- Email: [email protected]
Thermal compatibility of dry sliding tribo-specimens
Wear 251 (2001) 939–948
Thermal compatibility of dry sliding tribo-specimens Hisham A. Abdel-Aal∗ Department of Mechanical Engineering Technology, York Technical College, P.O. Box 5481, Charlotte, NC 28299, USA
Abstract This paper proposes that thermally-dominated failure of sliding contacts is likely to occur at a temperature that is intrinsic to the particular material. The value of that temperature depends, essentially, on the sliding speed, the temperature coefficient of conductivity and the number of contacting asperity pairs. Based on this conjecture, a set of thermal compatibility equations is proposed and a method of constructing a so-called temperature operation maps (TOMs) is demonstrated. The TOM is a plot that predicts the onset of thermally-dominated failure during rubbing. Failure temperatures predicted by the TOM for an aluminum (AL-A356)–steel SAE 52100 sliding pair were compared to experimentally determined temperatures, reported elsewhere, and were found to be in good agreement. As such, it is proposed that thermal matching of sliding materials should be attempted, such that the maximum expected temperature rise, due to friction heat, is equal to or is in close proximity to the critical temperatures of each of the rubbing materials as predicted by the particular TOM. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Tribology; Dissipation capacity; Desorption
1. Introduction The idea that tribo-failure is related to the efficiency of heat removal during sliding is fundamental to tribo-analysis. Indeed, this idea formed the well-known postulate of Blok [1,2] concerning the constancy of scoring temperatures. Wang [4] based on experimental observations reasoned that due to heat accumulation, the hardness of a rubbing solid will decrease and a wear regime transition, from soft-abrasive to hard-abrasive, will take place at an approximately constant temperature. Bian et al. [3] recognized that the thermally-induced change in a sliding system is an important factor that determines the tribological integrity of a rubbing material. These authors attempted to link scuffing to frictional heat input. Meng [5], suggested that for every sliding system there is a critical frictional power input. If operation conditions are such that the frictional power input to the system is less than the critical limit, scuffing (or tribo-failure in general) may be avoided. Whereas, if that input exceeds the critical limit thermal tribo-failure most likely will take place. The ideas of Meng were modified by Jeng [6] who considered that scuffing is the consequence of the disorientation and the desorption of the protective films formed during sliding due to the intensity of frictional heat input exceeding a critical limit. Alpas and coworkers [7–11] Abbreviations: FIH, friction-induced heat; HDC, heat dissipation capacity; TOM, temperature operation map ∗ Tel.: +1-334-576-3053. E-mail address: [email protected] (H.A. Abdel-Aal).
performed a series of experiments to investigate the sliding of aluminum on steel for a wide range of nominal loads and sliding speeds. These authors observed that the wear transition (from severe to mild) is associated with a marked drop in the heat transfer ability of the material within the mechanically affected layer. It was established [12] that the thermal load acting on the rubbing solids may be different than their heat dissipation capacity (HDC). It was also shown [13] that the HDC is a strong function of the so-called thermal effusivity (W/m2 ◦ C s1/2 ). This parameter is a derived quantity given by k B = ρkCp = √ α The effusivity also termed as the coefficient of heat penetration is a measure of the resistance of a solid to an abrupt change in its thermal state. This change, in the case of sliding friction is caused by the relatively cooler moving asperity establishing sudden contact with the relatively more hot stationary asperity (heat source). Thus, the following were concluded [14]: 1. The efficiency of friction-induced heat (FIH) dissipation, and hence thermal energy accumulation, is strongly related to the effusivity of the rubbing pair and its respective variation at higher temperatures. 2. Materials that posses a high room temperature effusivity associated with a minimal variation in the effusivity at higher temperatures are more capable of resisting an
0043-1648/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 3 - 1 6 4 8 ( 0 1 ) 0 0 7 0 5 - 0
940
H.A. Abdel-Aal / Wear 251 (2001) 939–948
Nomenclature Ac B C K qa Qgen ra tc U U
area of conduction (m2 ) thermal effusivity (W/m2 ◦ C s1/2 ) specific heat (J/kg◦ C) reference thermal conductivity (W/m◦ C) rate of instantaneous heat dissipation by an individual pair of contacting asperities (W) rate of total heat generation at the surface (W) radius of the contact spot between two contacting asperities (m) duration of a single contact cycle (s) sliding speed (m/s) normalized sliding speed
Greek symbols α thermal diffusivity of the material (m2 /s) β temperature coefficient of the thermal conductivity (◦ C−1 ) ηth thermal efficiency of a contact spot µ coefficient of friction Θ max maximum potential temperature rise at the end of the contact cycle (◦ C) instantaneous temperature rise at the center Θ sur of contact spot between two asperities (◦ C) ρ mass density Subscripts cr critical FR failure m moving op operation s stationary abrupt change in their thermal state due to liberation of FIH. Whence, such materials are more likely to display favorable high temperature wear resistance. The findings of [12–14] lead to the conjecture that it is possible to minimize the destructive influence of friction-thermal loads by the careful choice of materials on the basis of their thermal effusivity [15]. This paper, therefore, addresses the feasibility of this conjecture through proposing a thermal compatibility criterion for the choice of rubbing solid pairs.
2. Heat dissipation through asperity contacts The role of asperity contacts in maintaining efficient heat removal is fundamental to the preservation of the tribological integrity of rubbing pairs. An asperity may be viewed as a device that regulates the flow of heat from the contact interface (according to the intrinsic thermal properties of the rubbing pair) to the bulk of the rubbing materials thereby maintaining a balance between heat generation and heat dis-
sipation. Now, if the amount of friction-induced heat at the interface exceeds the heat removed by the asperities, a situation where heat accumulation within the sub-contacting layers may take place. Such situation is clearly undesirable since it is conducive to thermal failure. The amount of heat generated at the interface is a function of the external sliding parameters (velocity and nominal load) and the mechanical properties of the rubbing pair. Heat removal on the other hand is, in essence, an explicit function of the thermal properties of the rubbing pair and the thermal history at the interface. Thus, the balance between heat generation and heat removal may be regarded, on a fundamental level, as a function of the thermal properties and the response of the rubbing materials to the thermal conditions at the interface (i.e. the respective temperature-induced change in the thermo-mechanical properties of the rubbing pair). Consequently, maintaining that balance will depend greatly on the intrinsic ability of the rubbing material to accommodate an abrupt change in its thermal state due to the release of frictional thermal energy.
3. Heat dissipation efficiency 3.1. Thermal loading of a contact spot Contact between sliding solids occurs at numerous real contact spots, which are the source of frictional heat generation. If the contacts are well separated, so that the interaction of their temperature field is negligible, the heat flow into each solid is the sum of the heat inputs for each individual contact treated separately. If the contacts, n, are closely packed, a heat input at one will cause a secondary heat input at all the others. This secondary heat input represents the reaction of the other asperities to the thermal loading of a single asperity in their close proximity. Whence, the thermal loading at a given contact is a super position of 1. a heat input, q1 at the ith contact of radius a, where, q1 =
Qgen π a2
2. a heat input q2 uniformly distributed over the nominal area, i.e. q2 = (n − 1)
Qgen Anom
Thus, the principal effect of interaction between the temperature fields of the contacts is the introduction of the term q2 . This quantity represents the temperature rise of the ith contact due to the influence of the other (n − 1) contacts. Physically, the term q2 may be interpreted as the alleviation of the surface temperature due to the contribution of the thermal resistance of the nominal contact area.
H.A. Abdel-Aal / Wear 251 (2001) 939–948
The total heat conducted from the contacting surface per unit time depends on the number of contacts established between the rubbing surfaces and the amount of heat dissipated by a “single” asperity contact. The number of contacting asperities depends on the mechanical properties of the sliding surfaces and on the surface preparation method. The mechanical properties of the rubbing pair may be affected by the thermal history of the interface thereby affecting the number of contacts. However, for the purpose of the present analysis the number of contacting asperities will be considered independent of the thermal history of the interface. Therefore, assuming that each asperity pair dissipates an equal amount of heat, the thermal efficiency of the surface, with respect to the total applied thermal load may be written as nqa ηth = (1) Qgena where Qgena = q1 + q2 . Naturally, a thermal crisis may be avoided if nqa > Qgena , i.e. if ηth > 1. Whence, the critical thermal regime is defined, for the purpose of this work, as 0 < ηth < 1.
(5)
From Eq. (5) the temperature rise at the contact surface (where z = 0) for the moving and the stationary solids is deduced as t t θm = Qm Fm , θs = Qs Fs (6) π(ρCK)m π(ρCK)s where the subscripts ‘m’ and ‘s’ denote the moving and the stationary solids, respectively and the functions Fm and Fs are given by
Fs = erf( a2 )[erf(x +
Eq. (2) allows the use of the constant conductivity solutions to calculate the variable conductivity heat flux. This is achieved through the introduction of the term (1 + βθ) in Eq. (2) to act as a correction factor that accommodates the effect of the temperature rise on the thermal conductivity. The temperature rise due to the influence of a square heat source of side (a) which moves with a uniform velocity U along the X-axis is given by [17] a/2 x+b/2 Q (ρC)3/2 0 dτ Θ= ρC (π K)3/2 τ (t − τ )3/2 −a/2 x−b/2 ρC (x−x −U (t−τ ))2 +y 2 +z2 exp − dx dy 4(t−τ ) K (3) To integrate Eq. (3), introduce the following variables: (4a) (4b)
(4d)
exp(−λ2 )
The true amount of heat transferred through the contact spot (normal to the nominal contact surface) depends on the temperature gradient, and the actual value of the thermal conductivity of the contact spot at the particular temperature. This is given by [16] (see Appendix A). ∂Θ qa (θ ) = −k0 {1 + βθ } (2) Ac ∂Z Z=0
(4c)
substituting from Eq. (4) in Eq. (3), the temperature rise may be written as a t Θ =Q erf πρCK 2 a a × erf x + − U − erf x − − U 2 2
Fm = erf( a2 )[erf(x+ a2 − U ) − erf(x −
3.3. The amount of heat dissipated by a single asperity contact (qa )
dτ = −2ω dω ω = (t − τ )1/2 , (x − x ) ρC K , dx = −2ω ξ= dξ 2ω K ρC
ρC K , dy = 2ω dη K ρC (z − z ) ρC U ω ρC λ= , U = 2ω K 2 K
y η= 2ω
3.2. Efficiency of heat dissipation of an asperity pair
941
a 2 ) − erf(x
−
a 2 )]
a 2
− U )]
(7a) (7b)
The functions Fm and Fs incorporate the influence of the operating conditions on the temperature rise. Note that Fm includes the influence of the sliding speed whereas, Fs represents the influence of the size of contact (which is a direct function of the applied load and the mechanical properties of the sliding pair). To determine the heat quantities, Qm and Qs , and thereby the actual temperature rise at the surface, it is necessary to introduce the so-called heat partition factor Φ. Therefore, it is assumed that ΦQ heat units would go to the moving solid and (1−Φ)Q units will go to the stationary solid. It is important at this stage to distinguish between two heat quantities. The first is Qgena while the second is Q. The term Qgena represents the amount of heat generated at the interface of the contact spot due to friction. This quantity, as stated earlier, is a function of several variables among which are: the hardness of the material, the surface preparation method, the geometry of the contact, the sliding speed and the coefficient of friction. As such, Qgena represents the actual thermal loading of the contact spot. That is, the amount of heat to be removed from the surface principally by conduction, through the contact spot. The second quantity, Q, on the other hand, represents the amount of heat actually conducted through the contact spot. Unlike Qgena , the heat quantity Q is essentially a function of the intrinsic thermal properties of the material as implied by Eq. (2). Note that according to Eq. (2), the true amount of heat conducted through the contact spot is a function of the thermal conductivity and the temperature
942
H.A. Abdel-Aal / Wear 251 (2001) 939–948
gradient normal to the plane of contact. This later quantity was shown to vary in proportion of the thermal effusivity [15] which despite being a derived quantity is a function of properties that are intrinsic to the rubbing material. In this sense therefore, the true amount of heat conducted per unit area of the contact spot, Q, is intrinsic to the particular material and the generated heat Qgena is extrinsic to the material, whence the distinction between the two quantities. The amount, Q, is a true representative of the so-called heat dissipation capacity of the material (HDC). This parameter, as shown elsewhere [12] varies in relation to the temperature rise at the interface of the rubbing pair in a manner that is similar to the variation in the amount Q. However, Q, and thereby the HDC, is not necessarily equal to the actual thermal loading acting on the contact spot. The difference between these two quantities mainly takes place if the rate of heat production at the interface is different than the rate of heat removal, and due to the effect of the thermal history of the contacting layers on the thermal transport properties of the rubbing solid. When the rate of heat removal from the interface is less than the rate of heat production thermal accumulation within the contacting layers will take place. This will be displayed as a lag between the influence of the heat source on the surface of contact and the response of the material to that influence. In such a case, the time duration of the heat source compared to the thermal diffusivity of the material assumes a critical role [14] that may aid the growth of protective oxides at the interface. Detailed discussion of this effect, however, is considered to be out of the scope of this work. Now by imposing temperature continuity across the contact (i.e. equating the expressions for θ m and θ s ) the partition factor Φ is deduced as Φ=
Bm Fs Bm Fs + B s Fm
(8)
whence, the amount of heat going into the moving and the stationary contacts is evaluated as Qm =
B m Fs Q , Bs Fm + B s Fs
Qs =
B s Fm Q B s Fm + B s Fs
(9)
Substituting from Eq. (9) in Eq. (6) the surface temperature is deduced as F s Fm Q t Θsur = (10) Bm F s + B s F m π Fig. 1c depicts the behavior of these functions with respect to the incremental time of contact. This is the time within the contact cycle normalized by the total time duration of the entire contact cycle. Values were calculated for the sliding speed of 1 m/s (again for mild steel on mild steel). Note that the function Fm relatively displays more variation with the incremental time of contact than the function Fs . Fig. 1a and b depicts the behavior of the functions Fm and Fs at a range of nominal operation conditions (loads and speeds). The values plotted in each figure were calculated for the
Fig. 1. Behavior of the characteristic functions, Fm and Fs at different sliding conditions: (a) behavior with respect to sliding speeds (b); behavior with respect to the load (c); behavior with respect to the time of contact.
sliding of mild steel AISI 1020 on like material. Moreover, each of the curves was evaluated at the time of maximum area of contact (i.e. at the moment during a contact cycle when the moving contact spot is completely atop of the stationary spot).
H.A. Abdel-Aal / Wear 251 (2001) 939–948
It is noted that, Fig. 1a, at high sliding speeds (U ≥ 5 m/s) both functions tend asymptotically to an equal value (approximately 2). This implies that heat partition at high speeds will be an exclusive function of the thermal properties of the rubbing pair regardless of the applied nominal mechanical load. A similar conclusion, however, may not be generalized for the behavior of both functions with respect to the applied nominal load (Fig. 1b). At the same speed, as shown in figure, the values of both functions are not necessarily equal.
4. The compatibility criterion Differentiating Eq. (5) with respect to λ and making use of Eqs. (7) and (10), the amount of heat penetrating normal to the contact spot is written as qa = −f (λ, t) B0 (1 + βm,s θsur )θsur Ac
(11)
943
Eq. (15) are those at which both mating materials are predicted to perform efficiently. That is for either of the temperatures determined from Eq. (15), the heat transfer ability of each rubbing member will be at least equal to the respective applied thermal load. Clearly, the condition for a solution to physically exist depends on the quantity under the square root in Eq. (15) being positive (indicating a real number). This, in turn, depends on the sign of the β coefficient, which is related to the variation of the thermal conductivity of the material with temperature. That is if the thermal conductivity decreases or increases with temperature. As such, the condition for the critical temperatures Θ 1,2 to exist may be expressed as 1+
4βm,s Fs,m Qgen βm,s Fs,m + Bs,m Fm,s
≥0
(16)
Now, making use of Eqs. (10) and (13), Eq. (16) may be written as (after simple manipulation), nFs s ≥ Θcr √ 4 π |βs |
where the subscripts ‘m’ and ‘s’ are used for the moving and the stationary contacts, respectively and,
nFm m , ≥ Θcr √ 4 π |βm |
2 f (λ, t) = − √ exp(−λ2 ) t
Eqs. (16) and (17) are compatibility conditions which may be invoked to match sliding material pairs. Moreover, Eq. (17) defines the limits of application, temperature wise, for a rubbing member according to the mode of motion (i.e. stationary or moving). The domain of permissible operation temperatures may thus be written as
To avoid a thermal crisis within the contacting surfaces, all the heat generated at the contact interface has to be transferred through the contact spots. As explained elsewhere [12], this condition will only be satisfied if the heat transfer capacity, of each of the moving and the stationary solids, is equal to their respective loads, i.e. when nqm a = ΦQgena ,
nqsa = (1 − Φ)Qgen
(12)
Substituting from Eqs. (8) and (11) in Eq. (12) we may write Bm,s Θsur (1 + βm,s Θsur ) =
Bm,s Fs,m Qgen Bm Fs + B s Fm
(13)
where Qgen
√ Qgen t = nAc f (λ)
Eq. (13) may be rearranged in the quadratic form 2 + Mm,s Θsur + Nm,s = 0 Lm,s Θsur
(14)
where the coefficients L, M and, N are given by Lm,s = Bm,s βm,s (Bm,s Fs,m + Bs,m Fm,s ) Mm,s = Bm,s (Bm,s Fs,m + Bs,m Fm,s ) Nm,s = −Bm,s Fs,m Qgen The solution of Eq. (14) is straight forward and is given by 4βm,s Fs,m Qgen 1 1 ± 1+ (15) Θ1,2 = − 2βm,s 2βm,s Bm,s Fs,m + Bs,m Fm,s where the subscripts 1 and 2 denote the first and second roots of quadratic Eq. (14). The temperatures Θ 1,2 , given by
nFm m m Θop ≤ Θcr ≤ √ 4 π |βm |
(17)
(18a)
and, nFs s s ≤ Θcr ≤ √ Θop 4 π |βs |
(18b)
Eqs. (18a) and (18b) imply an interesting interdependence between the external operation conditions (load and speed), number of contacts and the variation in the thermal conductivity of the material. It follows that if the number of contacts increases or, if on the other hand the thermal conductivity of the particular material displays minimal variation with temperature, the critical temperature will be higher and vice versa. The reason being less thermal loading in the first case and preservation of the thermal integrity (i.e. the ability of the material to transport heat efficiently) of the material in the second. Naturally, there are trade-offs in the first case since an increased number of contacts may lead to more wear. Eqs. (10), (18a) and (18b) can be used to construct a temperature operation map (TOM), the goal of which is to predict the limits of applicability of a given material for a rubbing application under a pre-known set of operation conditions. This may be achieved by superimposing a surface temperature plot, Eq. (10), for a given range of speeds and loads on the plots of Eqs. (18a) and (18b) for the same range of loads and speeds. The points of intersection between the
944
H.A. Abdel-Aal / Wear 251 (2001) 939–948
Naturally there will be differences between the failure temperature of the moving and the stationary contact. This will induce an additional thermal load on the stationary contact, and may accelerate thermal failure. Whence, matching sliding materials should be attempted such that the critical temperatures of both materials are in close proximity to each other and to the maximum expected surface temperature rise m = Θ s ). This condition will be satisfied (that is Θmax = Θcr cr only if βm Fm = Fs βs
(19)
Eq. (19) complements the compatibility Eqs. (10), (18a) and (18b) and may be applied to a practical rubbing system depending on the unknown variable (the mating material or the operation conditions). For example, if the range of operational loads and speeds are known beforehand, the functions Fm and Fs may be readily evaluated and the materials may be matched with the aid of the ratio of the β coefficients. Alternatively, when both materials are decided upon, based on strength considerations for example, the ratio of the β coefficients, which is intrinsic to the materials, will dictate the operational speeds necessary for efficient thermal performance. In this case Eq. (19) is written as a a a Fm = erf erf X+ −U − erf X− −U 2 2 2 βm (20) = Fs βs
Fig. 2. Construction of TOM: (a) extraction of critical operation temperatures for a particular nominal load; (b) construction of temperature operation contours for a range of nominal loads.
plot of Eq. (10) and that of Eq. (18) will represent the critical temperatures beyond which thermal efficiency will be sacrificed. These may be used to construct another map of operation temperatures for each material pair for a stationary or moving application. A schematic of such plots is given as in Fig. 2a and b. Fig. 2a illustrates the first step, i.e. superimposing a surface temperature plot (calculated from Eq. (11) for the sliding of SAE52100 on aluminum 6061 at a nominal load of 5 N and different speeds. The contour of critical temperatures calculated from Eq. (18b) for the same range of speeds and at the same nominal loads are also plotted in the figure. The loci of the critical operational temperature is given by joining the points of intersection between the surface temperature curves and the particular critical temperature curve. The results are then plotted in (Fig. 2b) which represents the operation temperature map, the procedure is then repeated for different loads to acquire more data.
Note that the quantity a is a function of the size of contact (which in turn depends on loads and mechanical properties), and the diffusivity of the materials (recall Eq. (4)). These are known a priori whence, the only unknown in Eq. (20) will be the sliding speed U . The TOM is a plot that predicts the onset of thermallydominated failure during the sliding of two contacting materials. As such, a TOM is distinct from that of the so-called frictional heating maps that predict the surface temperature rise in the rubbing of two materials.
5. Results and concluding remarks To validate the proposed thermal compatibility equations, a TOM for an aluminum (AL-A356)–steel SAE 52100 pair was constructed for the experimental conditions reported by Wilson and Alpas [11]. The properties of the materials used in the calculations were extracted from Touloukian et al. [18], and are summarized in Table 1. The hardness values of the aluminum alloy, and the coefficient of friction were extracted from [11] as, H = 0.78 MPa and µ = 0.3, respectively. The sizes of the contact junctions in the sliding of several aluminum–steel pairs were determined by Wilson and Alpas [11] using metallographic techniques. The measurements were subjected to statistical analysis by the
H.A. Abdel-Aal / Wear 251 (2001) 939–948
945
Table 1 Properties of the materials used in calculations Material
K (W/m ◦ C)
ρ (kg/m3 )
Cp (J/kg ◦ C)
β (◦ C−1 )
B (W/m2 ◦ C s1/2 )
Θ melt (◦ C)
Aluminum AL-A356 Steel SAE 52100 Mild steel AISI 1020 Hematitea AISI 1020–hematiteb
150 45 52 8.39 8.58
2700 7810 7600
896 485 455
−0.000316 −0.0004 −0.0004236 −0.00079 −0.001058
19662 13055 13409.55
582 1410 1450
5446.985c
1760
a
Thermal diffusivity: 0.45E−5 m2 /s. Equivalent values. c Thermal effusivity based on an equivalent value of 0.3074E−05. b
same authors and reported as Fig. 6 of reference [11]. This data was extracted from the respective references and was used in this work to estimate the number of contacting asperity pairs, n, by means of the model equations developed by Ashby et al. [19]. The results of the calculations along with the extracted juncture radii are presented as Table 2. The resulting TOM for the aluminun–steel pair is shown in Fig. 3a. Surface temperatures, θ s were calculated from Eq. (10), whereas, the critical temperatures were evaluated from Eq. (18a) for the moving contact (steel) and from Eq. (18b) for the stationary contact (aluminum). The solid lines labeled θ crs pertain to the critical temperatures of aluminum, whereas, the dashed lines, labeled θ crm pertain to the critical temperatures of steel. Six sliding speeds were used in the calculation (0.25, 0.6, 0.8, 1, 1.2 and 1.4 m/s at a nominal load of 5 N) of these only the surface temperatures for the speeds reported in [11] are shown in the figure. The points of intersection between the surface temperature curves and each of the curves representing the critical temperatures of the moving and the stationary contact are joined with the heavy lines labeled θ FRS and θ FRM . These represent the contour of the temperatures at which each of the contact displays thermal failure. Each of these points were then plotted against the time within the contact cycle to determine the predicted failure threshold for the particular nominal loading (5 N). This plot is given as Fig. 3b The failure temperatures predicted from the TOM for aluminum are in good agreement with those calculated by Wilson and Alpas [10] and those reported by Zhang and Alpas [9]. For example at sliding speeds of 1.2 and 1.4 m/s the TOM predicts failure temperatures of around 130 and 145◦ C, respectively. These are in good agreement with the values reported in [9,10] of around 125◦ C. It is interesting
to note that Alpas and coworkers [9,10] reported that this temperature range represents the critical threshold beyond which a transition in the wear rate (from mild to severe) takes place. It is noted also that for the same sliding speed and load failure represents temperatures for steel (moving contact) are higher than those for aluminum (stationary contact). This is contrary to intuition as the thermal conductivity
Table 2 Summary of the number of contact asperities used in temperature evaluation for the aluminum (AL-A356)–steel SAE 52100 sliding pair Speed (m/s)
Fs (N)
rj (m)
ra (m)
n
0.6 0.8 1 1.2
150 130 100 100
4.25E−6 2.00E−5 4.00E−5 6.75E−5
4.18E−6 1.96E−5 3.90E−5 6.58E−5
117 7 3 2
Fig. 3. Temperature operation map for an aluminum (AL-A356) stationary–bearing steel (SAE 52100) moving, rubbing pair: (a) temperature map for a nominal normal load of 5 N at 0.6, 1, 1.2, and 1.4 m/s; (b) temperature failure contours for the aluminum–steel rubbing pair.
946
H.A. Abdel-Aal / Wear 251 (2001) 939–948
of aluminum is almost three times that of steel (150 W/m ◦ C for the former and 45 W/m ◦ C for the later. As such, according to these conductivity values, aluminum is expected to be capable of transporting (dissipating) a higher thermal load than that of steel. However, the failure temperatures may be may be explained by comparing the values of the thermal effusivity and the values of the thermal capacity (product of mass density and heat capacity) of both materials (refer to Table 1). Note that the thermal effusivity of steel is lower than that of aluminum, this indicates that heat will penetrate less in its subsurface layers (i.e. aluminum will be more resistant to an abrupt change in its thermal state). As explained elsewhere [15] the material with higher thermal effusivity will tend to retain the conducted thermal energy within the thermally affected layer. Thus, leading to an intensified local thermal accumulation which will further hinder the heat transport ability of the contacting layers and its immediate substrate. Thus, thermal failure is likely to happen at a lower temperature than that of steel. Moreover, the thermal capacity of aluminum is approximately 0.65 that of steel. This indicates a higher ability for steel to redistribute the transported energy more so than aluminum. Early failure of aluminum will impose a secondary thermal load, representing the residual thermal energy, on the steel contact. This secondary load may provide sufficient energy to surpass the energy barriers necessary for diffusion or oxidation or may, in some cases, lead to accelerated failure. This point remains a subject of ongoing investigations. However, it is worth mentioning that preliminary results published elsewhere [14] indicates a strong correlation between the change in the slope of the effusivity–temperature curve and the accelerated failure. Fig. 4 depicts the TOM for a mild steel AISI 1020 sliding pair under the influence of a 20 N nominal load. Three cases were considered: (a) both sliding surfaces are clean; (b) the stationary surface is clean while the moving surface is coated with a layer of hematite of thickness 5 m and (c) both sliding contacts are coated with the 5 m hematite layer. Calculations for the oxidized contacts were based on effective thermal properties evaluated as the logarithmic mean of the properties of the oxide and the metal substrate. The resulting temperatures (surface and critical were normalized by the melting temperatures of clean mild steel). The TOM of the steel pair bears some similarity to that of the aluminum steel pair. In particular, the moving contact tends to fail at a relatively lower temperature than that of the stationary contact. This behavior, however, may not necessarily be related to the thermal properties and their respective variation as they are equal for both contacting surfaces for cases (a) and (c). The results instead may be related to the influence of Fm , i.e. to the effect of the sliding speed. It is conjectured that there exists a correlation between the intensity of energy injected through the contact, the rate of energy injection and, the interval of time necessary to redistribute that energy in the bulk of the material and thermal failure. That is, a relation between the rate of thermal local-
Fig. 4. Temperature operating map for a mild steel (AISI 1020) rubbing against like material. Plots are evaluated at a nominal load of 20 N: (a) both sliding surfaces are assumed to be clean; (b) the moving contact is assumed to be coated with a layer of hematite of thickness 5 m; (c) both surfaces are assumed to be coated with a layer of hematite 5 m.
ization and the time duration of a contact spot (hot spot). The details of such relation, however, remains a point of interest. At higher speeds, thermal failure for both contacts is noted to occur at the point the contours of the failure temperatures for each of the contacts coalesce. The presence of an oxide layer is noted to cause that temperature to be considerably lower for the oxidized surfaces than that for the clean surfaces (compare Fig. 4a–c). Such coalescence is
H.A. Abdel-Aal / Wear 251 (2001) 939–948
not observed when one of the surfaces is clean. Note also that the range of efficient thermal operation is considerably affected by the presence of an oxide layer on both surfaces. This is expected since the effective thermal properties of the clean surface are considerably higher than those of the oxidized material (refer to Table 1). Consequently, the ability of the oxidized surface to transport heat from the interface to the bulk is affected. As detailed elsewhere [13], such situation was found to be correlated to the self repairing properties and the tribological stability of the so-called protective glaze layers especially for Ni- and Co-based alloys. The sliding of many metallic pairs includes the transfer and deposition of wear particles from each of the rubbing solids to the mating counterparts. This is of particular importance in the sliding of aluminum–steel pairs where a transfer layer of aluminum is deposited on the steel sample. This layer is characterized by the formation of metallic flakes that transfer to the steel counterface. The transfer of the flakes will shift the boundary for the onset of the transition in the regime of wear. Moreover, the formation of such layer also implies the dominance of strain rate localization effects at the interface. Now, to accommodate the effects of that layer in a thermal analysis it is necessary to consider the so-called thermo-mechanical coupling effects on heat conduction. While the skeleton of the thermal model used in such a case will essentially remain unaltered, mathematical difficulties will arise due to the non-linear nature of the governing equations. A remedy for this situation is to use modified thermal properties the value of which is to be based on the premise that the conducting material is a deformable one. That is, the influence of mechanical dilatation on heat conduction will have to be included in the analysis. This leads to the definition of a derived quantity known as the “apparent thermal conductivity” that represent the actual value of the thermal conductivity of the solid under the combined effects of thermal and mechanical strains. Detailed presentation of the mathematical treatment for this case is considered beyond the scope of this work. However, preliminary results, to be published elsewhere [20], indicate that the actual thermal conductivity of a rubbing material depends on the direction and magnitude of the strain rate. That is, whether the rubbing material is strained in tension or in compression will have an effect on the local effective conductivity of the sliding material. It follows that if the material is undergoing tensile strain (positive strain rate) the local effective conductivity is reduced whereas if the material is undergoing compressive strains (negative strain rate) the local conductivity will be augmented. Now, the different layers of a rubbing solid (including the transfer layer) will be strained in different rates. It follows that the different contacting layers will exhibit different thermal behavior according to the local magnitude and direction of the strain rate. Moreover, a sliding contact spot will undergo different strain modes and rates, in the plane of sliding, at different points. For example the leading edge of the sliding contact will experience compressive
947
strains while the trailing edge will experience tensile strains [21]. These strains combined to the compressive strains normal to the direction of sliding will lead to a directional effect with respect to the local thermal conductivities. That is regardless of the nominal thermal properties of the sliding solid an anisotropic heat conduction behavior will be exhibited by any material element located within the contacting layers. The magnitude of that anisotropy will naturally depend on the local strain rates and their respective orientation with respect to the plane of sliding. Depending on the material and the sliding parameters (nominal load and speed) a solid that is nominally thermally isotropic may very well exhibit anisotropic thermal conduction, and thereby anisotropic thermal effusion. So that the local intensity of thermal accumulation will follow that anisotropy. This will alter the heat removal capacity of the contacting layers. Moreover, it will cause the density of the oxide and the oxidation rate at the surface to follow the local intensity of thermal accumulation. Therefore, different parts of the slider will attain the critical thickness necessary for wear protection at different times. This will cause the slider to exhibit different wear rates at different locations. Thus, in general, the leading edge will tend to build up the oxide layer slower than the trailing edge. As such, it is more likely that the leading edge, and vicinity, will tend to wear faster than the trailing edge. In all, however, this particular preposition is a subject of ongoing investigation.
6. Conclusions The ability of sliding materials to transport an applied thermal load depends greatly on their thermal properties. In particular, the thermal effusivity and the thermal capacity. Materials with high thermal capacity are capable of withstanding higher thermal loads. On the other hand, materials which posses a high thermal effusivity resist abrupt changes to their thermal state. This leads to a shallow thermally affected layer compared to a material of lower effusivity under identical thermal load. Mismatch in the thermal properties, especially the thermal effusivity may cause intensified thermal localization within the thermally affected layer of the material with higher effusivity. This subsequently causes thermal failure of that particular material and will result in a secondary thermal load being applied to the mating counterpart. Depending on the change of the effusivity of the mating material with temperature, the thermal secondary load may provide the energy necessary to surpass the energy barriers needed for protective oxide formation or accelerates thermal failure. It is conjectured that a critical temperature beyond which thermal failure is likely to occur exists for every material. The value of that temperature depends on the sliding speed, and on the temperature coefficient of conductivity, β. Based on this conjecture, a set of thermal compatibility equations
948
H.A. Abdel-Aal / Wear 251 (2001) 939–948
were proposed and a method of constructing a so-called TOM was demonstrated. Failure temperature predicted by the TOM for an aluminum (A356)–steel SAE 52100 sliding pair were found to be in good agreement with experimentally determined temperature reported elsewhere. Thermal matching of sliding materials should be attempted such that the maximum expected temperature rise is equal to or is in close proximity of the critical temperatures of each of the rubbing materials.
Appendix A. The amount of heat dissipated by a single asperity The instantaneous amount of heat flowing away from the contact spot into a solid in the time [t1 < t < t2 ] is given by, t2 ∂θ q= (A.1) k (r , t) dA dt t1 −t2 t1 A ∂η Heat flow within the contacting layers is predominantly one-dimensional. As such, Eq. (A.1) may be written, for a nominally flat contact spot in the X–Y plane, in the familiar form, ∂U qa = −k (A.2) Ac ∂Z z=0 In a sliding solid, the maximum temperature is reached at the nominal contact surface. This implies that the thermal properties of the material at the surface, depending on the temperature difference between the surface and the bulk, will be different. To incorporate the effect of the temperature-induced variation of the thermal properties on the surface temperature the so-called Kirchoff’s transformation is applied. This transformation acts as a correction to the constant property solution of the heat equation. Thus, if the constant conductivity surface temperature is given by 2qgen αt 1/2 Us (t) = (A.3) k0 π Then the variable thermal conductivity solution will be given as, 1 Θs (Z, t) = [{1 + 2βUs (Z, t)}1/2 − 1] β
(A.4)
Differentiating Eq. (A.4) with respect to the depth Z we may write, ∂U ∂θ {1 + 2βU (Z, t)}−1/2 (A.5) (Z, t) = (Z, t) ∂Z ∂Z Z=0 Substituting from Eq. (A.4) into Eq. (A.5), the relation between the variable and the constant conductivity temperature gradient assumes the form, ∂U ∂θ {1 + βθ (Z, t)}−1/2 (A.6) (Z, t) = (Z, t) ∂Z ∂Z Z=0 Finally, substituting Eq. (A.6) into Eq. (A.2) the temperaturedependent amount of heat penetrating to the contacting layer of the rubbing solid takes the form, ∂θ qa (θ ) = {1 + βθ (Z, t)} (A.7) (Z, t) ∂Z Z=0 Eq. (A.7) is Eq. (2) in the main text. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
H. Blok, Cong. Mondial Pétrole, 2me Cong. 3 (1937) 471–486. H. Blok, in: P.M. Ku (Ed.), NASA S.P. 237, 1970, pp. 153–248. S. Bian, S. Maj, D.W. Borland, ASTM JTEVA 24 (1) (1996) 12–19. N. Wang, STLE Trib. Trans. 32 (1) (1989) 85–90. J. Meng, Fric. Wear Machines 14 (1960) 222–237. H. Jeng, ASME J. Tribol. 120 (1998) 829–834. S. Wilson, A.T. Alpas, Wear 212 (1997) 41–49. J. Zhang, A.T. Alpas, Mater. Sci. Eng. A161 (1993) 273–284. J. Zhang, A.T. Alpas, Acta Mater. 45 (1997) 513–528. S. Wilson, A.T. Alpas, Wear 196 (1996) 270–278. S. Wilson, A.T. Alpas, Wear 225/229 (1999) 440–449. H.A. Abdel-Aal, Wear 237 (2000) 145–151. H.A. Abdel-Aal, ASME J. Tribol. 122 (2000) 657–660. H.A. Abdel-Aal, Int. J. Thermal Sci., 46 (6) (2001), in press. H.A. Abdel-Aal, Revue Generale de Thermique 38 (1) (1999) 15–27. H.A. Abdel-Aal, Int. Comm. Heat Mass Transfer 24 (2) (1998) 241– 250. H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, 1959. Y.S. Touloukian, R.W. Powell, C.Y. Ho, M.C. Nicolaou, Thermal Conductivity, Plenum Press, New York, 1973. M.F. Ashby, J. Abulawi, H.S. Hong, STLE Tribol. Trans. 34 (1991) 577–587. H.A. Abdel-Aal, Compètes Rendus de l’Académie des Sciences-series IIB, in press. J.A. Greenwood, Proc. Inst. Mech. Eng. J. Eng. Tribol. 210 (1996) 281–284.
Thermal compatibility of dry sliding tribo-specimens Hisham A. Abdel-Aal∗ Department of Mechanical Engineering Technology, York Technical College, P.O. Box 5481, Charlotte, NC 28299, USA
Abstract This paper proposes that thermally-dominated failure of sliding contacts is likely to occur at a temperature that is intrinsic to the particular material. The value of that temperature depends, essentially, on the sliding speed, the temperature coefficient of conductivity and the number of contacting asperity pairs. Based on this conjecture, a set of thermal compatibility equations is proposed and a method of constructing a so-called temperature operation maps (TOMs) is demonstrated. The TOM is a plot that predicts the onset of thermally-dominated failure during rubbing. Failure temperatures predicted by the TOM for an aluminum (AL-A356)–steel SAE 52100 sliding pair were compared to experimentally determined temperatures, reported elsewhere, and were found to be in good agreement. As such, it is proposed that thermal matching of sliding materials should be attempted, such that the maximum expected temperature rise, due to friction heat, is equal to or is in close proximity to the critical temperatures of each of the rubbing materials as predicted by the particular TOM. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Tribology; Dissipation capacity; Desorption
1. Introduction The idea that tribo-failure is related to the efficiency of heat removal during sliding is fundamental to tribo-analysis. Indeed, this idea formed the well-known postulate of Blok [1,2] concerning the constancy of scoring temperatures. Wang [4] based on experimental observations reasoned that due to heat accumulation, the hardness of a rubbing solid will decrease and a wear regime transition, from soft-abrasive to hard-abrasive, will take place at an approximately constant temperature. Bian et al. [3] recognized that the thermally-induced change in a sliding system is an important factor that determines the tribological integrity of a rubbing material. These authors attempted to link scuffing to frictional heat input. Meng [5], suggested that for every sliding system there is a critical frictional power input. If operation conditions are such that the frictional power input to the system is less than the critical limit, scuffing (or tribo-failure in general) may be avoided. Whereas, if that input exceeds the critical limit thermal tribo-failure most likely will take place. The ideas of Meng were modified by Jeng [6] who considered that scuffing is the consequence of the disorientation and the desorption of the protective films formed during sliding due to the intensity of frictional heat input exceeding a critical limit. Alpas and coworkers [7–11] Abbreviations: FIH, friction-induced heat; HDC, heat dissipation capacity; TOM, temperature operation map ∗ Tel.: +1-334-576-3053. E-mail address: [email protected] (H.A. Abdel-Aal).
performed a series of experiments to investigate the sliding of aluminum on steel for a wide range of nominal loads and sliding speeds. These authors observed that the wear transition (from severe to mild) is associated with a marked drop in the heat transfer ability of the material within the mechanically affected layer. It was established [12] that the thermal load acting on the rubbing solids may be different than their heat dissipation capacity (HDC). It was also shown [13] that the HDC is a strong function of the so-called thermal effusivity (W/m2 ◦ C s1/2 ). This parameter is a derived quantity given by k B = ρkCp = √ α The effusivity also termed as the coefficient of heat penetration is a measure of the resistance of a solid to an abrupt change in its thermal state. This change, in the case of sliding friction is caused by the relatively cooler moving asperity establishing sudden contact with the relatively more hot stationary asperity (heat source). Thus, the following were concluded [14]: 1. The efficiency of friction-induced heat (FIH) dissipation, and hence thermal energy accumulation, is strongly related to the effusivity of the rubbing pair and its respective variation at higher temperatures. 2. Materials that posses a high room temperature effusivity associated with a minimal variation in the effusivity at higher temperatures are more capable of resisting an
0043-1648/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 3 - 1 6 4 8 ( 0 1 ) 0 0 7 0 5 - 0
940
H.A. Abdel-Aal / Wear 251 (2001) 939–948
Nomenclature Ac B C K qa Qgen ra tc U U
area of conduction (m2 ) thermal effusivity (W/m2 ◦ C s1/2 ) specific heat (J/kg◦ C) reference thermal conductivity (W/m◦ C) rate of instantaneous heat dissipation by an individual pair of contacting asperities (W) rate of total heat generation at the surface (W) radius of the contact spot between two contacting asperities (m) duration of a single contact cycle (s) sliding speed (m/s) normalized sliding speed
Greek symbols α thermal diffusivity of the material (m2 /s) β temperature coefficient of the thermal conductivity (◦ C−1 ) ηth thermal efficiency of a contact spot µ coefficient of friction Θ max maximum potential temperature rise at the end of the contact cycle (◦ C) instantaneous temperature rise at the center Θ sur of contact spot between two asperities (◦ C) ρ mass density Subscripts cr critical FR failure m moving op operation s stationary abrupt change in their thermal state due to liberation of FIH. Whence, such materials are more likely to display favorable high temperature wear resistance. The findings of [12–14] lead to the conjecture that it is possible to minimize the destructive influence of friction-thermal loads by the careful choice of materials on the basis of their thermal effusivity [15]. This paper, therefore, addresses the feasibility of this conjecture through proposing a thermal compatibility criterion for the choice of rubbing solid pairs.
2. Heat dissipation through asperity contacts The role of asperity contacts in maintaining efficient heat removal is fundamental to the preservation of the tribological integrity of rubbing pairs. An asperity may be viewed as a device that regulates the flow of heat from the contact interface (according to the intrinsic thermal properties of the rubbing pair) to the bulk of the rubbing materials thereby maintaining a balance between heat generation and heat dis-
sipation. Now, if the amount of friction-induced heat at the interface exceeds the heat removed by the asperities, a situation where heat accumulation within the sub-contacting layers may take place. Such situation is clearly undesirable since it is conducive to thermal failure. The amount of heat generated at the interface is a function of the external sliding parameters (velocity and nominal load) and the mechanical properties of the rubbing pair. Heat removal on the other hand is, in essence, an explicit function of the thermal properties of the rubbing pair and the thermal history at the interface. Thus, the balance between heat generation and heat removal may be regarded, on a fundamental level, as a function of the thermal properties and the response of the rubbing materials to the thermal conditions at the interface (i.e. the respective temperature-induced change in the thermo-mechanical properties of the rubbing pair). Consequently, maintaining that balance will depend greatly on the intrinsic ability of the rubbing material to accommodate an abrupt change in its thermal state due to the release of frictional thermal energy.
3. Heat dissipation efficiency 3.1. Thermal loading of a contact spot Contact between sliding solids occurs at numerous real contact spots, which are the source of frictional heat generation. If the contacts are well separated, so that the interaction of their temperature field is negligible, the heat flow into each solid is the sum of the heat inputs for each individual contact treated separately. If the contacts, n, are closely packed, a heat input at one will cause a secondary heat input at all the others. This secondary heat input represents the reaction of the other asperities to the thermal loading of a single asperity in their close proximity. Whence, the thermal loading at a given contact is a super position of 1. a heat input, q1 at the ith contact of radius a, where, q1 =
Qgen π a2
2. a heat input q2 uniformly distributed over the nominal area, i.e. q2 = (n − 1)
Qgen Anom
Thus, the principal effect of interaction between the temperature fields of the contacts is the introduction of the term q2 . This quantity represents the temperature rise of the ith contact due to the influence of the other (n − 1) contacts. Physically, the term q2 may be interpreted as the alleviation of the surface temperature due to the contribution of the thermal resistance of the nominal contact area.
H.A. Abdel-Aal / Wear 251 (2001) 939–948
The total heat conducted from the contacting surface per unit time depends on the number of contacts established between the rubbing surfaces and the amount of heat dissipated by a “single” asperity contact. The number of contacting asperities depends on the mechanical properties of the sliding surfaces and on the surface preparation method. The mechanical properties of the rubbing pair may be affected by the thermal history of the interface thereby affecting the number of contacts. However, for the purpose of the present analysis the number of contacting asperities will be considered independent of the thermal history of the interface. Therefore, assuming that each asperity pair dissipates an equal amount of heat, the thermal efficiency of the surface, with respect to the total applied thermal load may be written as nqa ηth = (1) Qgena where Qgena = q1 + q2 . Naturally, a thermal crisis may be avoided if nqa > Qgena , i.e. if ηth > 1. Whence, the critical thermal regime is defined, for the purpose of this work, as 0 < ηth < 1.
(5)
From Eq. (5) the temperature rise at the contact surface (where z = 0) for the moving and the stationary solids is deduced as t t θm = Qm Fm , θs = Qs Fs (6) π(ρCK)m π(ρCK)s where the subscripts ‘m’ and ‘s’ denote the moving and the stationary solids, respectively and the functions Fm and Fs are given by
Fs = erf( a2 )[erf(x +
Eq. (2) allows the use of the constant conductivity solutions to calculate the variable conductivity heat flux. This is achieved through the introduction of the term (1 + βθ) in Eq. (2) to act as a correction factor that accommodates the effect of the temperature rise on the thermal conductivity. The temperature rise due to the influence of a square heat source of side (a) which moves with a uniform velocity U along the X-axis is given by [17] a/2 x+b/2 Q (ρC)3/2 0 dτ Θ= ρC (π K)3/2 τ (t − τ )3/2 −a/2 x−b/2 ρC (x−x −U (t−τ ))2 +y 2 +z2 exp − dx dy 4(t−τ ) K (3) To integrate Eq. (3), introduce the following variables: (4a) (4b)
(4d)
exp(−λ2 )
The true amount of heat transferred through the contact spot (normal to the nominal contact surface) depends on the temperature gradient, and the actual value of the thermal conductivity of the contact spot at the particular temperature. This is given by [16] (see Appendix A). ∂Θ qa (θ ) = −k0 {1 + βθ } (2) Ac ∂Z Z=0
(4c)
substituting from Eq. (4) in Eq. (3), the temperature rise may be written as a t Θ =Q erf πρCK 2 a a × erf x + − U − erf x − − U 2 2
Fm = erf( a2 )[erf(x+ a2 − U ) − erf(x −
3.3. The amount of heat dissipated by a single asperity contact (qa )
dτ = −2ω dω ω = (t − τ )1/2 , (x − x ) ρC K , dx = −2ω ξ= dξ 2ω K ρC
ρC K , dy = 2ω dη K ρC (z − z ) ρC U ω ρC λ= , U = 2ω K 2 K
y η= 2ω
3.2. Efficiency of heat dissipation of an asperity pair
941
a 2 ) − erf(x
−
a 2 )]
a 2
− U )]
(7a) (7b)
The functions Fm and Fs incorporate the influence of the operating conditions on the temperature rise. Note that Fm includes the influence of the sliding speed whereas, Fs represents the influence of the size of contact (which is a direct function of the applied load and the mechanical properties of the sliding pair). To determine the heat quantities, Qm and Qs , and thereby the actual temperature rise at the surface, it is necessary to introduce the so-called heat partition factor Φ. Therefore, it is assumed that ΦQ heat units would go to the moving solid and (1−Φ)Q units will go to the stationary solid. It is important at this stage to distinguish between two heat quantities. The first is Qgena while the second is Q. The term Qgena represents the amount of heat generated at the interface of the contact spot due to friction. This quantity, as stated earlier, is a function of several variables among which are: the hardness of the material, the surface preparation method, the geometry of the contact, the sliding speed and the coefficient of friction. As such, Qgena represents the actual thermal loading of the contact spot. That is, the amount of heat to be removed from the surface principally by conduction, through the contact spot. The second quantity, Q, on the other hand, represents the amount of heat actually conducted through the contact spot. Unlike Qgena , the heat quantity Q is essentially a function of the intrinsic thermal properties of the material as implied by Eq. (2). Note that according to Eq. (2), the true amount of heat conducted through the contact spot is a function of the thermal conductivity and the temperature
942
H.A. Abdel-Aal / Wear 251 (2001) 939–948
gradient normal to the plane of contact. This later quantity was shown to vary in proportion of the thermal effusivity [15] which despite being a derived quantity is a function of properties that are intrinsic to the rubbing material. In this sense therefore, the true amount of heat conducted per unit area of the contact spot, Q, is intrinsic to the particular material and the generated heat Qgena is extrinsic to the material, whence the distinction between the two quantities. The amount, Q, is a true representative of the so-called heat dissipation capacity of the material (HDC). This parameter, as shown elsewhere [12] varies in relation to the temperature rise at the interface of the rubbing pair in a manner that is similar to the variation in the amount Q. However, Q, and thereby the HDC, is not necessarily equal to the actual thermal loading acting on the contact spot. The difference between these two quantities mainly takes place if the rate of heat production at the interface is different than the rate of heat removal, and due to the effect of the thermal history of the contacting layers on the thermal transport properties of the rubbing solid. When the rate of heat removal from the interface is less than the rate of heat production thermal accumulation within the contacting layers will take place. This will be displayed as a lag between the influence of the heat source on the surface of contact and the response of the material to that influence. In such a case, the time duration of the heat source compared to the thermal diffusivity of the material assumes a critical role [14] that may aid the growth of protective oxides at the interface. Detailed discussion of this effect, however, is considered to be out of the scope of this work. Now by imposing temperature continuity across the contact (i.e. equating the expressions for θ m and θ s ) the partition factor Φ is deduced as Φ=
Bm Fs Bm Fs + B s Fm
(8)
whence, the amount of heat going into the moving and the stationary contacts is evaluated as Qm =
B m Fs Q , Bs Fm + B s Fs
Qs =
B s Fm Q B s Fm + B s Fs
(9)
Substituting from Eq. (9) in Eq. (6) the surface temperature is deduced as F s Fm Q t Θsur = (10) Bm F s + B s F m π Fig. 1c depicts the behavior of these functions with respect to the incremental time of contact. This is the time within the contact cycle normalized by the total time duration of the entire contact cycle. Values were calculated for the sliding speed of 1 m/s (again for mild steel on mild steel). Note that the function Fm relatively displays more variation with the incremental time of contact than the function Fs . Fig. 1a and b depicts the behavior of the functions Fm and Fs at a range of nominal operation conditions (loads and speeds). The values plotted in each figure were calculated for the
Fig. 1. Behavior of the characteristic functions, Fm and Fs at different sliding conditions: (a) behavior with respect to sliding speeds (b); behavior with respect to the load (c); behavior with respect to the time of contact.
sliding of mild steel AISI 1020 on like material. Moreover, each of the curves was evaluated at the time of maximum area of contact (i.e. at the moment during a contact cycle when the moving contact spot is completely atop of the stationary spot).
H.A. Abdel-Aal / Wear 251 (2001) 939–948
It is noted that, Fig. 1a, at high sliding speeds (U ≥ 5 m/s) both functions tend asymptotically to an equal value (approximately 2). This implies that heat partition at high speeds will be an exclusive function of the thermal properties of the rubbing pair regardless of the applied nominal mechanical load. A similar conclusion, however, may not be generalized for the behavior of both functions with respect to the applied nominal load (Fig. 1b). At the same speed, as shown in figure, the values of both functions are not necessarily equal.
4. The compatibility criterion Differentiating Eq. (5) with respect to λ and making use of Eqs. (7) and (10), the amount of heat penetrating normal to the contact spot is written as qa = −f (λ, t) B0 (1 + βm,s θsur )θsur Ac
(11)
943
Eq. (15) are those at which both mating materials are predicted to perform efficiently. That is for either of the temperatures determined from Eq. (15), the heat transfer ability of each rubbing member will be at least equal to the respective applied thermal load. Clearly, the condition for a solution to physically exist depends on the quantity under the square root in Eq. (15) being positive (indicating a real number). This, in turn, depends on the sign of the β coefficient, which is related to the variation of the thermal conductivity of the material with temperature. That is if the thermal conductivity decreases or increases with temperature. As such, the condition for the critical temperatures Θ 1,2 to exist may be expressed as 1+
4βm,s Fs,m Qgen βm,s Fs,m + Bs,m Fm,s
≥0
(16)
Now, making use of Eqs. (10) and (13), Eq. (16) may be written as (after simple manipulation), nFs s ≥ Θcr √ 4 π |βs |
where the subscripts ‘m’ and ‘s’ are used for the moving and the stationary contacts, respectively and,
nFm m , ≥ Θcr √ 4 π |βm |
2 f (λ, t) = − √ exp(−λ2 ) t
Eqs. (16) and (17) are compatibility conditions which may be invoked to match sliding material pairs. Moreover, Eq. (17) defines the limits of application, temperature wise, for a rubbing member according to the mode of motion (i.e. stationary or moving). The domain of permissible operation temperatures may thus be written as
To avoid a thermal crisis within the contacting surfaces, all the heat generated at the contact interface has to be transferred through the contact spots. As explained elsewhere [12], this condition will only be satisfied if the heat transfer capacity, of each of the moving and the stationary solids, is equal to their respective loads, i.e. when nqm a = ΦQgena ,
nqsa = (1 − Φ)Qgen
(12)
Substituting from Eqs. (8) and (11) in Eq. (12) we may write Bm,s Θsur (1 + βm,s Θsur ) =
Bm,s Fs,m Qgen Bm Fs + B s Fm
(13)
where Qgen
√ Qgen t = nAc f (λ)
Eq. (13) may be rearranged in the quadratic form 2 + Mm,s Θsur + Nm,s = 0 Lm,s Θsur
(14)
where the coefficients L, M and, N are given by Lm,s = Bm,s βm,s (Bm,s Fs,m + Bs,m Fm,s ) Mm,s = Bm,s (Bm,s Fs,m + Bs,m Fm,s ) Nm,s = −Bm,s Fs,m Qgen The solution of Eq. (14) is straight forward and is given by 4βm,s Fs,m Qgen 1 1 ± 1+ (15) Θ1,2 = − 2βm,s 2βm,s Bm,s Fs,m + Bs,m Fm,s where the subscripts 1 and 2 denote the first and second roots of quadratic Eq. (14). The temperatures Θ 1,2 , given by
nFm m m Θop ≤ Θcr ≤ √ 4 π |βm |
(17)
(18a)
and, nFs s s ≤ Θcr ≤ √ Θop 4 π |βs |
(18b)
Eqs. (18a) and (18b) imply an interesting interdependence between the external operation conditions (load and speed), number of contacts and the variation in the thermal conductivity of the material. It follows that if the number of contacts increases or, if on the other hand the thermal conductivity of the particular material displays minimal variation with temperature, the critical temperature will be higher and vice versa. The reason being less thermal loading in the first case and preservation of the thermal integrity (i.e. the ability of the material to transport heat efficiently) of the material in the second. Naturally, there are trade-offs in the first case since an increased number of contacts may lead to more wear. Eqs. (10), (18a) and (18b) can be used to construct a temperature operation map (TOM), the goal of which is to predict the limits of applicability of a given material for a rubbing application under a pre-known set of operation conditions. This may be achieved by superimposing a surface temperature plot, Eq. (10), for a given range of speeds and loads on the plots of Eqs. (18a) and (18b) for the same range of loads and speeds. The points of intersection between the
944
H.A. Abdel-Aal / Wear 251 (2001) 939–948
Naturally there will be differences between the failure temperature of the moving and the stationary contact. This will induce an additional thermal load on the stationary contact, and may accelerate thermal failure. Whence, matching sliding materials should be attempted such that the critical temperatures of both materials are in close proximity to each other and to the maximum expected surface temperature rise m = Θ s ). This condition will be satisfied (that is Θmax = Θcr cr only if βm Fm = Fs βs
(19)
Eq. (19) complements the compatibility Eqs. (10), (18a) and (18b) and may be applied to a practical rubbing system depending on the unknown variable (the mating material or the operation conditions). For example, if the range of operational loads and speeds are known beforehand, the functions Fm and Fs may be readily evaluated and the materials may be matched with the aid of the ratio of the β coefficients. Alternatively, when both materials are decided upon, based on strength considerations for example, the ratio of the β coefficients, which is intrinsic to the materials, will dictate the operational speeds necessary for efficient thermal performance. In this case Eq. (19) is written as a a a Fm = erf erf X+ −U − erf X− −U 2 2 2 βm (20) = Fs βs
Fig. 2. Construction of TOM: (a) extraction of critical operation temperatures for a particular nominal load; (b) construction of temperature operation contours for a range of nominal loads.
plot of Eq. (10) and that of Eq. (18) will represent the critical temperatures beyond which thermal efficiency will be sacrificed. These may be used to construct another map of operation temperatures for each material pair for a stationary or moving application. A schematic of such plots is given as in Fig. 2a and b. Fig. 2a illustrates the first step, i.e. superimposing a surface temperature plot (calculated from Eq. (11) for the sliding of SAE52100 on aluminum 6061 at a nominal load of 5 N and different speeds. The contour of critical temperatures calculated from Eq. (18b) for the same range of speeds and at the same nominal loads are also plotted in the figure. The loci of the critical operational temperature is given by joining the points of intersection between the surface temperature curves and the particular critical temperature curve. The results are then plotted in (Fig. 2b) which represents the operation temperature map, the procedure is then repeated for different loads to acquire more data.
Note that the quantity a is a function of the size of contact (which in turn depends on loads and mechanical properties), and the diffusivity of the materials (recall Eq. (4)). These are known a priori whence, the only unknown in Eq. (20) will be the sliding speed U . The TOM is a plot that predicts the onset of thermallydominated failure during the sliding of two contacting materials. As such, a TOM is distinct from that of the so-called frictional heating maps that predict the surface temperature rise in the rubbing of two materials.
5. Results and concluding remarks To validate the proposed thermal compatibility equations, a TOM for an aluminum (AL-A356)–steel SAE 52100 pair was constructed for the experimental conditions reported by Wilson and Alpas [11]. The properties of the materials used in the calculations were extracted from Touloukian et al. [18], and are summarized in Table 1. The hardness values of the aluminum alloy, and the coefficient of friction were extracted from [11] as, H = 0.78 MPa and µ = 0.3, respectively. The sizes of the contact junctions in the sliding of several aluminum–steel pairs were determined by Wilson and Alpas [11] using metallographic techniques. The measurements were subjected to statistical analysis by the
H.A. Abdel-Aal / Wear 251 (2001) 939–948
945
Table 1 Properties of the materials used in calculations Material
K (W/m ◦ C)
ρ (kg/m3 )
Cp (J/kg ◦ C)
β (◦ C−1 )
B (W/m2 ◦ C s1/2 )
Θ melt (◦ C)
Aluminum AL-A356 Steel SAE 52100 Mild steel AISI 1020 Hematitea AISI 1020–hematiteb
150 45 52 8.39 8.58
2700 7810 7600
896 485 455
−0.000316 −0.0004 −0.0004236 −0.00079 −0.001058
19662 13055 13409.55
582 1410 1450
5446.985c
1760
a
Thermal diffusivity: 0.45E−5 m2 /s. Equivalent values. c Thermal effusivity based on an equivalent value of 0.3074E−05. b
same authors and reported as Fig. 6 of reference [11]. This data was extracted from the respective references and was used in this work to estimate the number of contacting asperity pairs, n, by means of the model equations developed by Ashby et al. [19]. The results of the calculations along with the extracted juncture radii are presented as Table 2. The resulting TOM for the aluminun–steel pair is shown in Fig. 3a. Surface temperatures, θ s were calculated from Eq. (10), whereas, the critical temperatures were evaluated from Eq. (18a) for the moving contact (steel) and from Eq. (18b) for the stationary contact (aluminum). The solid lines labeled θ crs pertain to the critical temperatures of aluminum, whereas, the dashed lines, labeled θ crm pertain to the critical temperatures of steel. Six sliding speeds were used in the calculation (0.25, 0.6, 0.8, 1, 1.2 and 1.4 m/s at a nominal load of 5 N) of these only the surface temperatures for the speeds reported in [11] are shown in the figure. The points of intersection between the surface temperature curves and each of the curves representing the critical temperatures of the moving and the stationary contact are joined with the heavy lines labeled θ FRS and θ FRM . These represent the contour of the temperatures at which each of the contact displays thermal failure. Each of these points were then plotted against the time within the contact cycle to determine the predicted failure threshold for the particular nominal loading (5 N). This plot is given as Fig. 3b The failure temperatures predicted from the TOM for aluminum are in good agreement with those calculated by Wilson and Alpas [10] and those reported by Zhang and Alpas [9]. For example at sliding speeds of 1.2 and 1.4 m/s the TOM predicts failure temperatures of around 130 and 145◦ C, respectively. These are in good agreement with the values reported in [9,10] of around 125◦ C. It is interesting
to note that Alpas and coworkers [9,10] reported that this temperature range represents the critical threshold beyond which a transition in the wear rate (from mild to severe) takes place. It is noted also that for the same sliding speed and load failure represents temperatures for steel (moving contact) are higher than those for aluminum (stationary contact). This is contrary to intuition as the thermal conductivity
Table 2 Summary of the number of contact asperities used in temperature evaluation for the aluminum (AL-A356)–steel SAE 52100 sliding pair Speed (m/s)
Fs (N)
rj (m)
ra (m)
n
0.6 0.8 1 1.2
150 130 100 100
4.25E−6 2.00E−5 4.00E−5 6.75E−5
4.18E−6 1.96E−5 3.90E−5 6.58E−5
117 7 3 2
Fig. 3. Temperature operation map for an aluminum (AL-A356) stationary–bearing steel (SAE 52100) moving, rubbing pair: (a) temperature map for a nominal normal load of 5 N at 0.6, 1, 1.2, and 1.4 m/s; (b) temperature failure contours for the aluminum–steel rubbing pair.
946
H.A. Abdel-Aal / Wear 251 (2001) 939–948
of aluminum is almost three times that of steel (150 W/m ◦ C for the former and 45 W/m ◦ C for the later. As such, according to these conductivity values, aluminum is expected to be capable of transporting (dissipating) a higher thermal load than that of steel. However, the failure temperatures may be may be explained by comparing the values of the thermal effusivity and the values of the thermal capacity (product of mass density and heat capacity) of both materials (refer to Table 1). Note that the thermal effusivity of steel is lower than that of aluminum, this indicates that heat will penetrate less in its subsurface layers (i.e. aluminum will be more resistant to an abrupt change in its thermal state). As explained elsewhere [15] the material with higher thermal effusivity will tend to retain the conducted thermal energy within the thermally affected layer. Thus, leading to an intensified local thermal accumulation which will further hinder the heat transport ability of the contacting layers and its immediate substrate. Thus, thermal failure is likely to happen at a lower temperature than that of steel. Moreover, the thermal capacity of aluminum is approximately 0.65 that of steel. This indicates a higher ability for steel to redistribute the transported energy more so than aluminum. Early failure of aluminum will impose a secondary thermal load, representing the residual thermal energy, on the steel contact. This secondary load may provide sufficient energy to surpass the energy barriers necessary for diffusion or oxidation or may, in some cases, lead to accelerated failure. This point remains a subject of ongoing investigations. However, it is worth mentioning that preliminary results published elsewhere [14] indicates a strong correlation between the change in the slope of the effusivity–temperature curve and the accelerated failure. Fig. 4 depicts the TOM for a mild steel AISI 1020 sliding pair under the influence of a 20 N nominal load. Three cases were considered: (a) both sliding surfaces are clean; (b) the stationary surface is clean while the moving surface is coated with a layer of hematite of thickness 5 m and (c) both sliding contacts are coated with the 5 m hematite layer. Calculations for the oxidized contacts were based on effective thermal properties evaluated as the logarithmic mean of the properties of the oxide and the metal substrate. The resulting temperatures (surface and critical were normalized by the melting temperatures of clean mild steel). The TOM of the steel pair bears some similarity to that of the aluminum steel pair. In particular, the moving contact tends to fail at a relatively lower temperature than that of the stationary contact. This behavior, however, may not necessarily be related to the thermal properties and their respective variation as they are equal for both contacting surfaces for cases (a) and (c). The results instead may be related to the influence of Fm , i.e. to the effect of the sliding speed. It is conjectured that there exists a correlation between the intensity of energy injected through the contact, the rate of energy injection and, the interval of time necessary to redistribute that energy in the bulk of the material and thermal failure. That is, a relation between the rate of thermal local-
Fig. 4. Temperature operating map for a mild steel (AISI 1020) rubbing against like material. Plots are evaluated at a nominal load of 20 N: (a) both sliding surfaces are assumed to be clean; (b) the moving contact is assumed to be coated with a layer of hematite of thickness 5 m; (c) both surfaces are assumed to be coated with a layer of hematite 5 m.
ization and the time duration of a contact spot (hot spot). The details of such relation, however, remains a point of interest. At higher speeds, thermal failure for both contacts is noted to occur at the point the contours of the failure temperatures for each of the contacts coalesce. The presence of an oxide layer is noted to cause that temperature to be considerably lower for the oxidized surfaces than that for the clean surfaces (compare Fig. 4a–c). Such coalescence is
H.A. Abdel-Aal / Wear 251 (2001) 939–948
not observed when one of the surfaces is clean. Note also that the range of efficient thermal operation is considerably affected by the presence of an oxide layer on both surfaces. This is expected since the effective thermal properties of the clean surface are considerably higher than those of the oxidized material (refer to Table 1). Consequently, the ability of the oxidized surface to transport heat from the interface to the bulk is affected. As detailed elsewhere [13], such situation was found to be correlated to the self repairing properties and the tribological stability of the so-called protective glaze layers especially for Ni- and Co-based alloys. The sliding of many metallic pairs includes the transfer and deposition of wear particles from each of the rubbing solids to the mating counterparts. This is of particular importance in the sliding of aluminum–steel pairs where a transfer layer of aluminum is deposited on the steel sample. This layer is characterized by the formation of metallic flakes that transfer to the steel counterface. The transfer of the flakes will shift the boundary for the onset of the transition in the regime of wear. Moreover, the formation of such layer also implies the dominance of strain rate localization effects at the interface. Now, to accommodate the effects of that layer in a thermal analysis it is necessary to consider the so-called thermo-mechanical coupling effects on heat conduction. While the skeleton of the thermal model used in such a case will essentially remain unaltered, mathematical difficulties will arise due to the non-linear nature of the governing equations. A remedy for this situation is to use modified thermal properties the value of which is to be based on the premise that the conducting material is a deformable one. That is, the influence of mechanical dilatation on heat conduction will have to be included in the analysis. This leads to the definition of a derived quantity known as the “apparent thermal conductivity” that represent the actual value of the thermal conductivity of the solid under the combined effects of thermal and mechanical strains. Detailed presentation of the mathematical treatment for this case is considered beyond the scope of this work. However, preliminary results, to be published elsewhere [20], indicate that the actual thermal conductivity of a rubbing material depends on the direction and magnitude of the strain rate. That is, whether the rubbing material is strained in tension or in compression will have an effect on the local effective conductivity of the sliding material. It follows that if the material is undergoing tensile strain (positive strain rate) the local effective conductivity is reduced whereas if the material is undergoing compressive strains (negative strain rate) the local conductivity will be augmented. Now, the different layers of a rubbing solid (including the transfer layer) will be strained in different rates. It follows that the different contacting layers will exhibit different thermal behavior according to the local magnitude and direction of the strain rate. Moreover, a sliding contact spot will undergo different strain modes and rates, in the plane of sliding, at different points. For example the leading edge of the sliding contact will experience compressive
947
strains while the trailing edge will experience tensile strains [21]. These strains combined to the compressive strains normal to the direction of sliding will lead to a directional effect with respect to the local thermal conductivities. That is regardless of the nominal thermal properties of the sliding solid an anisotropic heat conduction behavior will be exhibited by any material element located within the contacting layers. The magnitude of that anisotropy will naturally depend on the local strain rates and their respective orientation with respect to the plane of sliding. Depending on the material and the sliding parameters (nominal load and speed) a solid that is nominally thermally isotropic may very well exhibit anisotropic thermal conduction, and thereby anisotropic thermal effusion. So that the local intensity of thermal accumulation will follow that anisotropy. This will alter the heat removal capacity of the contacting layers. Moreover, it will cause the density of the oxide and the oxidation rate at the surface to follow the local intensity of thermal accumulation. Therefore, different parts of the slider will attain the critical thickness necessary for wear protection at different times. This will cause the slider to exhibit different wear rates at different locations. Thus, in general, the leading edge will tend to build up the oxide layer slower than the trailing edge. As such, it is more likely that the leading edge, and vicinity, will tend to wear faster than the trailing edge. In all, however, this particular preposition is a subject of ongoing investigation.
6. Conclusions The ability of sliding materials to transport an applied thermal load depends greatly on their thermal properties. In particular, the thermal effusivity and the thermal capacity. Materials with high thermal capacity are capable of withstanding higher thermal loads. On the other hand, materials which posses a high thermal effusivity resist abrupt changes to their thermal state. This leads to a shallow thermally affected layer compared to a material of lower effusivity under identical thermal load. Mismatch in the thermal properties, especially the thermal effusivity may cause intensified thermal localization within the thermally affected layer of the material with higher effusivity. This subsequently causes thermal failure of that particular material and will result in a secondary thermal load being applied to the mating counterpart. Depending on the change of the effusivity of the mating material with temperature, the thermal secondary load may provide the energy necessary to surpass the energy barriers needed for protective oxide formation or accelerates thermal failure. It is conjectured that a critical temperature beyond which thermal failure is likely to occur exists for every material. The value of that temperature depends on the sliding speed, and on the temperature coefficient of conductivity, β. Based on this conjecture, a set of thermal compatibility equations
948
H.A. Abdel-Aal / Wear 251 (2001) 939–948
were proposed and a method of constructing a so-called TOM was demonstrated. Failure temperature predicted by the TOM for an aluminum (A356)–steel SAE 52100 sliding pair were found to be in good agreement with experimentally determined temperature reported elsewhere. Thermal matching of sliding materials should be attempted such that the maximum expected temperature rise is equal to or is in close proximity of the critical temperatures of each of the rubbing materials.
Appendix A. The amount of heat dissipated by a single asperity The instantaneous amount of heat flowing away from the contact spot into a solid in the time [t1 < t < t2 ] is given by, t2 ∂θ q= (A.1) k (r , t) dA dt t1 −t2 t1 A ∂η Heat flow within the contacting layers is predominantly one-dimensional. As such, Eq. (A.1) may be written, for a nominally flat contact spot in the X–Y plane, in the familiar form, ∂U qa = −k (A.2) Ac ∂Z z=0 In a sliding solid, the maximum temperature is reached at the nominal contact surface. This implies that the thermal properties of the material at the surface, depending on the temperature difference between the surface and the bulk, will be different. To incorporate the effect of the temperature-induced variation of the thermal properties on the surface temperature the so-called Kirchoff’s transformation is applied. This transformation acts as a correction to the constant property solution of the heat equation. Thus, if the constant conductivity surface temperature is given by 2qgen αt 1/2 Us (t) = (A.3) k0 π Then the variable thermal conductivity solution will be given as, 1 Θs (Z, t) = [{1 + 2βUs (Z, t)}1/2 − 1] β
(A.4)
Differentiating Eq. (A.4) with respect to the depth Z we may write, ∂U ∂θ {1 + 2βU (Z, t)}−1/2 (A.5) (Z, t) = (Z, t) ∂Z ∂Z Z=0 Substituting from Eq. (A.4) into Eq. (A.5), the relation between the variable and the constant conductivity temperature gradient assumes the form, ∂U ∂θ {1 + βθ (Z, t)}−1/2 (A.6) (Z, t) = (Z, t) ∂Z ∂Z Z=0 Finally, substituting Eq. (A.6) into Eq. (A.2) the temperaturedependent amount of heat penetrating to the contacting layer of the rubbing solid takes the form, ∂θ qa (θ ) = {1 + βθ (Z, t)} (A.7) (Z, t) ∂Z Z=0 Eq. (A.7) is Eq. (2) in the main text. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
H. Blok, Cong. Mondial Pétrole, 2me Cong. 3 (1937) 471–486. H. Blok, in: P.M. Ku (Ed.), NASA S.P. 237, 1970, pp. 153–248. S. Bian, S. Maj, D.W. Borland, ASTM JTEVA 24 (1) (1996) 12–19. N. Wang, STLE Trib. Trans. 32 (1) (1989) 85–90. J. Meng, Fric. Wear Machines 14 (1960) 222–237. H. Jeng, ASME J. Tribol. 120 (1998) 829–834. S. Wilson, A.T. Alpas, Wear 212 (1997) 41–49. J. Zhang, A.T. Alpas, Mater. Sci. Eng. A161 (1993) 273–284. J. Zhang, A.T. Alpas, Acta Mater. 45 (1997) 513–528. S. Wilson, A.T. Alpas, Wear 196 (1996) 270–278. S. Wilson, A.T. Alpas, Wear 225/229 (1999) 440–449. H.A. Abdel-Aal, Wear 237 (2000) 145–151. H.A. Abdel-Aal, ASME J. Tribol. 122 (2000) 657–660. H.A. Abdel-Aal, Int. J. Thermal Sci., 46 (6) (2001), in press. H.A. Abdel-Aal, Revue Generale de Thermique 38 (1) (1999) 15–27. H.A. Abdel-Aal, Int. Comm. Heat Mass Transfer 24 (2) (1998) 241– 250. H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, 1959. Y.S. Touloukian, R.W. Powell, C.Y. Ho, M.C. Nicolaou, Thermal Conductivity, Plenum Press, New York, 1973. M.F. Ashby, J. Abulawi, H.S. Hong, STLE Tribol. Trans. 34 (1991) 577–587. H.A. Abdel-Aal, Compètes Rendus de l’Académie des Sciences-series IIB, in press. J.A. Greenwood, Proc. Inst. Mech. Eng. J. Eng. Tribol. 210 (1996) 281–284.