliding) and the normal con-ductivity (conductivity in the plane normal to the slidingplane), i.e. the ratio Kxx/Kzz, whereas the second is the ra-tio Kxx(Uslid)/Kxx (0.3 m/s). This represents the change in

the lateral conductivity as a function of speed and is cal-culated by dividing the lateral conductivity at each slidingspeed by the conductivity at 0.3 m/s, since this speed was thestarting sliding speed in the experiments. As such, the ra-tio Kxx(Uslid)/Kxx (0.3 m/s) represents the evolution of thelateral conductivity at the different sliding speeds.

Fig. 7a is a plot of the ratio Kxx/Kzz for the differentsliding speeds at 50 N. The value Kxx/Kzz is plotted againsta non-dimensional temperature T/Tmax, where Tmax is the


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maximum temperature reached in a loading cycle at a par-ticular sliding speed. It is noted that the lateral conductivityincreases with respect to the normal conductivity in directproportion at the sliding speed. That is, as the speed of slid-ing increases the ratio Kxx/Kzz also increases.

Fig. 7b represents the variation of the ratio Kxx(U)/Kxx

(0.3) at different sliding speeds. Again this ratio is plottedagainst the non-dimensional temperature T/Tmax. Consistentwith the trend ofFig. 7a, the total conductivity increases withincrease in the speed. It is to be noted, however, that alloyTi6Al4V exhibits an increase in the thermal contributionto the conductivity, i.e. the term (1 + T) > 1. This indi-cates that the thermal contribution to the lateral conductivityovercomes the mechanical dilatation component. Moreover,the increase in the ratio Kxx/Kzz implies the decrease of theHDC in the normal direction. For verification, the RRH forthe Ti alloy was calculated at the different sliding speeds(see Appendix B). The amount of heat generated was takento be a function of the surface temperature, i.e.

Qgen(T) = H(T)Uslid (22)The results of the calculations are presented in Fig. 8a, wherethe RRH of titanium is plotted against the non-dimensionaltemperature, T/Tmax. The general trend of the curves indi-cates that the RRH increases as the sliding speed increases.This is due to two factors: the increase of the amount of heatgenerated as the speed of sliding increases and; the degrada-tion in the thermal conductivity due to the mechanical cou-pling effect (strain rate effect) especially in the lateral plane(plane of sliding). Akin to this degradation, the HDC in thenormal direction decreases as the sliding speed increases.

This applies a higher lateral thermal load which, for this par-ticular case, cannot be dissipated instantaneously. As such,congestion and thereby, lateral heat accumulation will takeplace. The forgoing analysis is confirmed by Fig. 8b, wherethe ratio of the normal HDC at different speeds relative tothe HDC at the initial speed, 0.3 m/s, is plotted. Interest-ingly, it is noted that the degradation in the HDC and theincrease of the RRH are more significant past 0.5 m/s whichis the transition speed (compare the trend of each curve inFig. 8a and b).

The RRH depends on a group of factors (see Appendix B).For this quantity to be minimized or not at a given speeddepends on the corresponding change in the HDC. The in-crease in the HDC meanwhile depends on which of thethermal-induced or the mechanical-induced perturbation inthe effective conductivity would dominate. The generatedheat also has partial influence on the behavior of the RRH atdifferent speeds. In general, heat generation at the surface isa function of the mechanical properties. These, on the otherhand, are functions of the temperature. As such, the higherthe speed, the higher the temperature rise and thus a declinein the mechanical properties. This leads to less heat gener-ated at the surface. On the surface, however, strain harden-ing takes place. This partially counteracts the effect of thetemperature on the mechanical properties. To this effect the

Fig. 8. Variation of the ratio of residual heat (RRH) and heat dissipationcapacity (HDC) of the titaniumtool steel rubbing pair at different slid-ing speeds. (a) RRH of titanium is plotted against the non-dimensionaltemperature, T/Tmax. (b) Evolution of the heat dissipation capacity in thenormal direction (HDCzz) with the speed of sliding.

fluctuation in the heat generated at the surface may be im-material. All things considered, however, the general trendof the RRH with speed will depend on non-linear influences,the detailed study of which is a subject of current investiga-tion.

From the foregoing, it may be concluded that the tran-sition in wear from oxidation to delamination is triggered,in essence, by the change in the normal thermal conduc-tivity Kzz and the associated degradation of the total HDC.The change in the conductivity, therefore, is not to be con-sidered as a mere consequence of the thermal events at theNCI. Rather, the thermal events at the interface are trig-gered by the degradation in the conductivity. In particular,the change in the thermal conductivity will cause the direc-tional constriction of the HDC of the material. Dependingon the degree of constriction, a shift in the relative magni-tude of the directional components of the applied thermalload. The shift effect will intensify when the normal HDC(where the strongest temperature gradient is expected) is


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impaired. This situation will lead to a significant instanta-neous, additional, lateral thermal load. This load representsthe residual normal heat that is in excess of the normalHDC of the material. Depending on the lateral thermal con-ductivity and the associated lateral HDC, the newly appliedthermal load may be dissipated or may be, on the other

hand, tend to accumulate. When the accumulation occurs, arelatively soft zone will form. This zone will be marked bya relatively low resistance to shear. The extent of materialsoftening, and thereby the degree of shear resistance, willdepend on the amount of heat accumulated laterally. If theamount of heat accumulated is less than the critical amountneeded to instigate delamination, the oxidative wear regimewill be dominant else delamination will operate. Thus, inview of this analysis, the change in the thermal conductiv-ity may be viewed as an intrinsic trigger that decides whichenergy consumption mechanism will take place (oxidationor delamination). It is acknowledged that other mechanicaloccurrences are associated with oxidation or delamination.

However, in order for these mechanical occurrences to takeplace a supply of kinetic energy has to be available. Thisenergy is provided, indirectly, through the work done by the

Fig. 9. Behavior of the influential thermodynamic parameters with respect to each other. (a) Variation of the rate of change in heat generation at the surfacewith non-dimensional surface temperature T/Tmax. (b) Variation of the rate of surface temperature rise with the non-dimensional surface temperatureT/Tmax. (c) Variation of the rate of change in heat generation at the surface with the rate of surface temperature rise. (d) Three-dimensional surface plotthat depicts the variation of all the thermodynamic parameters with respect to each other.

friction force through the energy accumulation mechanism.The main catalyst for this accumulation process is the changein the thermal conductivity and the variation in the balancebetween conductive to diffusive effects within the MAZ.

4.3. Thermodynamic implications

The variation in the balance between the conductive tothe diffusive effects also reflects on the continuity of the rel-ative motion between the two surfaces. This is because inthe sliding of one body over another, energy must be con-tinuously supplied to the sliding system or relative motionwill cease. The attainment of zero relative motion can beviewed as a form of an equilibrium state of the sliding sys-tem. Now, the energy supplied to maintain relative motion isdissipated in the sliding body system and produces entropy.As such, the equilibrium states, and thereby the thresholdsof transition in wear regimes, may be thought of as linkedto the entropy production of the system. In particular, the

equilibrium states and transition thresholds could be thoughtof as characterized by local maximum or minimum entropygeneration states.


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898 H.A. Abdel-Aal / Wear 254 (2003) 884900

Entropy production depends on the ratio between the heatadded to the temperature (Q/T). The change in entropy,meanwhile, depends on the difference between the value ofthis ratio at two states undergone by the system. It followsthat the points at which the rate of entropy production ismaximum or minimum will depend on the time derivatives

Q and T (i.e. the rates of change in heat production and rateof temperature rise at the surface and the surface temper-ature itself). All of these quantities, however, are stronglyrelated to the balance between the heat generated at the sur-face and the heat diffused in each of the rubbing pair. Thatis, the balance between conduction and diffusion.

To complete the analysis, two additional quantities werecalculated for the same experimental conditions of Table 2.These being, the rate of change in heat generation at thesurface (Q) and the rate of temperature rise at the surface(T) where,

Q

=Q

t =Q

T T

t

,

T

=T

tFig. 9a-d detail the relationship between these quantities andthe temperature rise at the surface for each of the speeds usedin the experimental work of Molinari. Fig. 9a and b depictthe relation between Q, T and the surface temperaturenormalized by the maximum surface temperature rise fora particular speed. Both figures display similar trends. Inparticular, both figures contain three zones: in the early partsof the cycle (up to T/Tmax 0.15) the gradients of Qand T are so steep. A transitional zone follows between(T/Tmax 0.20.4) and, finally toward the end of the cycle,uniform Q and T are encountered. The same trend isapparent in the plot that depicts the variation in

Q with

respect to T (Fig. 9c) except that the slopes are opposite insign to those noted in Fig. 9a and b. It is to be noted herethat, in contrast to intuition, higher rates of change in surfacetemperature do not require corresponding higher increasein the rate of heat generation at the surface. In fact, it isapparent from Fig. 9c that after a specific value a uniformor quasi uniform change in the rate of heat production at thesurface is sufficient to sustain higher rates of change in thesurface temperature rise.

An interesting trend is noted in Fig. 9d, which depicts thevariation in the three parameters Q, T, and T/Tmax rela-tive to each other. Note the local minima and maxima points

in the surface (observe the shaded points). The local slopeof the surface at these points is equal to zero which impliespotential possible thermodynamic equilibrium (maximumentropy generation) of maximum instability (minimum en-tropy generation). The correlation of these points to actualtransitions in wear regimes for the particular tribo system isof interest and remains a point of ongoing research.

5. Conclusions

This work presented an alternative theory for the ther-mal events that take place during the sliding of metals. The

present theory is based on considering the coupling be-tween the mechanical and thermal states of a sliding solid.The consequence of that coupling is that thermal conduc-tion in a sliding solid will be influenced by the coupledthermo-mechanical state at the surface and within the MAZ(local strain rates and temperature difference).

The influence of the local strain on conduction was de-duced and a so-called apparent thermal conductivity wasdefined. This parameter, which is sensitive to the local dis-tribution of the strain rate, represents the actual thermalconductivity of the solid under coupled thermo-mechanicalloading. Due to the variation in the direction and magni-tude of the strain rate, thermal conduction in sliding mayexhibit an anisotropic behavior (even when the solid is nom-inally thermally isotropic). This behavior was shown to bea key factor in the dissipation of frictionally induced ther-mal loads. In particular, it was shown that the anisotropy ofthermal conduction may cause the flow of heat to be locallycongested. This causes thermal energy accumulation within

the contacting layers. This, in turn, results in making avail-able the energy needed by the reactive gases to jump thebarriers and form a protective layer. Thermal energy accu-mulation was correlated to the transition in the mechanismof wear. This was achieved by the analysis of wear data fora titaniumtool steel sliding pair.

It was postulated that the decrease in the HDC of the solidcauses the formation of a soft zone. This zone is marked byrelatively low shear resistance. This resistance to shear isclosely related to the behavior of the so-called HDC of thematerial. When the HDC decreases to a critical value thesoft zone will have the lowest resistance to shear. This will

cause the delamination of oxide flakes, thus instigating thetransition in the dominant mechanism of wear.

Appendix A. Extraction of the strain rate from wear

rate data

The volumetric wear rate W is given as:

W = dWdS

(A.1)

in which dW is the volume loss (A dZ) per unit sliding dis-

tance dS or:

W = A dZUslid dt

(A.2)

where Uslid (=dS/dt) is the sliding velocity, A the initialcontact area and dZ the vertical displacement of plasticallydeformed material.

The vertical displacement of plastically deformed mate-rial, dZ, in the deformed zone per unit time, dt, can be con-sidered equivalent to the engineering strain rate, e, where

e = dZZ0

1

dt(A.3)


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Substituting Eq. (A.3) in Eq. (A.2), the wear rate is expressedas:

W = AZ0Uslid

e (A.4)

where AZ0, the volume of the deformed zone and U, the testvelocity are constants. As such, if the width of the deformedzone is measured and W is provided, the strain rate may bedetermined. Eq. (A.4) was applied to the data of Straffeliniand Molinari to extract the strain rates.

Appendix B

B.1. Calculation of the HDC and RRH

for the Ti6Al4V alloy

1. The size of the contact spot is determined from plastic

contact theory, i.e.

= 1

N

H

1/2(B.1)

where Nis the nominal load, and Hthe hardness of thesoftest material (Ti in the present case).

2. The total time of contact is determined from:

tc = 2aUslid

(B.2)

3. Calculate the mechanical contribution to the thermalconductivity:

(a) The strain rate in the normal direction is determinedas explained in Appendix A, whereas the strain ratein the plane of sliding is approximated by:

exx = Uslida

(B.3)

(b) The strain rates are substituted in Eq. (15) to de-termine the effective thermal conductivity of the Tialloy, along with an estimation of2T from lineartheory.

(c) The effective thermal conductivities Kxx and Kzzare used in Eq. (16) to determine the temperatureversus time of contact.

B.2. Calculation of the heat dissipation capacity (HDC)

The HDC is given by:

HDC = K

D0

(1 + s) 3KbKe/2T2

(1 + s)tc(B.4)

Substituting the values of K and e in the x and z directionsyields the directional quantities HDCx and HDCz, respec-tively. The total HDC is given by:

HDCtot = H DCxx + HDCzz (B.5)

B.3. Calculation of the ratio of residual heat (RRH)

(a) The heat generated at the interface is calculated from:

Qgen(s) = 3Uslidy(s) (B.6)where s is obtained from Eq. (16).

(b) The RRH is calculated from:

RRH =

HDCtot Qgen(s)Qgen(s)

(B.7)

where is the heat partition factor.

References

[1] R.B. Waterhouse, P.E. Taylor, High temperature fretting and wear oflike metallic contacts, Rev. High Temp. Mater. 4 (1980) 259298.

[2] F.H. Stott, G.C. Wood, The influence of oxides on the friction andwear of alloys, Trib. Int. 10 (1978) 211218.

[3] Y. Saito, K. Mino, Elevated temperature wear maps of X-40 and

Mar-247 alloys, ASME J. Trib. 117 (1995) 524528.[4] H.A. Abdel-Aal, The correlation between thermal property variation

and high temperature wear transition of rubbing metals, Wear 237 (1)(2000) 145151.

[5] H.A. Abdel-Aal, On the influence of tribo-induced super-heating onprotective layer formation in sliding metallic pairs, Rev. Gn. Therm.40 (2001) 6.

[6] H.A. Abdel-Aal, On the connection of thermal dilatation to protectivelayer formation in fretting tribo-specimens, Wear 247 (1) (2000)7886.

[7] H.A. Abdel-Aal, Correlating thermal dilatation to protective layerformation in fretting wear, Int. Commun. Heat Mass Trans. 28 (1)(2001) 97106.

[8] H.A. Abdel-Aal, On the influence of thermal properties on wearresistance of rubbing metals at elevated temperatures, ASME J. Trib.

122 (3) (2000) 657660.[9] G. Burns, Solid State Physics, Academic Press, San Diego, CA, 1990.

[10] A.M. Hofmeister, Mantle values of thermal conductivity and thegeotherm from phonon lifetimes, Science 283 (1999) 16991706.

[11] D. Rosenthal, The theory of moving sources of heat and itsapplication to metal treatments, Trans. ASME (1946) 849866.

[12] J.C. Jaeger, Moving sources of heat and the temperature of slidingcontacts, Proc. R. Soc. NSW 76 (1942) 203224.

[13] H.A. Blok, Theoretical study of temperature rise at surfaces of actualcontact under oiliness lubricating conditions, in: Proceedings of theGeneral Discussion on Lubrication and Lubricants, I. Mech. E.,London, 1937, pp. 222235.

[14] R. Holm, Electrical Contacts Theory and Application, Springer,Berlin, Germany, 1967.

[15] J.F. Archard, The temperature of rubbing surfaces, Wear 2

(1958/1959) 438455.[16] W. Nowacki, Thermoelasticity, Pergamon Press, London, 1965.[17] H. Jeffrey, The thermodynamics of an elastic solid, Proc. Cambr.

Phil. Soc. (1930) 101106.[18] W.A. Day, Heat Conduction within Linear Thermoelasticity, Springer,

New York, 1984.[19] D.Y. Tzou, Macro to Micro Scale Heat Transfer, Taylor and Francis,

Philadelphia, PH 1995.[20] S.P. Clark Jr., Trans. Am. Geophys. Union 38 (1957) 931.[21] M.Q. Brewester, Thermal Radiative Transfer and Properties, Wiley,

New York, 1992, pp. 229230, 374382.[22] F. Wooten, Optical properties of solids, Academic Press, San Diego,

CA, 1972.[23] U. Brydsten, D. Gerlich, G. Bckstrm, J. Phys. C Solid State Phys.

16 (1983) 143.


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900 H.A. Abdel-Aal / Wear 254 (2003) 884900

[24] M. zisik, M. Necati, Boundary value problems of heat conduction,Dover Publications, New York, 1989.

[25] H.A. Abdel-Aal, Error bounds of variable conductivity temperatureestimates in frictionally heated contacts, Int. Commun. Heat MassTrans. 25 (1) (1998) 99108.

[26] K.L. Johnson, Contact Mechanics, Cambridge University Press, UK,1984.

[27] F.A. Moslehy, M.A. Seif, S.L. Rice, Measurement of surfacedisplacements in a tribological contact, Exp. Mech. 30 (1990) 4955.

[28] M.A. Seif, P.J. Mohr, F.A. Moslehy, S.L. Rice, Deformation andstrain fields in pin specimens in sliding contact by laser speckle andmetallographic techniques, ASME J. Tribol. 112 (1990) 506512.

[29] H.A. Abdel-Aal, M.A. Seif, Thermal strains in nominally flat drysliding contacts, Mech. Res. Commun. 22 (1) (1995) 5966.

[30] M.A. Seif, H.A. Abdel-Aal, Temperature fields in dry sliding contactby a hybrid laser-speckle technique, Wear 181183 (1995) 723729.

[31] M.A. Seif, Surface displacements in a tribological contact, Ph.D.Thesis, University of Central Florida, Orlando, FL, 1988.

[32] H.A. Abdel-Aal, Temperature fields in dry sliding contact by a hybridlaser-speckle strain analysis technique, Masters Thesis, TuskegeeUniversity, AL, USA, 1994 (also published as, UMI MS 13-57220,Ann Arbor, MI, USA, 1994).

[33] H.A. Abdel-Aal, On a method to deduce friction-inducedtemperatures from thermal strain measurements in the dry slidingof metallic pairs, Int. Commun. Heat Mass Trans. 25 (5) (1998)609618.

[34] H.A. Abdel-Aal, The deduction of friction induced temperatures fromstrain measurements in the dry sliding of metals, Rev. Gn. Therm.38 (2) (1999) 158174.

[35] Y.S. Touloukian, R.W. Powell, C.Y. Ho, M.C. Niclaou, Thermo-Physical Properties of Matter, vol. 1, Plenum Press, New York, 1973.

[36] G. Straffelini, A. Molinari, Dry sliding of Ti6AL4V alloy asinfluenced by the counterface and sliding conditions, Wear 236 (1999)328338.

[37] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, second ed.,Oxford University Press, Oxford, 1959.

kinetics of friction and thermal energy

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