E L S E V I E R Wear 216 t Iq~)8) 41-59

WEAR |

On friction-induced temperatures of rubbing metallic pairs with temperature-dependent thermal properties

H.A. Abdel-Aal *. S.T. Smith D~Tuirmrent ~I Mechanie.I ~.)r,gim'erin¢ Tire Uuiver~itr tqNvrlh Carolina al Clrarhme. Charhme. NC 28223. USA

Received I0 June Ib'97; accepted 27 November 1¢~7

A b s t r a c t

This paper investigates the temperature rises fi~r dry sliding systems when the variation in the t~rmal conductivity with tem~ralur¢ is taken into account. For the purl~se of the analysis, it hgs been assumed that the thermal conductivity of the robbing materials vary linearly with temperature. Accordingly. materials are classified into three categories ba~d on that variation: materials ftn" which the conductivity drop with temperature elevation ( class a ); materials for which the conductivity increases with temperature elevation (class b); and materials fat which the conductivity-temperature curve has an inflation Ixfint ( class c I. The variable conductivity temperatures are o~ained by applying the so called "Kirchofftranstbrmation" to the fundamental solution of the heal equation. The results indicate that the behavior of the conductivity with temperature is significantly influential to the magnitude of the temperatures reached by the mlxbing pair. Fat a variety of sliding pairs analyzed in this work, significant variatmn between the constant and the variable conductivity predictions were fimnd. Far example, the temperature rise for a mild steel ~ AISI 1020) rubbing pair, sliding at 6 m/s and 30 N m~minal load. predicted by the variable ¢~aaluctivity solution is about 30c,~ higher than that predicted using a constant conductivity solmion. It is also shown that the estimates of the heat c ~ l g d through the surface may be in error ( by about 30-4(}c~ . ) if based on a constam conductivity solution. Such behavior has direct effects on the thickness of the thermally aflected subsurface layer ( the so-called thermal skin ). and the thermal distortion of the contact interface. The error intr~gluced in the estimates of the temperature rises for class c materials is shown to be prol~)rtional to the ratio between the inflation to the melting temperatures of the moving solid. ~[.~ 1998 Elsevier Science S.A. All rights reserved.

Keywords: Conductivhy: Temperature: Kirchoff Iransfiwmatilm

I . I n t r o d u c t i o n

Most surface temperature analysis has been based on the

pioneering work of Blok I I I and Jaeger 12 t. Blok and Jaeger assumed that a heat source is applied at a single insulated region of the surface, and that it had been acting for a suffi- ciently long time. The same assumptions essentially apply to the work of Archard [ 3 I, who simplified the formulation of the theory for both slowly moving and rapidly moving con- tacts. Refinements of the theory continued mostly being based on the solutions outlined by Carslaw and Jaeger 141. This later work was used extensively to study transient tempera- ture distributions in the vicinity of heat sources. Cameron et al. 151 and Sinclair 161 modified Jaeger 's solution by con- sidering multiple heal sources. Kuhlmann-Wilsdorf [7.8[ developed an approximate solution for a single source at intermediate velocities ( between the slow and fast velocities of Archard's work). Marscher [91 developed a computer

* Corresponding author, c-mail: Haabdela(-'UNCC.cdu

0043-1648/98/$[9.00 ~q 19~}8 Elsevier Science S.A. All righl.s rc~:r*,cd. PII S0U43- i (~48 ( 97 )00286-X

program that calculates transient surface temperatures due to multiple interacting heat s(mrces, and considers both one- dimensional and three-dimensional heat flow. Barber i IOi introduced an asymptotic .,adution for the short time transient heat equation ha~d again on heat source models. Kuhlman- Wi]sdorf I I I I pre~ntedexplicitequations for tbecalculation of the flash temperatures at plastically deformed circular con- tact spots due to friction and Joule heal. A review of the development of the theory has been published by Kennedy I 121 and more recently by Pervereza and Balakin 113 [.

Despite the abundance of information in the literature. frictional heating calculations entrain considerable uncertain- ties resulting from the ill-definod boundary conditions at the interlace. This is mainly due to the ~nsitive interdependence of the factors affecting the penetration of the heat flux into the hulk of the rubbing pair and the contact parameters 1141. In sliding, parameters such as velocity and nominal normal loads may be characteristic to a particular sliding situation: other parameters such as thermal conductivity, diffusivity.

actual contact stress, and thermal expansion will change as a

42 H.A. AhdeI-AaL S. 7:. Smith / Weur 216 ( i~J98 J 41-5~

function of the inteffacial temperature. The temperature, in turn. develops with the contact time and is affected by the change of sliding conditions due to friction-induced heating. This infers that the equation that governs the conduction of heat in a sliding-friction system is nonlinear in nature. How- ever. to avoid mathematical difficulties, several authors apply a linear heat equation to frictional heating problems. As such, the temperature estimates thai are based on the linearization of frictional heating problems inherently contain uncertainties.

h has been shown I t 51 that the behavior of the thermal conductivity with temperature is influential to the contact temperature ri~. In parlicular, if the thermal conductivity of the rubbing material reduces with increasing temperature (e.g.. low carbon steels and cast iron), the real temperature ri.~ is higher than that predicted by a constant conductivily solution. Conve~ely. if the conductivity of the rubbing mate- rial increases with temperature (e.g., stainless and nickel steels l, the real temperature rise is lower. This situation affects the estimates of the real quantity of heat transmitted through the surface. Consequently. by assuming thai the ther- mal conductivity is independent of temperature, an order of magnitude error in thermal distortion estimates may result 116i.

Surprisingly. considering the potential influence of the var- iation in the thermal properties wi th temperature on friction and wear, inve~;tigations that deal with a variable property heat equation within the realm of friction studies are scarce, In fact° the only work known to the authors is thai of Ling and Rice j 17 l- The approach of Ling and Rice was to idenzi fy two functions: the first,f, represents the variation in the heal capacity of the solid (product of the density and the specilic heat) with temperature: whereas the second, g. represents the variation in the thermal conductivity of the solid with tem- perature. Subsequently, they assessed various temperature- behavior combinations of these functions. They concluded thai the dominant function is the g function that pertains to the thermal induced variation in the conductivity. Addition- all)', the parameter (.Ig) ~;~ was identilicd as an indicator for when il is necessary to consider the temperature dependence of the relevant thermal properties. One problem not addressed by Ling and Rice was the development of the interfaeial teinpe-atures when materials with an intlation fKfint in the temp .~rature-conduetivily relation (e.g.. titanium and zinc) rub a gainsl each other, Moreover. the effect of Ihe tempera- :,ire dependence of the thermal properties on the partition : f frictional heat among the rubbing members was not considered.

The objective of the current work is to investigate the effect of ihe variatmn in the thermal conductivity with temperature on the contact temperature risc~, [=or tile purpose of the snai- l.sis, it has been assumed thai the thermal conduclivily of the rubbing materials vary linearly with temperature. Accord- ingly, materials arc classilied into three categories based on that variation. Tbe~: being materials fiw which the conduc- tivity drop with temperature elevation ( class a ): materials for

which the conductivity increases with temperature elevation (class b): and materials for which the conductivity-temper- ature curve has an inflation point (class c),

The thermal analysis of the temperature rises is twofold. Firstly, we seek the constant conductivity temperature rise due to a point source moving on a semi-infinite space ( with no internal heat generation). Subsequently. we obtain the transient temperature rises due to a uniformly distributed continuous heat source that slides with a uniform velocity on a half space. Secondly. the obtained solutions are modified to accommodate the temperature dependence of the thermal conductivity. This is achieved by applying the so-called Kir- choff transformation.

Results of this analysis indicate that when the thermal conductivity of the material decreases with temperature, the actual contact temperature rise will be higher than that pre- dicted on the basis of a constant conductivity solution and vice versa. When the thermal conductivity of the material has an inflation point, the actual temperature rise may increase ( or decrease ) with time until it reaches a critical point beyond which it drops (or increases). For a variety of sliding pairs analyzed in this work. significant variation between the con- slant and the variable conductivity predictions were found. For example, the temperature rise for a mild steel (AISI 1020) rubbing pair, sliding at 6 m/s and 30 N nominal load. predicted by the variable conductivity solution is about 30% higher than that predicted using a constant conductivity solu- tion. h is also shown that the estimates of the heat conducted through the surface may be in error (by about 30-40c/¢ ) if based on a constam conductivity solution. Such behavior has direct elTects on the thickness of the thermally affected sub- surface layer (the so-called thermal skin), and the thermal distortion of the contact interlace.

2. Thermal analysis of sliding pairs

When sliding occurs between two confiwming solids heat will be generated. The amount of that heat depends mainly on the sliding parameters ( speed, load, actual contact stress, etc. ). The heat liberated due to rubbing represents the work of the friction force tess the amount oi" work actually con- sumed in plastically detbrming the surface asperities and pan of the underlying layers (the mechanically affected layer). A s such, sliding friction entails internal heat generation due to tile plastic defi~rmation of the sub-contact layers. This implies that locally, the mechanical, especially the hardness and the yield strength, as well as the thermal properties may vary with temperature. This variation has implications for the size and number of the contact spots, and for the ratio between the real area of contact to the apparent area of contact.

The variation in the hardness of the rubbing pair during contact is a deterministic function of the temperature rise. The nature of this variation is rather complex as it incorpo- rates several interdependent factors. These include, work hardening, sliding speed, and temperature elevation effects.

H.A. AhdeI-AaL $.7", Smith I Wear 216 ( 1998~ 41-39 43

The local strain rates (~U,t~d/r;,) in sliding friction are high: typically they range from 10 to ]Ot'/s. At the~ rates, metals respond in a way which is best modeled by low- temperature plasticity ( 'dislocation glide') rather than creep even at high temperatures [ 18l. A con~quence of this is that increasing the strain rate would cau~ the hardness to increase. Now. since heat generation is associated with slid- ing. and the effect of heat is to cause the hardness to decrease. one would expect the hardness to drop. The question that arises here. however, is concerned with the magnitude of that drop. That is, for a given temperature ri,,¢, is the final hardness higher or lower than the bulk room temperature hardness of the material'?

It has been reported I 19-21 [ that the microhardness of the asperities may be work hardened to double the value of the hulk hardness. Now, with the assumption that heat is gener- ated at the tips of the contacting asperities, the following scenario (drawn along the lines of Ref. 1221 ) may be con- ccived, At the interface, the relative velocity between the two sides is accommodated by plastic shearing within the deform- ing layer on the softer side. Correspondingly. the local plastic shear strain rate of the deforming layer can be very high. So thai the increase in the yield strength and the hardness with strain rate can heroine signilicant (twice the bulk hardness as mentioned earlier). When heat is generated, the hardness drops from that hi[:ll hardness to a lower value that depends on the temperature gradient, it is unlikely in this .~enario that the final hardness wonld be lower than the original Iroom temperature) bulk hardness of the material unless a consid- erably high temperature rise is encountered. Such consider- ations, caused Lim and Ashby ! 23 l, for example, to conclude that both heat and strain rate effects would ronghly cancel and that no serious change in hardness would occur ( espe- cially for steels).

A counter argument, to the contrary, may be formulated on account of wear and heat penetration effects. Consider the following alternative scenario in which initially, a single point of contact, point one, between the sliding solids is considered, Due to sliding, a small amount of heat is generated. The heat transferred away from the contact spot will set a temperature gradient in both solids, rising toward the point of contact. The resultant thermal stresses will cau~ the point to rise above the immediate surroundings. The point will also wear and eventually, to the level of the surroundings. Suppose that another point, point two, is brought into contact as a result of either the disengagement or the wearing down of point one. The new point will share part of the load and the heat input to the first [x)int will he reduced. As a result of heat heing conducted into the new point, the point will expand and it will bear a higher load. Meanwhile. due to the continuous penetration of heat, the surface and the hulk temperature are rising and the bulk hardness is decreasing accordingly. If sliding continues, the expanded spots where the pressure is now concentrated will wear down until contact appears else- where. The new contact spots proceed to heat. expand, and carry the load, The old ones relieved from load. c~)l. contract

and .separate. This process of localized peeling off of the surface and exposing new contact asperitiescontinnes, Every newly formed contact spot. however, engages the surface with thermal and mechanical properties (including harcbtess and yield strength) that reflect the thermal history of the contact layer. So that. with the evolution of contact lime, the resultant hardness (due to thermal and hardening effecLs) may be lower than the room temperature bulk hardness of the material.

Con~quendy. any physically comprehensive attempt to analyze the true temperature rise should ineoqmfate the effects of temperature elevation on the relevant mechanical and physical properties, This approach, however, will not he strictly adhered to in the current work as the main focus of the analysis is to study the effecLs of the variation e t a single property, namely the thermal conductivity as the temperalure ri~s. It follows that the mechanical properties to be consid- ered are temperature independent (except for one computed example presented in Section 9).

To calculate the temperature history of an asperity contact, it is necessary to know the time-dependent geometry of the contact, the energy dissipated at the contact and the lhennal properties of the material involved.

The geometry of the contact may be approxima~d by a .~ries of spherically topped asperities all having the rome radius of curvature although with a distribution of heights. The~ assumptions imply that the asperities are toughly of the same size. that any one contact has a circular cross ~ction, and that the contact cross section remains circular Ihronghont any one contact cycle. Under these idealized conditions lhe problem reduces to the study of the temperature r i~ due to a sliding of a moving sphere over a stationary sphere. When the moving sphere engages the stationary sphere, the real area of contact slarts to grow. When the moving sphere is directly above tim stationary sphere, the area of contact is at maxi- mum. As the moving sphere slides away. the area of contact gets incrcmenlally smaller until it is reduced to zero,

In the current work. we shall consider that the heat gener- ated during sliding originates at a mathematically flat piano at the exact interface between the contacting asperities ( Fig. I ). This may not be strictly realistic from a physical point of vk:w i 22 I. However+ such an assumption is adequate within the current mathematical formulation. Additionally, it is assumed that the coefficient of friction depends only on the velocity and not on the temperature and that the conlact stress is either elastic or equal to the hardness of the bulk of the sorer material. Whence, the frictional heat that is liberated duc to rubbing is expres~d as,

Q=I.tN U,,,,tA,,/A~ ) , l )

The analytical relationships for thecalculation of the radius of the real area of contact are given as 124 I.

[- N 1'/-" ""=L ~ J (2a)

44 H.A. A&IeI-AaL .T.T, Smith/Wear 216 flOOR)41-59

(a) (b) (¢)

(3 Direction of sliding

a~.~al ~vlnlaet.sp~ ~ J dec'non of caat~l d~slartlon

Fig. I. TL: model of contact used in lhe current work. C-ntact asperities art, c(msidered as spherical caps having the same radius of curvature t solid lines ). v,.ilh a circular cross ~-ction. The area of contact changes with motion: a. alter sliding a linite distance: b, moving asperily amp ofthe slatkmary asperity, contact area is at maximum: c. after sliding a distance. 2a. cantaet area just reduced to zero. Dashed lines represent actual events in sliding. When sliding starls. Ihe aspefilie.', deform plaslica~ly and the ,contact sl~t is distorled.

/

t = 0 t = t' t = t X r

./ ?.." X

Fig. 2. The gcomctr2, of the mo~ ing heat source problem.

~ X

N i/~ , :b,

LE,.j where the subscripts , p and e denote plastic and elastic contact condi t ions , respect ively.

2. L C~mtacl temperature risex

At the sl iding contacts , the heat source m o v e s a long the

surface and, therefore, the temperature rise at a g iven point changes with t ime. If a Canes i an c tmrdinate sys tem is set s tat ionary wi th respect to the heal source and the semi- int ini te

m e d i u m is a l lowed to move relat ive to the heat source, the

heat tvanster equat ion govern ing the temperature field near

the heat .source takes the form.

otV "-f-)= i~H +U.haVH ( 3 ) iat

The solution of Eq. (3) may be obtained by adding the contribution.~ of an inlinite number of small instantaneous sources placed one behind the other at infinitesimal intervals of time along the direction of motion ( Fig. 2 ). I f the direction of motion is taken as the x-axis, then the contribution of an inlinitesimal instantaneous point source ofstrengthdQplaced at a d is tance a'~ from the origin at a t ime t ' may be expressed

as 125 I.

i d,' 4k ( r r i t - t ' )) ~i,~ j (t_t,).~/,-

o

Nexp - - ~ .%'--.lt-l--!U,hdd/) 2+.V2+Z2

(4 )

HA. AbdeI-AaL S,7:. Smith I Wear 216 ( 1998~ 41-59

When the point source moves with a uniform velocity and the heat generation is a known function oft ' . the temperature contribution of a point source becomes.

! df 4~r/2 ~ ( t - l ' l ~:"

t ] . r -x~- fU, ,d dr') + y 2 + , 2

t)

Now. let

t - t ' =¢~'. x - U , , d = ~ . ~2 +y_-+:2 = R 2 (6)

There follows.

Op t v / ' ~ 0 expl-~U,,, , ,~ I 2 , r ~/2k

' }] fd~" A R-" , , X ~2exp - - ~ ~.~-U,L,: ' - (7)

Steady state temperatures can be determined by ~tting I in Eq. (7) to infinity, whence:

~7~.= Qr----expl-AU.,.,( R+~) I (8)

45

dr" = - 2mdm. dx' ~-- (edL:. dv = 2 . "~mdv I ( I I )

Substituting Eqs. ( I0) and ( I I ) into Eq. (9), the cou.,~am conductivity temperature. 19 (z.y.O.t), may be written as:

~.t.,'.,-..,= ~ Jd., • " 1121

X l ! @ 0 exp{-(~'-~ +v/-" )}d~'d'r/,

Oh,,(X'.y.O.II = ~ ! dw

(13)

x ~ f ( I - @ ) O exp{-(~"~ +Tu') jd~dl~],, R

for bodies I and !i. respectively. The quantity of heat received by each of the rubbing members is found from the continuity of the temperature across the interface. Now, assume that the temperature ri~s of the bulk in each of the ruboing materials were zero. Then. the imposed condition across the contact band takes the form:

~,{ x'.~:O.l )=O,t( x'.y.O.t ) (14)

2.2. Transient sue'ace lemperulure rises due to a t'otltinuon.~ iOl~t'nl ~war soterce

When the heat source has moved only for a finite interval of time, the temperature field may not be at a steady state. In this case. the fundamental point source solution, Eq. (4). has to be integrated with respect to time. Thus, the temperature of a point P(x, y, z) at time t is obtained by the summation of the temperature rises caused by the heat 0d.~dydt at each instant of time from t = 0 to t. Whence, the temperature, at the surlace Z=O. is expressed as.

Oh(x.y.O.! l= 4,r ~/~ /~ J ( t - t ' ) ~¢~

!f[ ] X exp -~.~..{l.~-.~'l-~+y -'} d.Vdy (9)

Where x' has been substituted for ( t'~ + U.,,d( t - t ' ) ). Eq. (9) has an imaginary singularity at ( z - t' = 0 ). To overcome this. we introduce the following new variables:

(.~-.r ' ) ~2A yV/~.A , o = ( 1 - t ' ) 1:2. ~= - - . r/ I 10)

2m 2m

Then,

3. I[k.pendmce of the thermal cmdtgtivity oa tem~rature

Al l materialsdisplay a variation in the thermalcond~dvity at different temperatures. Under normal sliding conditions. this variation may he classified into three l~sic categories. The~ are summarized in Table I. The variation in the con- ductivity with temperature may be re~e~nted in one of two ways. The first is to average the values of the conductivity at different temperatures. For example, if the surface of the

Table I The cla-'e.itication of malcrials aczaa-ding to Ih¢ varlalion in c,mdnctivity wilh tcmj~ralum

Material ~lass Omducfivity behavior with temperature

C']a~s a

Class h

C'l~s~ c

Condncfi~ily drops wilh [cmpcralurc elevation ( e.g.. ¢arl~m steels, sapphire, and zirconium ) Conductivity incma.'~s wilh Icmpcr'aturc ~lcva6on ( e.g.. slainless stccl.~, duralumin, and ca.sl iron The tcmperalurc--conduclivily curve includes an inltalion poinl. Thai is. the conductivity of Ihc malcrial incre;~s ( or drops) with lcmperuture, reaches a maximum f or a minimum I. ihen drops ( or increa.'cs ): c.~.. litanium, zinc and vanadium )

4 6 H,A. 4hdeI-Aal, S.T. Smith / Wear 216 ( 199~.141-59

sliding material is at a temperature (9_~ whereas, the bulk of the material is at a temperature H.. We may deline an average effective thermal conductivity of the form.

1 ~ k d ~ (15) k~,~- H,-H: HI

The ~cond method is to assume a linear variation of the conductivity with temperature, i.e,,

k(O)=k.( l + /3 .0 ) (16)

Note that ko is the thermal conductivity at a "r~ference" tem- perature that is not necessarily zero. In fact. the conductivity at zero or at room temperature may be interchangeably used as a reference conductivity with less than O. I C/r error intro- duced in the value of/3126 !-

4. Variable conductivity solution

When the thermal properties of the solid vary with the temperature, but remain independent of position, the heat equation governing the conduction o1" heat in the material is nonlinear. To change the dependent variable in Eq. 13 ), we apply the Kirchoff transfi}rmation 127.281. To this effect deline a ne~ dependent variable. U, such that.

(k((.9) dH U= j - - ~ . (17} ii

Substituting Eq. (17) into Eq. 13l gives,

av- 'U= ijff_U +U,,,,~U (18) at

Eq. I 18) is still nonlinear as the diffusivity, at. is a function of temperature. Now. there are two quantities that contribute to the value of the diffusivily, the thermal conductivity, and the thermal capacity ( product of the density ,'md the specilic heat ). For metals. Ihe specilic heat i;;creases with temperature elevation, whereas, the density decreases slightly ( by a factor of 5 × I0 ~/°C 12OI ). Con.~quently. the change in the ther- mal capacity would tend to somewhat oll~et the effect of the variation in the conductivity with temperature. As a result. the change in the conductivity is two to three folds the cor- responding change in the di ffusivity, For example, the change in the diffusivity of Duralumin at 6(X)°C is approximately - 0.04¢7r of that at r(~)m temperature: while the corrcH'~md- ing change in conductivity is approximately O. 13q, of that at nmm temperature. Essentially, the same applie~ to steels, where the change in the diffusivity at 9(X)°C ( fi)r example I compared to the n~ml temperature ~alue is almost one third of the corresponding change in conductivity 13111. So that the dominant thermal influence at the interface, surface Z--- O. is the thermal conductkity. This was also implicated in the v.'ork of Storm 1291 and Ling and Rice 117t. Admittedly.

however, one cannot afford to neglect the diffusivity variation if an accurate estimate of the penetration depth ( thickness of the thermally a fleeted layer) is desired. This thickness clearly depends on the point wise variation in the diffusivity which in turn depends on the temperature. Thus. if the variation in the thermal diffusivily, c~, with temperature is neglected com- pared to the variation of the conductivity, Eq. ( 18 ) becomes a linear partial differential equation. This may be solved using the usual techniques provided that the boundary conditions for the problem are also transformed.

Frictional heating problems are usually solved with bound- ary conditions of the second type [ 31 ]. These are used when the magnitude of the heat flux along the boundary surface is prescribed. That is at the boundary surface j,

k(P)) ~ =]; ( R.t) 119)

Eq. (19) is nonlinear as the conductivity depends on tem- perature. This boundary condition may be transformed into a linear prescribed heal flux condition by means of the Irans- fi)rmation given by Eq. (17). Whence. the transformed boundary conditions assume the fi)rm,

aU k ,, i ' h =Ji(/¢,t ) (20)

at the boundary surface j. The solutions of Eq. ( 17 ). subject to the boundary condition Eq. 120). yields the function U. To obtain the values of the variable conductivity tempera- lures. (:), we apply the inverse transformation,

I ~b(~.,t= ~(¢i+2t~U-l) (21)

Note that as Eq. (21) is based on a linear variation in the thermal conductivity, the application of Eq. ( 171 is restricted by the behavior of Ihe product 2flU. When the conductivity increases with temperature, class B material, the flcoefficient is positive and the quantity ( I + 2/3U) ~/-' may be evaluated fi)r any U value. For a class A material, however, the c(~tlicient is negative. So that the quantity ( I -2 f lU) ~'-' is rendered imaginary beyond a critical U value above which the product 2/3U grows larger than unity. This critical point delines when the inverse transformation breaks down. As such. we may detinc the limits for the application of the transformation as.

I :'=>/3 > - ~ ( 22 )

2U

When it is anticipated that the maximum U value would exceed the criti,:al value, the analysis may be attempted numerically (iinile dift~rence .... finite element, etc.) or by the application of an inverse transforntathm that is based on a higher order regression of the conductivity variation, such as the quadratic form used in Ref. 132 I. However. even with such enhancement, im analytical closed fiwm inverse trans- fiwmation is generally unattainable 133 ]. This is because the

H.A. AhdeI-AaL S.T Smith / l, Veor 216 ( 1998141-59 4 7

inversion process may entail the integration of a system of coupled simultaneous ordinary differential equations.

5. Heat partit ion

To analyze the thermal processes in the vicinity of an individual contact spot. a heat partition factor. (b. is intro- duced. Such that. ~Qr heat units flow into body I. and I I - q ' )Q, units flow into body. II. In developing an expres- sion for 4). it is normally assumed I 1.21 that the thermal properties are constant. However. the temperature rise influ- ences the thermal properties and thereby the partition of fric- tional heat. To obtain an expression for the heat partition factor, when the thermal properties vary with temperature. the actual temperature, O, ( and not the function U) has to be continuous at all points of contact 134 I. That is at the surface Z= 0 the temperature continuity condition is:

(~d R.i ) = (~u( R.t ) (23)

Substituting Eq. ( 21 ) into Eq. ( 231. we may write.

I -;11V/l + 2 0 , / ~ , - I )= ~-: t 1/I + 2 0 , , [ 3 , , - I ) (24)

where the dependence on (R.I) is omitted 1br brevity. Eq. (24) indicates that the heat partition depends on position and on time. Now let the apparent area be divided into m × m subdivisions. So that. after substituting from Eqs. (12) and (13) for the value of 6)~ and 69,,. the imposed condition at the surface, for each time step. takes the form:

I/3.( I +A,/3,@ F,Q)" : - /3, , I',~' l,

=1/3,( I + A.,/3,,I I - q~}F:0)"-' -/3, I',;' I . (25)

where.

A,= 2 ~ A . = 2 2 ~ . rr -~k;" ¢r ~J-'k. (2On)

r,-- j" j" e~pl -t ~: +¢ J}d~ d~ ,, (2oh) R

F, = f f exp{- I~ - '+ ,= I}d~d, I i26c) R

Note that ¢~ in Eq. 125) is assumed constant only lbr a particular subdivision ( i.j ) and not for the entire surface of contact. A binomial series expansion of the terms in paren- theses in Eq. ( 25 ) yields.

I {I+A~tQF, qJ}":: I+ ~A,~,QF,~b- ... 127a)

{ I+A.~ .QF. ( I _ </~)} v:

I • I - = 1 + ~_Auu~uuQF. _ - ~A,o~,nQF:~I~... (27b)

Substituting Eqs. (27a) and ( 2761 in Eq+ (25), and neglect- ing higher order terms, we may write:

[ ' T ¢ , + ;~ ,¢ i , ,A,QF,4~- . . . -¢hl

-"-" 1281

,.1:. Finally. upon substituting from Eqs. (26a). (20b) zlnd 126¢) the variable conductivity heat partition function. { (kl at each subdivision takes the form.

c 4 l i _ I bl,.,,-I{~/~p,,C,,F.} ×{~,p,C,~,+ Vk?p.c,;r;}- 'ry ~_~)

where the conductivity varies with temperature. As such [ ~ ! is a nonlinear function that varies implicitly with temperature. +,'he .solution of Eq. (29) is obtained numerically step by step. Note that this expression incorporates a coupling belween the thermal capacity (product of the density and the specific heat) of the mating materials. This fails in line with the ob~rvation of Ling and Pu [ 35 ) that the heat capacity of one or both rubbing members has an eric:or oil the ¢ominuity of the temperature at the interface through its ipfluem'e on the conditions of heat removal.

6. Temperature ¢aknlation

A computer program, written in FORTRAN-Ch'~. is u~d to integrate the equations developed in the preceding sections. The program incorporates a .segment that computes the num- ber of the contacting ~.sperilies and the coefficient of friction by using the equations developed in Ref. 123 ). The number of contacting asperities, according to these equations, is assumed to be a function of the nominal load. the radius of a single contact asperity ( assumed as IO/~m), the nominal area of contact ( assumed to be a circle of radius 1.5 m) and the sliding speed. The coefficient of friction, however, isassumed to depend only on the sliding speed but not on the tempera(are rise. The program also assumes that the total friction beat is due the interaction of n pairs of contacting asperities at the interface ( with each pair dissipating an equal amount of fric- tional heat).

To compute the temperature development at the contact spot. Eqs. ( I ). (12) and (13) are used to calculate the con- stant conductivity solution, the function U; this. in turn. is transibnned by means of Eq. ( 21 ) to yidd the variable con- dactivity temperatures. This procedure is carried out numer- ically by integrating the respective equations at infinitesimal time steps At. obtained by subdividing the time of contact T¢ ( see Appendix B).

Fig. 3 illustrates the algorithm of computations. To start the calculations an initial guess for the constant conductivity heat partition fizctor is supplied by assuming that the frictional

4~, H.A. AbdH-AaL S.T. Smith / Wear 216 (1908) 41-59

( a ) ( START )

I Time of Contact

T~

. . . . . L . . . . . s

Determine ~al area of contact Ar

calculate number of contact asperities n

calculate time o f contact

I Determine time step

-IJ solve eqs. 12A3,14

evaluate actual Ihermal properties [

apply inverse e n transformation "1 (21)[

I calculate actual heat partition I

. I ", [ calculate actual contact l

tsmpcratures I

h, time s t e p . ~ continue )

I ) Fi~. 3. Flow churl of the ¢omputalional procedure.

heat is partitioned in the ratio of the thermal conductivities of the two materials, i.e..

~ , , = kl kl +k u (30)

This initiai guess is used in Eqs. (12) and (13) to obtain ;~ preliminary value for the constant conductivity temperatures of the rubbing pair. The~, in turn, are used to evaluate the actual thermal conductivities of each of the rubbing materials, by evaluating Eq. (16). At this po[.m an approximate estimate of the actual heat partition factor q~ may be readily evaluated. Whence, the actual amount of heat going into each of the rubbing pair is known, and an improved estimate of the con- stant conductivity temperature is obtained. Subsequent to this step the inver.~ transformation. Eq. (21 L is applied L~ the

Table 2 The: pgol'~-nic~, of the: malerial.,, u.~d in ¢alculalhm. ~. [ 3K I

current value of the constant conductivity temperature thus the variable conductivity temperature O is obtained. This procedure is carried out itcrativcly at each time step until the solution converges to yield the temperatures of the contact band.

7. Materials and computation conditions

The rubbing pairs selected tor temperature evaluation were chosen to represent the three ~mups identified in Section 3 al~we. Table 2 provides a summary of the physical properties of the selected materials. The variation in the thermal con- ductivity with temperature was modeled by means of Eq. (16). using both the conductivity at 0°C and at 20°(: as

Material K p ( ' a x 10 ~ H i I Wlm "(') (kgtn]~.~ IJ /kg"C) (m: l~) (GPal

AISI 1020' 51 7~X) 4~0 0.97 1.04 Sapphire" 4n 398o 758 1.326 19.6 SS AISI .'~gl HN" 14.~ 79(X) 477 I).395 1.035 Titanium ( TI-8 AL- I MO- I V )' 21.9 451~) 522 0.932 2.853 Zinc" 121 .I) 714~.) 3K5 0.44 0.4 AIS1521(NP 37.0 7(~JO 4~5 1).956 l-L,5

'Sur, cr.~riin I¢ltcr.,. ~'~,igna~¢ maIcrial ¢l',.,,s a.~ 1o the ¢la.~xiiieatiorl of th~ eurren! ~,ork. H ~ d n ~ expressed a~ Ipre~urc.

H.A. AbdeI-Aal. S.1". Smith I Wear 216 f 1998) 41-59 49

Table 3 The values of reference and regression conductivities for rig material used in the calculations

Material k.. (Wlm °C) ("C)

AISI 1020 5t.S23 - {),O0(g~

Sapphire 28.2 - 0.rglo~ I SS AIS1304 HN 13.38 0.1~10874 AIS152 IIX} 35.4 - 0.0¢}04

reference conductivities. It was lbund, however, that for all the materials considered, the choice of either of them refer- ence conductivities is relatively immaterial as detailed else- where {26]. As such, it was decided to u ~ the 20°C conductivity as a reference value. The values of the reference conductivity kn, and the corresponding fl coefficients are pro- vided in Table 3.

Modeling the conductivity of class c materials is an inter- esting feature of the current analysis. Given the variation in the conductivity of such materials with temperature, it would be quite inaccurate to calculate a single line that repre~nts this variation (as this would affect the accuracy of any tem~ perature estimate based on the single line model ). To enhance the accuracy of the estimates we repre~nt the conductivity variation by two lines of different slopes. To demonstrate this procedure, consider titanium for which the conductivity reaches a minimum at 400°C, We shall consider that the conductivity temperature curve is composed of two linear segments of slopes/~lt and ~., respectively. The values of these coefficients is calculated by applying Eq. (25) to each temperature interval independently. Thus, the first slope, ,~l,, is valid for the interval, 0°C < (9< 400°C. Whereas the me- and,/3_~, is valid for the interval 400°C < (9< tO00°C. Simi- larly for zinc, which in contrast to titanium, has a maximum conductivity at 200°C, we model the conductivity tempera- tare relation by two lines that reptemnt the 0°C < 6)< 200°C and 200°C < t ')<400°C temperature intervals respectively. The values of the temperature coefficients for both titanium and zinc are given in Table 4. Note that using each of the coefficients in Table 4 to predict the rubbing temperatures would yield a different value tar the same slioing conditions. The most enhanced value would be expected to result when using the coefficients/3~ and B_,, depending on the tempera- tare rise, in a variable conductivity solution.

& Results aml d i s c m a m

In this.section, we present resultsthat illustrate the behavior of the temperature for the different combinations of sliding pairs chosen in this work. These represent the evolution of the maximum temperature rim at the center of a circular contact spot assuming plastic contact conditions.

The character of the variation of the temperature held and, particularly, of the surface temperature is determined by a number of facuws, including the change of friction power. peculiarities of the contact of the rubbing pair, size, distri- bution, and migration of the contact spots over the friction interface. These factors govern the kinetics of heating of the rubbing pair. All of them factors, however, are affected, mote or less, by temperature, and possibly by sliding .speeds, The temperature in itself is a time dependent quantity. Such that, in this sense, time may be considered a true independent variable in a sliding process. Whence tracing the temperature with time may lead to more insight into the thermal operations during sliding, One might argue, however, that the contact spot attains, roughly, its maximum temperature when it ha~ moved through its radius. Albeit. the maximum temperature attained by the contact spot is a cumulative quantity. That is, the temperature rims depend on the thermal histmy of the contact spot. This includes the histopj of the variation ofimth the physical and the mechanical properties of the r u l i n g pair. Con~qucmly, if the conductivity cocre~tion is aplflied step-by-step, it results in further reduction (or increase) in the maximum temperature rise. Moreover, the efficiency of heat removal at the surface is time dependent. Indeed, it was found 136] that the ratio of the heat dissipated through a single asperity to the total heat generated at the interface strongly depends on the time within the contact cycle (i.e.. the ratio t / t~) . In addition, for ~veral material combinations, the computations revealed that the maximum efficiency of heat removal, again for a single asperity, is always reached t h e n the ratio 6)/6)m~ lies between 0.25 toO.3 regardless of the value of the nominal load or the sliding speed. As such, it was elected to present the results as plots of the temperature rises vs. the time of contact (expressed in units of At). At any rate. several questions do arise. These pertain to: the magnitude of the correction factors involved; the effect of the conductivity variation on the penetration of heat throngh the microronghness; and the effect of this variation on the con- ditious of heat removal at the interface. Perhaps, mote impor- tantly, which effects are more dominant at the interface: ~ e degradation of the mechanical properties with temperature or

Table 4 The values of the temperature cocnicients o[ conductivity fi)r zinc and titanium

Coefficient Zinc Titanium

/3~,,. - O.{RX)2289 {FC < ~ < 6(X)~C

~ l).O0(.g}9159 O°C < ~< 2{XFC

fl_, - 0.(XX)3392 200°C < ~< 4(X)"C

- O.{XN)I 7 O°C < H< 12~g'C -0.0004464 IleC < 19< 400°C

0.1~O2668 40ITC < H< 12~C

50 H.A, AI~I¢I-A,d. 5,7". Smith / Weur 216 ¢ t ~tO 41-:5~

the variation of the thermal properties with temperature, In

l i g h t o f the resu l ts , s o m e o f t h e ~ q u e s t i o n s a re add ressed in

S e c t i o n s 8. I and 8.2.

8. I. Ef fec! o f the con&u' l iv i O' on the he luw ior o f the

t empera ture

Figs. ( 4 }-(6) depict the evolution of the interfacial tem- peratures for different sliding pairs. These being sapphire- tool steel ( AISI 52100). both of which are class A materials: mild s led (AISI 1020) pair, again class A materials: and a

stainless steel ( AIS1304 HN)-mi ld steel pair class B material rubbing a class A material, respectively.

We present three sets of computations in each figure. These correspond to: ( 1 ) the temperatures calculaled by means of a constant conductivity solution with room lemperature con-

ductivity ( denoted, k . ) : ( II ) the temperatures calculated by

means of a variable conductivity (denoted. kr): ( I l l ) the temperatures calculated by means of a constant conductivity

solution, however, using an average thermal conductivity obtained by linear interpolation between the conductivity of the bulk of the material and that at tbe surface ( denoted k,,~).

1,000

9OO

gO0

700

600

-- 400 E "~ 300

O' 0 2 4 {~ 8 IO

Rub time ~ At ( p sees ) Fig. 4. De,.eh~w~.nt of the contact temperatures ri~¢s I~.w a Sapphtre-Too] ...tee] 4 AISI .~2 II}O I sliding pair. Tcml~ruturcs are plotted cgainst the contact time in lime ~,tcp~ ~'./~ - I).52. ~ ~ 14. Load = 30 N./,'.,,~ = 2.54 rots. AT= 35,6 p...,, The-" c~mdithms of sliding match the cxperim~:ntal conditions of Quin and Wincr 137 ]. Note the maximum u:mp,:ralur¢ rise prcdi¢lcd hy a variable conductivity solution ~ > (.~ H)~C) match the measur~..mcnls ~,f Rd', 137 ] ( 0{~)~C < (')< I(XI0°C }.

1,100

880

¢*

~ ooo

E.

44o ~ Ku

E ~ K T

220 E

O o 2 4 6 8 ]o

Rub time ~ At ( p sec ) Fig. 5. Dc,,¢lopn,cat o f the conrail tcmwraturc with rubbing lime li,r a mild steel ~ AIS| IO2(I-AI.S] 1(1201 sliding pair. n = 43. l.t = (I.57. l.(~ad = 31) N. U.~,,, = fl m/.'.,. ~ ' = 5~.6 ~.,,. Nice Ih¢ dil ' tcr¢~¢ I~l~,c~.'n the coll~tanl conducd,~it), ~.olulk|ll (/~,, I. the constalli-avcrage condu~:ti'.ii)' ~,olulion (k~,,,), ai!.d the ~.'~.riablc

H.A. AhdeI.AaL S.T. Smilh /Weur 216 f 1998~ 41~59 51

750

] - - -~° I I - ° - ~'r I

450

0 - 0 2 4 6 8 I0

Rul) tinT¢ x A t ( p sets )

Fig. 6. Development of the c~mta,.'! lempvratarc v,+ith tubbing time fiw a ~,taiale.,,~, .,qc¢l ( ArSI 304 HN )-mild steel ( AISI lU21D tubbing Fair. -= 22./J. =O.31. Load= 15 N. U.,,,, = 12 m/~. A¢=57.6 p.~. The variable ¢onduetivily solution k~ predicts It)wet Iempetulure rises than Ih¢ con.slant c~mducti,~ity solulion. k,,>k/>~+,,,.

For calibration purposes, the load and the sliding speed fi)r the sapphire-tool steel pair (Fig. 4) were chosen to match the experimental conditions of Quinn and Winer 137 I. In their experiments a direct optical ob~rvation technique was used to estimate the temperatures of the hot spots in a Sap- phire-AISI 52100 sliding pair. Temperatures between 900°C and IO00°C were reported which are in lair agreement with the variable conductivity temperatures (closed symbols) computed in this study.

Fig. 5 is a plot of the temperatures developed at the contact spot between a mild steel rubbing pair. It will he ob~rved that the variable conductivity solution (closed symbols) yields higher temperatures than those predicted by a constant conductivity solution. So that, as zhc temperature increases. and thereby the conductivity drops, the ability of the sud'ace to dissipate the generated heat to Ihe bulk of the material is affected.

Fig. 6 depicts the temperatures developed in the ca.~ of a stainless steel-mild steel sliding pair. Here, we note that the variable conductivity temperature rises (open symbols) are lower than those which are predicted by means of a constant conductivity solution. Notice that lbr all the cases considered. thus far. the constant conductivity solution that uti}izes an average conductivity ( as defined above) yields considerably low temperatures regardless of the variation in the conductiv- ity. So that. the validity of any attempt to calculate the tem- peratures based on this average coefficient is questionable ( even in the capacity of a first orde~ approximation ).

Fig. 7a depicts the evolution of the temperature rises |or a titanium-tool steel sliding pair. The conductivity-tempera- ture behavior of titanium has three distinct regions: starting at room temperature the conductivity drops until it reaches a minimum at 600°C+ beyond this lemperaZure the conductivity

of titanium gradually increa,~s. As such. it is expected Ihat 'he variable conductivity ,~lution would yield temperatures higher than those predicted by lbe constant conductivity solu- tion in the interval 20°C < ~ < 600°C: and would yield lower temperatures beyond fiO0°C. [t will bc notioed that the con- stunt conductivity ~lufiou (dubbed L,) yields temperatures that are slightly lower than those obtained by a variable con- ductivity ~lution+ Noie that as the combination of the load and speed is relatively low ( IO IN. 4 m/s ) the maximum temperature rise does not exceed the 600°C marker. Thus, the temperatures display a behavior a.,; that of class a materials.

When Ihe developed contact temperatures envelopo t i c inflation point the material will behave as a linear combina- tion of materials of classes A and B. The temperature rises for this ca.~ are plotted in Fig. 7b for a nominal load of 30 N and a sliding speed of 3.25 m/s+ Temperatures were calcu- lated using three different values of the conductivity, room temperature conductivity, kr: "and two values for the conduc- tivity obtained from Eq. (16), first by substituting the average conductivity coefficient, k . . . . and ~condly by substituting the actual thermal conductivity modeled as two linear ~ g - ments in the manner explained above. Note that using the average temperature coefficient introduces a larger error in the temperature estimates compared to those obtained by using the actual conductivity. Note also. that at the inflation point the actual temperature is less than that predicted by a constant conductivity solution. This trend is not reflected in the temperature plots that are ba.~d on an average coefficienl. Consequendy the use of an average conductivity coefficient would result in the over estimation of the contact spot temperature.

Fig. 8 is a plot of the temperature development in zbe contact spot of a Ziuc-AISI IO20 sliding pair. sliding az a

52 tl.A. Abth'l-AaL S.T, Smith / Wear 216 (1998141-59

5O0

4O0

300

200

I00

q~

750

L -£

ca 450

E 3 ~

.E_

1511

(a)

----0-- Ku ----4t--- KT

2 4 6 8

Rub lime ~ At ( I.t see )

(b)

-,---+0-- kT I

2 4 6 8 10

Rub lime • AT . . s e e

Fig. 7. la) De~'clopmcnt of the ¢onl~t |ClYlp~.'ralur¢ I'iSCS. I~)r a titanium-mild steel | AlSI 1020) slldin+ pair. n=8..a =1).41. Load = IO N. U,+,a =4 m/s. x r = 56.4/~,,. The maximum lemperature ri+ dtm~ not exceed the inllation tcmpcrmurc fur titanium I fi(H)"C L Materials ~havc like ¢las+ a pairs (/~l > k,, > k.,,,. I. <hi Devdopmem of tl~ +omact temperature ri++ toe tilanium-mild steal sliding pair, = 17./.L + 0.43. Lo:,d = 30 N. U,,a ~-3.25 m/s. A¢---56.4 ,as. The Iclllpefalul~s cnclt)~.4.. Ih~ inllati;m len)pcrature 4 point/'+L The r~m lemp~.'r~lur¢ conslam ¢olldu¢livily soluthm I/~, 1, and Ihe averaged/3 ¢oel'licicnT. soluli0n (L~,.). Ihil Io predJ¢l the .,,hill in Ilk: I'~:havk+r o1' Ih¢ Iclnpcralurc. The IXVo ~ coetliciem s,dutiun (kr) reflect.,, this variatlun. Titanimn bch.',vcs as a linear combinalitm of ¢la.-,*s a and b malerials.

speed of 2.5 m / s under a nominal load of I0 N. The behavior of the conductivity of zinc with temperature is opposite to

that of titanium. That is the conductivity of zinc increases with temperature tbr the interval 20~C< 6 1 < 2 0 0 ° C and d¢crea.~s for the interval 200°C < 6) < 450°C. The same pro- ccdure followed for modeling the conductivity of titanium was u.~d here. Notice that the temperatures predicted by a ¢onstanl conductivity solution are slightly higher than those predicted by the variable conductivity solution. Past the infla-

tion tempcr'aturc ( 200°C ) the opposite is noticed. That is, the variable conductivity mlution yields higher values than those obtained by the constant conductivity solution.

The difference between the temperalures given by the con- slant and the variable conductivity solutions for zinc is not as pronounced relative to the corresponding difference in the sliding of titanium. This may be attributed to the moderate conductivity gradient of zinc with temperature ( ~O.OI W / m °C) as compared to that of titanium ( ~0 .06 W / m °C). Moreover, for a combination of low nominal loads and sliding speeds the temperatures developed may not be of a magnitude that triggers a considerable pointwise variation in the thermal conductivity. The difference between the constant and the variable conductivity temperatures estimates appears to be pmtx)rtional to the ratio of the inflation temperature O+,,t- to

H.il. AluIeI-AaL S. 7'. ,~m#h / Wear 216 ¢1~I~) 41-59 53

350

30O

,.--" 250

zoo

g 150

J .E I00

5O

F, mflmhm l~ m

0 2 4 6 g 10

Rub time x ~T Psec Fig. 8. Devclopmcnl of the conla¢l I¢.mpcratnr¢ wilh tubbing lime. fi~r a zinc-mild ~,1¢cl I AISI In20 ~ rubbing pair. Tcmpt:ratm¢s include ih¢ inAati~m poim of zinc ( 2(8)~C |. poim F. n = 36. p, = 0.396, Load = tO N, U,,a = 2.5 m/s. A~'= 53.4 ~'~. The difference between the rtmm temperature etmstant ctmductivily solutinn, k,, the averaged/3 c~:fli¢icnt .~flulion. k+,,. is nu~derat¢. The I'¢V¢~-/,~ cc~:l'ficicn! solution reflects Ihe varialion in Ih~ conductivity. Compare that sohtlhm t() Ihe respective solution in Fig. 7b.

the melting temperature 6),,,¢j. The smaller this ratio. 0.2886 and 0.307 for zinc and titanium respectively, the smaller the difference between the constant and the variable conductivity solutions.

The results pre.~nted in this ~c l ion suggest a close relation between the variation in the thermal conductivity and the predictions of temperature r i~s . Namely. if the thermal con- ductivity decrca~s with temperature (class A materials ) the true temperature rise (variable conductivity solution~ is higher than thai predicted by means of a constant conduclivity solution. If. however, the thermal conductivity increa.~s with temperature (class B material), the true temperature r i ~ is lower than that predicted by a constant conductivity .solution. In case of class C materials, the predictions of the variable conductivity solution depends on the sliding conditions. Thus, if the conditions of sliding are such that the maximum temperature r i ~ is lower than the inflation temperature: the material would behave as a class A material (or class B material ) according to the direction of change in the conduc- tivity ( increase or decrca~) . When the sliding paramete~ are such that the temperatures developed do envelop the infla- tion temperature: the material would behave as a linear com- bination of class A and B materials. The temperature behavior of class C materials, however, is anticipated to affect the heat dissipation to the hulk of the sliding pair in a more compli- cated manner than that expected for class A and B materials. This is due to the very nature of the variation in the conduc- tivity of such materials which, in one contact cycle, might work to increase the heat dissipation ( through the increa~ in conductivity) until the inflation temperature is reached. At this point, the conductivity drops and less heat is conducted away from the surface. This might be thermally destructive to the contacting layer (depending on the ability of the mating

material to dissipate heat). In this ~ n ~ , the choice of ther- mally compatible sliding Fairs should extend to consider the manner and the rate by which the thermal prope~ies, in gen- eral. vary throughout a range of anticipated temperature rises.

8.2. The combined ef fect o f the varialion o f bolh the hardness and the condzwtiri~" o n the contac! s ign temper+~tures

This ~ct ion provides a qualilativc assessment to the role of lhe decrea~ of hardness, as compared to thal o f the vari- ation of conductivity, in determining the contact temperature r i~s. The ca.~ to be considered here is that of the rubbing o f stainless steel (AISI 304 HN) turbine blades, against a ~a l of the .~me material. According to the classification of the current work. this cam repre~nLs the sliding of two class B materials. The constant conductivity ~lut ion was originally computed by Marche r 191, for a speed of 304 m/s and a nominal load of 0.74 N. Mar.~her, obtained his results for the maximum temperature r i~s at the center of a ,square asperity. Firstly, he integrated the 3-D fundamcnlal ~ lu l ion of the heat equation. The results, dubbed by Ref, 19] as 3-D interaction arc plotted in Fig, 9 as closed circles. He realized thai the 3-D ~lut ion predicted temperatures that are in excess of the n~lting points of the materials involved. This was mainly due to the constant load. constant heat flux assampron and, the assumption that the real area of contact was corn- p o n d of one lumped asperity that dissipated all the heat. This led Marcher to link the actual contact load to the yield strength of the materials, then to consider that the yield strength drops with temperature in the following exponential form.

+ry( O)--- o-~, e x p - A I O + 273) ( 31 )

54 H.A. AhdeI.A,L 5.T. Smidt / Wear 216 (1998) 41-5~

2.000

I .BOO /

l , ~ 3.D mlerttt'a,n t-ansztml _ ~,A~ "~-

~ c.+m.wantctmdu c ,ra,zt!., trmlx~t'ature,~ curreclcd

8oo ~ ,

200 . cur ,~" ~ t .' l'¢wletIllm wllh /i.mttvrea/urt.

O-- I , , .

0 4 8 12 16

R u b T i m e * l 0 " 5 { s e c

Fig. 9. The variation of the peak Ilash telnpcralurcs, at dilfereal rub intervals, fi~r stainless sleel ( AISI 3IM HN ) rubbiilg pair. The 3-D uncorrected temperatures and Ilk: constant coqductivity ~,olutitm~. ! closed symbols | were calculated hy Marscher I t} 1. The constam conductivily solution was calculated by assuming that Ih¢ load is prolx~rtional to the yield strength ~r,. The yield strength '.'.'as a~.sumed to decrease exponentially with temperature. This has lowered the temperatures signiticaatly. The variable conductivity ~,olulion ~ open symbols) was calculated using the ( KircholT transfilrmation). Nixie the effect of the condactb, ity variation on t~" Icmra:ralurc.

Thus, accounting for the themml soflening, associated w i t h

heat generation. The resulls, for this case. are plolled in solid .squares in Fig. 9. II will be noticed thai the Iherma] softening has a major influence on the contact temperatures. So that. the temperature rises are considerably lower thai the 3-D interaction estimates. Moreover. a steady state solution is reached earlier in the contact cycle.

The application of the conductivity correction to the vari- able yield strength temperalures results in further reduction of the contact temperature rises l open squares in Fig. 9). Notice that the conductivity variation does not al'fect Ihe time that elapses until a steady state tcmperalure prevails. As such, for this parlicular case. we may infer thai the variation in conductivity lends to augment the reduction of the contact temperature rises, This result, however, is a consequence of the conductivity behavior with tempemlure, In conlrasl, if'the rubbing pair belongs to class A materials the drop in conduc- tivity will cau,~ an increase in the contact temperature rises. The extent of this increase will depend on the sliding condi- lions and on the gradienls of the conduclivily with lempera- lure. It is unlikely, however, Ihat Ihe temperature rise caused by the drop in Ihe conduclivily would match the 3-D inter- aclion gdulion, it may he conceivable, under special condi- tions, that the drop in I~th the hardness and the conduclivity may be of equal importance fi)r class a materials. However, whether thermal softening dominales, in general, over the conductivity variation or vice versa, that should be deter- mined ca.~-by-ca~. The el'feet of the temperature depend- ence of the hardness on the steadiness of the contact temperature fi.~ retnains unafl;zcted, however, by the direc- lion of the conductivity variation.

8 3. l'~gect of the ;'uriati,n m c,nduclivitv , n the heat

comh.' led through the smJitce

The amount of heal conducted through the microroughness follows Fourier's law. This may be wrilten for the surface Z=O as.

q , ~ _ f J . i~(') q'~ , - -k--~-dA, (32)

.4,

When heat is assumed Io transfer where metal to racial contact is established, and the mechanical properties of the material remain independenl of the temperature the producl k il(-)/aZ governs the heal conduction.

To calculate Ihe temperature gradient at the surface.change in the diffusivily with lemperalure must be considered. How- ever, if only a qualitative eslimate is required, we may approx- imate the temperature gradient as,

[ ~ ~-),-o,, ;~Z , , - Z ,

I33)

Fig. 10a-~" depicl the temperalure rises normal to the con- tact spot, at successive rub times, fo r the sliding pair consid- ered in Fig. 9. The values plotted in Ihe ligures correspond to the single asperity interaction solution extracted from Ref. 191 (solid symbols) and the respective temperatures cor- rected lot the varialion in the conduclivity (open symbol.,;). The temperatures are plotted against the distance from the

rubbing surface, expressed in units o f asperity diameters. Notice that. despite the assumption of a constant diflhsivity, the variation in the conductivity leads to a slight exlension of the penetration depth. This extension slightly increases with

H.A. AbdeI-A,d. S.T. Smidt I W,,ur 216 ( lOqS.~ 41-59 55

+ !11(i

^ . .

i 2 l 4

I)ldinC¢ them (',mtKt ~rfil¢¢ m a,prnn dl~mltlet,

i ,2~)

(a) I.IO~

m+ t=-9.l#7 11# ^-~ ) +¢~

"rr~)

\ \ pm

ii I I ; 4

~,olrlu ~' t o 3 a~,l~'rhy diat:llclcP, duc h~ Ih r ~'~ltdU¢lit ilX ~'ariathm

time ( compare the extent of the variable conductivity gradi+ ents in each ligure), St) that the lirst approximation of Eq. (37) yields+ roughly, a constant gradient regardless ol time, As such, the main influence on the heal conducted through the suet:ace is clearly the variation in the conductivity with temperature. In tact, it can he shown that the actual heat conducted through the microconla¢ls is proporlional to the ratio of the actual to the room temperature conductivity. This ratio is given in Table 5, as the factor A for all the cases considered in the current work. Three different values were computed lot each case, These are as Ibllows: (a) A,,,..: the ratio of the conductivity calculated at the maximum temper- ature rise to the r(~t~m temperature conductivity, (b) ,~,+,~: the

Tahl¢ 5 V~IIuc.', t~l" the ¢~,~+duttivily hettt Itow CmTcctioll faclor~.

Rubbing Pair Figure h ...... ,~.,,. ,~,..

AISI 1020-AISI lU2I) Fig. 5 I)..535 t).724 0.348 AIS1304 HN-AIS130.4 HN FLg. 6 1.35X l./)87 O.~7fl Titanium-AISI 1020 Fig. 7a ().~3 O.tX)4 {)+Jh ¢} Tita,fium-AISI 1020 Fig. 7b {L82 O.t.;41 {).45 Zin¢-AISI 11)20 Fig. 8 1).¢~[~ ().¢;~)5s ~).5 Sapphirc-AIS152 Ilg) Fig. 4 0.56 n.78tr~ ().36

ratio of the average conductivity, as defined alxwe, to the rm)m temperature conductivity, and l c) A,..: the ratio of an efiective conductivity, obtained by summing the resistances of the thermally affected layer and the bulk of the material in parallel, to the rt~m temperature conductivity, h wil l be noticed thai when tim: conductivity drop~ with temperature. the amount of he:u conducted through the surface drops accordingly and vice versa. Note that the varialion in the conductivity is not reflected in the factors predicted by an el'fcctive conductivity. So that the amount of heal c(mdu¢led through the surface is always lower than thai at room tem- perature. As such+ the ~[eclion of a correction factor wi l l be conlined to the choice between A ...... and h+,,c. The effor intro- duced by selecting a particular correction factor depends on the slope of the temperature gradients; the gradients of the conductivity with temperature, value of 111¢/3 coefficients: and the temperature r i~s encountered.

The gradients of the surface and the subsurf~e tempera- lures depend strongly on the diffusivity and on the geometry of the contacting t~ l ies . So that without considering the.~ factors a general deduction is not possible, T ~ change in the conductivity with temperature, however, is a material p r o p erty that is reflected in the value of the/3 coefficient+ I f that coefficient is high, the di.~repancics between the correction

(c)

2r~,

ql

tksl+r.cr bl,m{'l,m~l .u.d~= +mlt~erm'dlar~rrr,

Fbs- H). The ~ffecl o f the Icmpcralurc dupcndr .L 'y o f lhe Cul+ducl i l i ly ou Ihc depth o f heal pcl!cl.ralion fi~r Il ia ~.lairlh:~,~. +.1c¢1 ruhbin~ pat, r ~+l'~Wll ill Fig. t). Cf, nglp;.IriM+ql h¢1~'¢cIi t i le c+.lllP.|all| ttmdtLcli~,ily 19~ ! ~:h~>,cd ~)'lSllm,~l~, ~ and Ih+: ~ ariabl¢ conduc[i ' , ity (open ,,)lnbtd+, i solutions+ loT ¢¢ms.cculiv¢ rub intervaL.+: ( u. ) ! = 4 5 / I . ' , . i b I t = + )2 / J+ . ( ¢ ) l = I ?,~ p.~. N~)I~.' t h a t ~.'x e l l u n d e r t h e ~N~+UlIIpIJOll o f ¢¢+nM+illl dil'Iru~i++ Jly l h e lh ickn¢~,+ o f t11¢ l h c r m a l t y a f l '~c l+iJ l a y e r exlcnd.~,

5h H.A. AhdeI.Aul. S.7". SmitlJ I Wear 216 (199~) 41-59

015

01 F1

0 . 0 5 . - "__ ' -~ -" :

o

-O05

r z _ - _ _ _ ~0.1

-015 ~ EavT

-0 2 0 2 4 b 8 10

Rub time, AT p s e e s

Fig. I I. The variation of Ih¢ error~, Ibr the Icmpcralure cstimales Ibr cl.'lss c malcrials ( Zinc'. Z. and Titanium. T). Errtn-s are evaluated Ibr the cases plotted in Fig. 7b and Fig. 8. rc~,pcctivcly. Errm" pmpagalion is opposite Io the canduclivity variation. Past the inflation t~fint ( F" I and F,, ) the errars are unifi~rm fi~r Zinc { E,., :rod/:.-,,,.., ). Whereas. the error.., fi~r litanium slightly increase.

of heat flow by means of A .... and that based on h ...... will be appreciable. To this effect, compare the corresponding factors toe the mild steel and the Sapphire-tool steel pair in Table 5. Conversely. if the/~ cocflicient is of a moderate value then the difference between the two predictions will not be as pronounced. For class C materials, the criterion would be. how close arc the values of the coet~cients ft,,,.,/3~, and/3:. If the.~ values are close then the choice of either h,,,,, or h ..... is probably immaterial ( compare the values of h ...... and A ..... tbrZ;, . , in Table 5 to the respective values offl~ ..... and fl_, in Table 4).

The errors involved in the calculation of the temperatures for class C materials are presented in Fig. I 1. These errors ,,,,'ere referenced to the variable conductivity solution. That is. the temperatures resulting from the variable conductivity solution, using two temperature coefficients, were considered the true temperatures at the contact Sl~t. Consequently. the difference between these temperatures and the temperatures obtained by means of a constant conductivity solution con- stitules the error in the temperature estimates. For example, consider the sliding of titanium at a nominal load of 30 N and a sliding speed of 3.25 m / s . The results fiw this particular ca.~ were ploned in Fig. 7b. The errors in the estimates, E,. of the temperatures that are based on room temperature con- ductivity would thus be given by.

E , = - - ( 3 4 ) P).j

Two .~ts of error estimates for each ol" zinc and titanium are plotted. The curves dubbed !',. represent the errors in the temperature estimates that arc based on a room temperature constant conductivity solution. Whereas. the effors thai result from the calculations that utilize an average temperature toe f-

ficient./3~,,: in a variable conductivity solution are dubbed

Note that the errors are inversely proportional to the behav- ior of the conductivity. That is, if the conductivity drops with temperature the errors are positive (overestimated contact

temperature). Conversely, i f the conductivity increases with temperature the errors are negative (under estimated contact temperature). It will be noticed that past the inflation tem- perature the errors are nearly constant. This may suggest that the maximum errors involved in the estimates are linked m

the inflation temperature. Recall that it was proposed, above, that the ratio of the inflation temperature to the melting tem- perature is the factor that determines the difference between the different estimates of the contact temperatures. However. this remains a subject for future investigations that involve a wider band of class C materials.

9, Concluding r emarks

An investigation of the evolut ion o f the contact tempera- lures o f rubbing pairs has been presented. It has been shown Ihat the treatment o f the thermal conductivi ty o f the rubbing pair as a nonvariant quantity may lead to errors in estimating the contact temperatures. In essence, all materials display different variation in the thermal conductivity at different temperatures. However. the classitication of materials adopted in this work is suitable for a broad range of practical applications.

l! is generally tedious to account for the variation of the thermal properties with temperature in a practical model. Part of this difficulty is due to the absence of exact analytical relations that describe the behavior of the properties with

H.A. AbdeI-AaL S,7". Smith I Wear 216 ¢ 1098) 41-5~ 57

temperature. Moreover. for practical considerations, a model q, that incorporates the temperature dependency of all the respective physical and thermal properties is. practically, r~ inadmissible, r.

The partition of frictional heat, in the present work. iueor- t porates a coupling between the thermal capacity and the f-)p thermal conductivity of the rubbing members. The tempera- ture dependence of the thermal conductivity is influential to 6)rs the magnitude of the contact temperatures. Moreover. this dependency affects the conditions of heat removal at the fgb interface for rubbing members of comparable sizes. Con~- quently, under the same sliding conditions, for a class A ([) rubbing pair. interracial heat effects are expected to be more 8, .... pronounced than the corresponding effects for a class B rub- 6),,,, bing pair. ~,,,..i

It was proposed that the error in the estimates of the contact (,9. temperature, is inversely proportional to the variation in the ('91, conductivity with temperature and is proportional to the ratio ~r of the inflation temperature to the melting temperature of the c p solid. However. this finding may be generalized with caution. ~P at present, as more data pertaining to class C rubbing pairs is a unavailable. In general, there remain some fundamental ques- B lions concerning the thermal compatibility of class C pairs. B,. Especially when each of the rubbing members belong to the ~, opposite family ( e.g.. a titanium-zinc rubbing pair). Such a /~,,,~. case remains a subject for future investigations.

A /z

P Ar /:

¢jr.~

I0. Nomenclature

A,, Apparent area of contact A, Real area of contact A, Area of conduction (Eq, (32)) C Specific heat E~,¢ Error in temperature estimates based on an

averaged coefficient of conductivity E~ Error in temperature estimates for class c materials Ev Compound m~vJulus of elasticity given by

Hertzian contact theory H Brinell hardness M Constant ( Eq. ( 31 ) ) N Normal nominal load Q Total heat generated Q Rate of heat generation S Composite radius of curvature 7", Time of contact U,,,d Sliding speed U Translbrmed temperature function Zp Maximum temperature penetration depth k Thermal conductivity kq, Reference thermal conductivity k .... Average thermal conductivity k, Room temperature conductivity kr Temperature dependent conductivity n Number of contacting asperities

Heat transferred normal to the contact spot by conduction Radius of a single contact asperity (IO/an) Radius of the nominal area of contact f L5 turn) Time Constant conductivity temperature rise for a point .~uree Steady state temperature for a point source .solution Constant conductivity temperature ri.,~e for a moving band .source Variable conductivity temperature Maximum temperature r i~ Inflation temperature (class c materials) Melting temperature Surface temperature r i~ ( Eq. ! 33) ) Bulk temperature rise (Eq. ( M ) ) True temperature rise ( Eq. (34) ) Heat partition factor Temperature dependent heat partition factor Thermal diffusivity Temperature coefficient of conductivity Temperature coefficients of c~mduetivity for class c materials Average temperature coefficient of conductivity ( class c materials) Constant ( Eq. (31 ) ) Converse of the thermal diffusivity (h = pC/k) Coefficient of friction Mass density Time increment for integration Strain rate Material yield strength

Appendix A.

The error introduced hy neglecting higher order terms in dw ~:wressiou fi,r heat partition

The expansion attempted in F.qs. (27a) and 127b) is that of the quanlity (1+2/3U) ~z-~ which upon rearrangement assumes the form I I +_-A/3Ftb. Q) ) j/" with the negative sign u~d for class a materials and the positive sign u~d for class b materials. Now. the maximum value of the product 2/3U (and thereby, the product AIJF(rP.Q) is at most unity for class a materials. Whereas. unity is a practical maximum value for class b materials.

Attempting to expand Ire .series for a class a material we would write.

{ I -A/3F(~L'(~)}'/-" ={I-OI}l/-" (A-l) -- ! -0 .5+0.125

Now if we truncate higher order terms Ire value of the .series would be 0.5. Whereas. the inclusion of higher order terms

58 H.A. Al,+h,t.Aal, S+F. Stnilh / Wear 216119~8; 41-59

( three terms say) yields the value of the series as 0.625, So that the maximum truncalion error is around - O. ! 25. Simi- larly, for A ela~s B material we may write,

I I + A/3F<</', 0 >l"-" =1 I + O { l l l ' " ( A - 2 ) = I + O.5-O, 125

Again, the truncation of higher order terms yields the value of the sedes as 1.5 as opposed Io a value of 1.375 that results from the summation of the first three lerms. So tllat the error in this case is O. 125. We note here that these values represent an upper tmund for the truncation error. Moreover, the values quoted here correspond to high temperatures (in excess of

I IO0°C Ion carl'~m steels ) that are encountered under a com- bination of high loads and sliding velocities, As such, the accuracy of the results would not be seriously affected under normal sliding conditions as those consi&.~'ed in the current work.

A p p e n d i x B.

Eq. ( B - 4 ) was used to obtain the time step Ar used for the integration of the temperature equalions.

13. The coeff ic ient o f frict ion

The equation for p was extracted from the work of Lira and Ashby 1231. In this work it was considered that the coellicient of friction varies with velocity, except al very high speeds when additional dependence on load is encountered. A s such the regression of several data sets lead these authors to the following equation.

/ , t=0 .78 -0 . t31 log,,(/;1 (B-5)

Where iS is a normalized velocity given as,

i~= U,,,,,t:, ( B - 6 ) fl/

Eq. ( B-5 ) was used to calculate the coeflicient of friction used in the current work.

Tile ('qel~tlioll.~" I~.~['tl !o ~'~th'Male the sliding pat~tt~leterx References

I I. T h e n u m b e r o f contac l ing asperi t ies

Lira and Ashby 123 ! developed the following equation I~r the number of contacting asperities.

. 2

L ':'J Where , /= " is u n o n d i m e n s i o n a l m~mina l l oad ~ i~¢n by+

/:.= N A,,H,. (B-2)

Eq. ~B-I f ,.,.as used m predict the number of conlacling pair.,, ot'asperilie,,, n. v.'ith ~,. = 1.5 X tO ~ m, and r , = IO "

nl.

12. T h e t ime o f eontac l 1",.

The lime of contact is a funetioq of the sliding speed, the number of contacting asperhies, and the real area Of contact. If the real art';.] of contact is composed of n asperities, then

the diameter, d. of each asperity is given by.

, 1 = ~ 4 N " ~ ' " ~ n l l . . j I B-31

When lhc heat som¢c move,, p, ilh a unilbrm velocily the time of contact t'~.'twcen two asperities, i~, - iven a.,,.

, l / 4N " ~ ' " 1~ = - - = I B-41

U ..... I , ~r, t t . , I /~. , , .J

I I I t:l. FlIt@. "rhL.tlrelieal ~llld% ;it" lemPeralun: ri,,e ~t ,,urfa¢,~s ~1l" a¢ltl:ll ¢ofil;lk'l tllldl.'r Ifilil}¢ss ¢ondilk~nn. Proc. |nM. Mech.. t:.ng+ (;choral piscus,,ion on I.llbri¢~.|lioll. Ills[. Mech. l':n".. I.ondorl ( 1937 ) pp. 222- 232.

[ 2 [ J.C. Jaeger. Mte, ing smlr¢c~, o f heat a,ld the te,npen,tun:.,, o f sliding ,'onta¢tn. Proc. R. So¢, N.SW. 76 ¢ 1942) 21)3-222.

131 J.F. Aruhard. The L,2mperaltlr¢.~ ~f rubbing surt'a,.:,a:,,. W|-AR 21 ltJ.<i~ t 43~455.

141 H.S+ ('~lrsl;iv.. J.C. Jaeger. Cimduclion ~ff Heal in Solid:,. Oxford Univ. Pr~-,.,.. I.ot|dol~. It;59.

151 A. ('st|errs. ]~N. (h+rdo~i. ( ;.T..~y ms. ('tmtad temperaltlre,, hl r~+ll- in~/~liding r, urfacc. Prec. R. S~c. l+ondtm A 286 11962 ) 45-61.

161 (i.B. Sin,:lair. (In muhiple mw,'i;Le so,,rc+~, o f hem and impli,~+,limp, It~t 11m, h l¢ll:qle~-illur¢,,. ASN'IE J. fh.'al Tn, nM'cr 116 t i t)O4) 230-233.

]7] D+ Ktdllmam1-Wilr.dorf. l)emyslifying flash temperalures: I. Analyli- ,.'al v\pression ba,,,.:d ml a simple me, d,,.,l. Mater. S¢i. l!ng. q3 1 It.IF,7 ) II17-1 17.

I nl I). Kuhhn:mn-Wil,,d~rf. L~¢m.wlifyine ll:,sh 1¢ml'~ramr¢,~: II. Fir~4

~irt.lcr iil+l+ro~,illl;.llilql fur pl:p, li¢ ¢lmla¢l. Maler. St:i. Ellg. l).~ | lqg7 j

101 WD. MarshL, r. A erilieale~.tgualkm ,+fthel]ash lemperatttre t.'~mCepl. ASI .I- l 'rcprinl. ,~ I -AM- td-3.

I IO] J.r. Barber. An as) mpt~+ti¢ ,,t+lution for r.lIi~rt-lillle tran~,ielll l~,¢at ¢()n t duclitm hetv.¢en I',~o similar conlacling bodies. Inl. J. ll+:,t Ma.',~,

I I I I I). Kuhlmann-Wil,,+h~rf. Samph.,caleulalion o f lla,,h lemperalur¢,.allt ,,ih ¢r ~gr~phil,.' elcclric ¢OliI:i¢I ~,liding o,I copper. WI-AR 1117 1191@, ) 71J-~L

1121 I:.IL Kenlled.,.'rherrl;+d and II'lerlt+~l+|cchallical cffectr, iildr)nlidino. WEAR IIH} q It)84~ 453.-.476.

1131 [).V. Pere'.¢n.'za. V.A. B,lakin. Di~,ldbuliun ol'hc~,l bctv.¢¢n the rub- binlg bodi¢,,. So'~. J. Friclkm %%'ear 13 q 3 ) 119~}217t)-87.

J 14l J.R I'l.arl~,r, The c,reduction t~l" heat l'r~m~ *,lidin~ solidr... Is. J. Ileal Nla~.x Trat~.l'¢r 13 l It.1741 X57-~6t,L

1151 II.A. Abd¢l-Aa]. A remark tm lh¢ ilash tcmlx'ralur¢ Ih¢oL',. Inl. Comm. tt¢:n't Ma,.,. Tranr, lk, r 24 12 ) 11997 ) 241-251}.

1161 tt.A. Abd¢t-Aal. A m~lc on the thermal di.,,torlitm of fri¢liOllafly healed C~qllltcf',,. Mech. Re,,. ('tllnlll. 22 I 1'9t)51 289-296+

H.A. ,Ihdtq-AaL S.T. Smid~ / Wear 2161H,'O,% 41 50 59

I t?l F.F.l.ii~g.J.C. Ricc. Surfacelemwratur,-'swilhten)wratum-depcml- cnt thermal propcrlics. ASL[: Trans. 9 119(+ I 195-2OI.

1181 H+J. t:m+l. M+F. A.,,hby+ I)cl'~rmatitm-M~:chanism Maps. "rh,." Plastic- ily and ('rccp of Metal.,,. PcrL:amm=. Oxfurd. 198".

I f01 J Pullen. J.B.P. Wjlliuln~,~n. On the ph=>tic contact of rough ,urfi=cc,,. Prec. R. S~.'. I.~md~m A 327 ~ 1972~ 150~173.

1201 T.H.C. (_'hilds. The persislence . l asperities m mdemati~m experi- mcnt~,. WEAR 25 ¢ 1973 ) 3-14.

i 21 ] J.A. Grcc~ ~ . 1 . Ct,ll~ct prc,.~,qr¢ IIn¢lu:lliOll>. Pr~.. Im, I. Mech. I'.n~.. J. Eng. Trih. 2 I|) I ] t~,~ i 2Nl-28J.

122 D. Kuhlm;nm-Wil~d~rt~ l).l). Makcl, N.A+ S~mdcrgaard. Rclincm~nt oi' I]~,h IClnp~.'ralur¢ cah:ulations, h~; F.A. Sehmidl. P.J. B[;la ~ I'd~,. i. Engill~:t.+¢~d Materials Ibr Ad+. ;n|c~.'d I:rK-li~,~ +.|11d ~,Vcilr APPllca6on,.. ASM Inll.. Mchll:, P:lrk. O1"1. IUS~. pp. 23-32.

I +'3 S.('. I.im. M.F. A~.hby. Wear m¢ch:nfi,,m nmp~,.o+co, icxx n,. 55. Aeta M+:hdl. 35 I 11 < lq871 I-2.t..

124 S,R. ('o~van. W.O. Wim:r, I:ri~.-thmal heatill" calcuhalion>,, in: P.j. Blau I Ed. I. Fricti~m. I_uhricali,n and Wear Tcchm,lt+;}. ASM tlandl~,,k. .Melal~, Park. OII. I~. 1'~3. pp. 3t)-44.

125[ I). ro,,Clllhal. "rhc thcor~ ol~'qt~, irlg norm:e,, ~d heat and d,. a[~plicaltt,i to metal 11-=2;.1|111~.'1|1~.o "rr:lll~. ,A..~II':. 1940. pp. ,~d.9-~6f'~.

1201 H.A. Al~d¢l-/t:d. I-rror bound.,, ol ~ariabl¢ condncli~it~ Icmperatur¢ ¢'4hnatcs in friction:all) heated conhlct',, hit. ('omm. H¢:II Ma,,~Tran~- I¢r. 2511 ) i 199~1 q(}- II)S.

127i M.N+ {),,i~Pik. Bozll|d;j.r~ v;.ihi,... Prohlcllls ,d' Ileal (',nducli.ul. D*~'.er Puhlicati=.~s. Ne~x York. ]q~q.

I -.281 W+F+ Amc,,. Noidincar Parlial |)il'l;:r,.,nlml Equati~ql', in I'~t'~gmecring. Acmicmic Pr,:,,s. Nex~ Y~wk. P'Jfi3.

12~1 M.I+. Slorln. Ileal c~mdtlcli=+ll hI ,,tropIc n|c¢;d,, j. APPL I'h}-,. +-2 ! 19511 +.j-Ill.

I?.OI Y.s.T.uh~kia=|+ R.W. P~mell, t '.Y. tit+. M.('. Ni~;,d;,m, "Fhcr+nal I)itlu,,ix'il). Plemim, Next "York. 1973.

1311 V.A. I]alakin+ I:ormatim~ and dintrihuti, ln o1 ht:~d il) Ih+: frictional ~.'o111~lCI lOl]C Lllldfr ~.'ontjjljonr+ olnollnlalionar% heal ¢x, ch;.n'~¢. %VIAR 72~ IOt41j [3j-141.

1321 J.tl. Kni,,:h|. J.R I)hiliP. tG'~a~;l ",oh~lh,),. hI +,,+l,w~tr di ff+.=,.i,l+, j. t~n~:. Malh. S I l l I IU?41 ~1~j-~27.

1331 M. Su.,'Llki. S. Mal.',tlinoro, .~. Mac++l,a. Nov,' anal}li,:a) in¢lh~+d t,.r a nolllinear dil'[llshm prohlcln, h|l. J. tlc;.ll M:|~', Franqcr !ff I IU771

1341 II.A. AbdcI-Aal. On Ihc db+lrihu|iu. *d" frieLiort-iitdu~.'¢d heal in Ihc dr} ~,litlin~ of mctatllc ,,,lid pair'.. I . t ('=mtm. II¢:n Ma,+ l-r:m,d~..r 24 I 7 ~ ( Itlt)7 ~ 9,~¢)-t.n;~.

I;51 F.F- l+in~.. S.l.. Pu. Probab[e inlerl~c tcmwralures or.,~dids "m sliding ¢¢mlacl. WEAR 7 ~ I~h%.L) 23.

13~1 tI.A. Al~J¢l-AaL S.T. Smith. Thermal compalibili|y eft .,,lidin[~ |rair.~. P , vc¢cding~, of the ASI.E I ~.~8 Annual Mce,n~+ l)etn~il+ MI. in pres~.

I]71 T.J.F. Quinn. W.O. Wince. An c.~rin~..nlal study ,r lh~ ' l~a - s~" ~..currin~ durin~ the ,~.idatitmal wear of h~l steel on sapphire. ASME J. Trih. IIF.} r I,~87 ~ 3i5-321L

[ 381 I..(', Th,lrla~.. Heat Tran:,l~:r. Table A-C- l I a I. 8"n. Prcnlk:c Hall. I.n~lc~t*~J ('llft,,. NJ. I~Y)J_

Bi+~raphies

Stuart Smilh c(>mpleted a -I-year apprenticeship in the rubber industry in 1981 befi~re oblaining his BSc and PhD degrees al lhe University o I Warwick. where he lectured in Engi- neering from It)87 to I (,~.)3. He now works at the Center for Free|skin Mctrology al the University of North Carolina at Charlotte. Hi.,, current interests are ultra-precision mechanism analy~,is, s l idcway design, low load contact mechanics and Ihc Mclr~)h~gy ~d" f ine mot ion mceh~,nical systems.

HL,,ham A. Abde I -Aa l . obta ined hi~; first degree in Mechan ica l

Eng ineer ing f rom the Un ivers i t y o f A l exand r i a Egyp t in 19X4. F o l l o w i n g his graduat ion, he was commiss ioned as u l irst l icu lenant in the Egyp l i an A i r force. Du r i ng hi.~ cmn-

miss ion , hc re turned to Ihe Univers i ty o f Alexandr ia where

he ob ta ined the dc~.ree o f Mas te r o f Sc ience in Mechan ica l

Engineering ( 19861. fi+llowed by a spec ia | d i p l o m a o f h i g h e r

gradualc s ludics in Fluid Mechan ic s ( i989)+ He j o i n e d the

Schtml o f Mechan ica l Engineer ing at Tuskegee Uni,~'crsity. AL. USA. as a research ass~. ' iate in 1¢~')2. whe re he ob ta ined the dcgrcc of Master o f Mechan ica l E , g i n e c r i n g ( I £)94 ). He

j<;ined Ihe d(~.'loral p rogram at Ihc Dcpar lment o f Mechan ica l Eng ineer ing at U N C C in the |~11 o f I¢F)6. where he c(mtinue~;

t<) wo rk under Ih¢ superv is ion o f Dr. Sluart T. Smi lh . His

currenl rcscttrch interests I',~us on the fundamenta l aspects

,d" dyuamic C¢)lllact o r ~liding sur faces and heal genera l ion in

frict ion and wear.