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Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science http://pic.sagepub.com/

An analytical model of thermal contact resistance based on the Weierstrass??Mandelbrot fractal function

S Jiang and Y Zheng Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2010 224: 959 DOI: 10.1243/09544062JMES1799 The online version of this article can be found at: http://pic.sagepub.com/content/224/4/959

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959

An analytical model of thermal contact resistance based on the Weierstrass–Mandelbrot fractal function

S Jiang? and Y Zheng School of Mechanical Engineering, Southeast University, Nanjing, People’s Republic of China The manuscript was received on 19 June 2009 and was accepted after revision for publication on 23 September 2009. DOI: 10.1243/09544062JMES1799

Abstract: A fractal model for analysing the thermal contact resistance (TCR) of rough surfaces is presented; it is based on the classical heat conduction theory and fractal geometry for the surface topography description, elastic–plastic deformation of contacting asperities, and size-dependent constriction resistance. Relations for the TCR in terms of contact load are obtained for heat conductive surfaces with known material properties and surface topography.With the real contact area being approximately 1 per cent of the apparent contact area or less, the microcontact area distribution has a dominant in?uence on the TCR. Useful design guidelines for heat contacts are extracted from the numerical results. The analytical results agree well with previous experiments. Keywords: machined joint surface, thermal contact resistance, fractal geometry

1

INTRODUCTION

Heat transfer between interfaces formed by the mechanical contact of two rough solids occurs in a wide range of applications: microelectronics cooling, spacecraft structure, infrared satellite detective technology, nuclear engineering, ball bearing, and heat exchangers. And improving the heat conduction from the micro electromechanical system is also a growing issue in the industry, since the size of electronic devices continues to decrease. When a compressive load is applied between two surfaces, the presence of surface roughness produces imperfect contact at their interface and thus bottlenecks the heat ?ow, which leads to a relatively high temperature drop across the interface. Analytical, experimental, and numerical models have been developed to predict thermal contact resistance (TCR) since the 1930s. Several hundreds of papers on TCR have been published that illustrate the importance of this topic and indicate that the development of a general predictive model is dif?cult. These dated theoretical models make use of contact mechanics techniques, which have since been proven to have

? Corresponding

author: School of Mechanical Engineering,

Southeast University, Jiangning District, Nanjing, Jiangsu 211189, People’s Republic of China. email: jiangshy660118@yahoo.cn

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signi?cant pitfalls. Currently, the statistical model is usually used for predicting TCR. The very popular Greenwood–Williamson (G–W) statistical model [1] has been shown to produce results by considering a ?at surface in normal contact with an equivalent rough surface comprising spherical asperities of constant radius, but the assumption of all the spherical asperities with the same radius is unreliable. Leung et al. [2] developed a predictive model of thermal contact conductance using a statistical mechanics approach; however, for this method, there exist some unreasonable assumptions: conical asperities with constant slope; and touching asperities undergo pure plastic deformation. Since statistical approaches based on the G–W model are limited by the dependence of statistical roughness parameters on the sampling length and resolution of the measuring instrument, they cannot provide unbiased information for the surface topography. Therefore, the scale dependence of contact parameters derived from the G–W model may affect TCR results. TCR problems consist of three different problems: geometrical, mechanical, and thermal. The heart of the TCR analysis is the mechanical part, and any solution for the mechanical problem requires that the geometry of the contacting surfaces be quantitatively described. The thermal problem of contact heat transfer is coupled with the mechanical problem of two solids in contact. Examples of this school include the works of Cooper et al. [3], Mikic [4], Thomas and

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Probert [5], and Bush et al. [6]. A more recent study is by Majumdar and Tien [7], who for the ?rst time applied the concept of fractal theory for characterizing the surface topography of the solids. In this regard, Majumdar and Tien have considered all the asperities plastically deformed, but neglected the constriction resistance associated with the roughness of the contacting surfaces. The structure of a rough surface is usually quite a disorder and is often assumed to be random. Nayak [8] proposed statistical parameters to characterize rough surfaces, whereas these parameters depend strongly on the sample size, instrument resolution, and experimental ?lter used to acquire the topography data. However, studies [9] have indicated that surface topography is a non-stationary random phenomenon for which the variance of the height distribution is related to the length of the sample and is therefore not unique; the use of conventional statistical parameters of surface roughness, such as the rms slope, height, and curvature, is not valid. The surface roughness contains roughness features at several length scales ranging from millimetres to nanometres. Thus, the main objective of this study is to obtain a new analytical model of TCR in vacuum based on the classical heat conduction theory and fractal geometry for the surface topography description, elastic–plastic deformation of the contacting asperities, and size dependence of the microcontacts constituting the real contact area. With the model, the effects of contact load and surface topography on the TCR of the joint rough surfaces were discussed, and the experimental data from previous studies for the TCR of machined joint surfaces was adopted to verify the theoretical model. 2 THEORETICAL BACKGROUND 2.1 Surface modelling

frequency distribution density considerations lead to γ = 1.5 [10]. The fractal roughness G is a height scaling parameter independent of frequency. A rougher surface is characterized by higher G values. The fractal dimension D determines the contribution of highand low-frequency components in the surface pro?le. Hence, high values of D indicate that high-frequency components are more dominant in the surface pro?le than low-frequency components. For a ?xed value of fractal dimension D, higher values of the fractal roughness G yield smoother topographies. 2.2 Contact modelling

Contact of two rough surfaces is equivalent to contact of a smooth (?at) half-space with reduced elastic mod2 2 ulus E = [(1 ? v1 )/E1 + (1 ? v2 )/E2 ]?1 , where v1 and v2 and E1 and E2 are the Poisson ratios and elastic moduli of the two surfaces, respectively, and a rigid rough surface with topography equivalent to those of the two rough surfaces. It is assumed that surface contact yields numerous circular asperity microcontacts, which are suf?ciently apart from each other in order for asperity interactions to be neglected as secondary. This is a reasonable approximation for relatively light contact loads where the real contact area A is a small percentage of the apparent contact area Aa . Based on these assumptions and knowledge of the mean contact pressure at asperity microcontacts and real microcontact area, the total contact load and total real contact area can be obtained using an integration procedure. For the simple cases of elastic and fully plastic asperity deformation, the contact load F and contact area a of a circular asperity microcontact with truncated area a are given by [10] 2(9?2D)/2 (ln γ )1/2 G (D?1) E ? (a )(3?D)/2 3π(3?D)/2 a ae = 2 Fe = and Fp = KYa ap = a (3a) (3b) (2a) (2b)

The scale-invariant parameters used in fractal geometry enable a realistic multi-scale roughness description. Fractal geometry can describe geometric features of various length scales and thus provides a means of characterizing asperities of a large range of sizes. Majumdar and Tien characterized engineering surfaces using a Weierstrass–Mandelbrot (W–M) fractal function, which can be written in a dimensionally consistent form as [10] z(x) = L G L

D?1 nmax

(ln γ )1/2

n=0

γ (D?2)n (1)

where the subscripts e and p denotes elastic and fully plastic deformation, respectively. The ratio of the hardness of the softer material to the corresponding yield strength Y is represented by K (typically, K = 2.8 [11]). The critical truncated contact area ac demarcating the elastic and fully plastic deformation regimes is [12] ac = 2(9?2D) 2(D?1) G 9π(3?D) E? KY

2 1/(D?1)

2πγ n x ? φ1·n × cos φ1.n ? cos L

ln γ

(4)

The scaling parameter γ controls the density of frequencies in the surface pro?le. Surface ?atness and

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Asperities with a > ac and a ac are in the elastic and fully plastic deformation regimes, respectively.

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The statistical distribution of the truncated microcontact area a can be written as [12] n(a ) = D (a )D/2 (a )?(D+2)/2 2 L (5)

Contact rigid flat surface (regarded as horizon)

Contact spot (regarded as island)

The total contact load P and the real contact area Ar can be determined by integrating the contribution of the elastic and plastic asperities. Hence, P and Ar can be obtained in the following two equations P= and Ar =

ac as ac as

Rough surface

Interstice (regarded as lake)

Fp (a )n(a )d(a ) +

aL ac

Fig. 2

Fe (a )n(a ) d(a )

(6)

Contact between a rough surface and a ?at rigid surface

n(a )a d(a ) +

aL ac

1 n(a ) a d(a ) 2

(7)

2.3 TCR of a single microcontact Some research [13] has shown that the radiation is lower than 2 per cent of total heat transferred at the metallic surfaces when the contacting temperature is below 900 K. Therefore the radiating is ignored in this article. As shown in Fig. 1, Mikik has obtained the single microcontact thermal constriction resistance Rc by the following equation [4] ψ(c/b) Rc = 2kc

1.5

described by the fractal geometry. Figure 2 schematically shows the contact process, which has widely been used for the geometric modelling of the actual contact between two rough surfaces. In this contact process, the rigid ?at surface is regarded as a lake horizon; the contact spot is like an island and the interstice is like a lake. Mandelbrot proposed that the number of islands N with area greater than a particular area s follows the power-law relationship N (s ) = sL s

Ds /2

(9)

So do the lakes N (su ) = sLu su

Ds /2

(10)

(8)

where ψ(c/b) ≈ (1 ? c/b) is the interface constriction ratio and 2/k = 1/k1 + 1/k2 is the combined thermal conductance. Mandelbrot proposed that the global surface has fractal property [14]; furthermore, from the present research it appears that the machined surfaces can be

For each truncated island area s , there would be a cavity area su to correspond with, that is, N (s ) = N (su ). Thus, from equations (9) and (10), it is found that sLu su = sL s (11)

z

The actual contact area of an elastically deformed microcontact is given by a = a /2. The total real contact area of a fractal domain Ar is related to the truncated area of the largest microcontact aL by Ar =

aL as

n(a )ad(a ) ac aL

(2?D)/2

k1

=

δ o k2 c b 0

D a 4 ? 2D L

?2

as aL

(2?D)/2

+1 (12)

and the total untouched area, Au , by Au = =

aLu aus

n(au )au d(au ) au aLu

(2?D)/2

D a 4 ? 2D Lu

?2

aus aLu

(2?D)/2

+1 (13)

Fig. 1

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Schematic of a single microcontact

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Dividing both sides of equation (13) by the apparent area of a fractal domain Aa with Aa = Ar + Ac rearrangement, gives

? Ar =

Substituting equations (5), (17a), and (17b) into equation (18) yields √ hc = (2/π)Dk(aL )1/2 aL as

D?1/2

Ar aL = Aa aL + aLc

(14)

?1/2 (D ? 1)(1 ? Ar )1.5

×

√

2

+ (1 ?

√

2)

Substituting equations (11) and (14) into equation (8) yields Rc =

?1/2 (1 ? Ar )1.5 2kr

aL ac

D?1/2

?1 (19)

(15)

The thermal contact conductance hc of a single microcontact is de?ned as

?1 hc = Rc

Hereafter, all area parameters will be normalized by the apparent contact area Aa and all dimensionless parameters will be denoted with an asterisk. ? The dimensionless thermal contact conductance hc is de?ned as

? hc =

(16)

3

APPROXIMATE ANALYSIS

hc 1/2 kAa √ 2/πD(aL? )1/2 = ?1/2 (D ? 1)(1 ? Ar )1.5 × √ 2 aL? as?

D?1/2

The objective of the approximate analysis is to determine the dimensionless parameters affecting the dimensionless contact load, real contact area, and TCR. An analytical solution for the TCR can be obtained based on the following simpli?ed assumptions: elastic or fully plastic asperity deformation. As mentioned in section 2.2, the equivalent contact model of two rough surfaces consists of a smooth elastic–plastic medium in contact with a rigid rough surface. The corresponding real contact area comprises discrete asperity microcontacts. As in the G–W model, the total TCR is assumed to be the sum of individual parallel resistances corresponding to the restriction resistances of the established microcontacts. According to the Mikik mechanism, the thermal contact conductance of a single microcontact in the elastic and fully plastic deformation regimes, hce and hcp , respectively, can be obtained by substituting equation (15) into equation (16) and using the appropriate contact area relations hce = √ and hcp = √ 2ka 1/2 π(1 ?

?1/2 Ar )1.5

+ (1 ?

√

2)

aL? ac?

D?1/2

?1 (20)

Likewise, the dimensionless contact load and real contact area are formulated as P? = = P Aa E KY E? + D a? 2?D L ac? aL?

(2?D)/2

?

as? aL?

(2?D)/2

2(9?2D)/2 3π(3?D)/2 ac? aL?

D (ln γ )1/2 G ?(D?1) (aL? )(3?D)/2 3 ? 2D

(3?2D)/2

× 1? P? =

for D = 1.5 D aL? 2?D ? as? aL?

(2?D)/2

(21a)

KY P = ? Aa E E × ac? aL?

(2?D)/2

2ka 1/2

?1/2 2π(1 ? Ar )1.5

(17a)

+ 2π?3/4 (ln γ )1/2 G ?1/2 (aL? )3/4 ln for D = 1.5 and

aL? ac? (21b)

(17b)

? Ar =

Ar Aa D a? 4 ? 2D L ac? aL?

(2?D)/2

The total thermal contact conductance hc is given by hc =

ac as

=

?2

hcp (a )n(a ) da +

aL ac

as? aL?

(2?D)/2

+1 (22)

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hce (a )n(a ) da

(18)

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From equations (4) and (20), it can be seen that the dimensionless thermal contact conductance depends on the fractal parameters D and G ? , the material properties E/Y and the smallest and largest truncated microcontact areas as? and aL? , respectively. For a continuum description, the size of the smallest microcontact should be greater than the atomic dimensions. For example, the diameter of the smallest truncated area as may be assumed to be equal to six times the lattice dimension of the contacting materials. For contacting surfaces of known surface roughness, the unknown parameters in equation (20) are aL? , which can be found implicitly from equation (21) as a function of ? the contact load P ? and Ar , which can be obtained by substituting the value of aL? in equation (22). 4 NUMERICAL ANALYSIS

100 G=10–7 G=10–9 G=10–11 10

Thermal contact resistance,R*

1

0.0

0.5

1.0

1.5

2.0

2.5 (×10–4)

3.0

3.5

Contact load, P*

Fig. 3

An accurate TCR analysis should take into account the elastic–plastic deformation regime and the sizedependent constriction resistance. Following the form of equations (6) and (7) for the total contact load and real contact area, respectively, the total TCR is given by ? R=?

N (as )

Dimensionless TCR R ? versus dimensionless contact load P ? for contacting rough surfaces with E/Y = 106, D = 1.3, and various values of G ?

? ?1

?1 Rci ?

(23)

i=1

where the TCR of the ith microcontact Rci is given by equation (9). Thus, the dimensionless total contact resistance is obtained as ? ?N (as ) √π(1 ? A?1/2 )1.5 1 r ? 1/2 R = ? = RkAa = ? hc 2(ai? )1/2 i=1

?1

??1 ? ? (24)

Equation (18) shows that the TCR is a function of the apparent contact area, the thermal conductivities of the materials, and microcontact areas ai . Therefore the TCR is also a function of the fractal parameters, contact load, and reduced elastic modulus-to-yield strength ratio. 5 RESULTS AND DISCUSSIONS

For a parametric study of the aforementioned dimensionless parameters, two-dimensional fractal surfaces were generated from equation (1) and contact simulations were performed for surfaces with apparent contact areas Aa = 1 μm2 and fractal dimension 1 < D < 1.5. The other surface topography parameters were selected to be 10?17 m G 10?13 m (i.e. 10?11 m G ? 10?17 m), γ = 1.5, L = 1 μm, Ls = 1 nm, and as = 1 nm2 , in accordance with reference [15]. Since the sample length L was chosen to be smaller than the

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upper limit of the sample length for fractal characterization, the entire apparent contact area can be described by fractal geometry. Unless otherwise stated, the results presented below are for thermal conductive materials with reduced elastic modulusto-yield strength ratio E/Y = 106. In the following, the results are limited to A? < 10?2 (i.e. the interaction effects between the asperities would be ignored as secondary). A lower limit of A? > 10?4 is used since an extremely small real contact area may result in a very small number of asperity microcontacts, which would be unrealistic. Results are presented to reveal the effect of surface fractal parameters on TCR. Figure 3 shows the dependence of the contact resistance R ? on the contact load P ? and fractal roughness G ? for fractal dimension D = 1.3. The termination of the curves with different G ? values in Fig. 2, as well as in the following ?gures, is in accord with the minimum and maximum contact loads corresponding to A? = 10?4 and 10?2 , respectively. For ?xed fractal parameters, the TCR decreases with increasing contact load due to the increase of the real contact area, whereas for a given contact load, the TCR increases with fractal roughness. Since G ? is a height scaling parameter, higher G ? values correspond to rougher surfaces, produce smaller real contact areas and hence higher TCR. To further investigate the role of surface topography, the dimensionless fractal roughness was ?xed (G ? = 10?7 ) and the fractal dimension was varied in the range of 1.1 D 1.4. Figure 4 shows the dependence of the TCR on contact load and fractal dimension. For a ?xed load, the rise of the fractal dimension increases the real contact area, which in turn decreases the TCR. It is expected that larger D values corresponding to

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S Jiang and Y Zheng

100 D=1.1 D=1.2 D=1.3 D=1.4 10

0.01

Full plastic deformation Elastic-plastic deformation Full elastic deformation

Thermal contact resistance,R*

Contact Area, A*

1E-3

1

1E-4 1

0 1 2 3 4 5 6

10 Thermal contact resistance,R*

100

Contact load, P* (×10– 4)

Fig. 4

Dimensionless TCR R ? versus dimensionless contact load P ? for contacting rough surfaces with E/Y = 106, G = 10?7 , and various values of D

Fig. 6 Dimensionless contact area A? versus dimensionless TCR R ? with different patterns of deformation

100 Thermal contact resistance, R*

E/Y=106 E/Y=288 E/Y=391

processes of deformation. The relation between A? and R ? indicates that different processes of deformation result in different microcontact area distributions, which in?uence the TCR directly. 6 MODEL VERIFICATION

10

1

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Contact load, P* (×10–4)

Fig. 5

Dimensionless TCR R ? versus dimensionless contact load P ? for contacting rough surfaces with D = 1.3, G = 10?7 , and various values of E/Y

smoother and denser surface topographies yield larger real contact areas and hence lower TCR. To illustrate the effect of the mechanical properties of the contacting surfaces on the TCR, results for different E/Y values and surfaces possessing similar roughness parameters (D = 1.3 and G ? = 10?7 ) are compared in Fig. 5. As shown in Fig. 5, for a given contact load, the TCR decreases with increasing E/Y . For ?xed contact load and reduced elastic modulus E, the TCR increases with the yield strength of the softer surface. For ?xed contact load, higher yield strength will result in a higher TCR due to smaller contact area. The relationship between the real contact area A? and the TCR is shown in Fig. 6. It can be seen that the TCR varies at the same contact area A? for different

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Figures 7(a) to (d) show a comparison of the present analysis with some experimental results obtained from previous studies [16]. The measurements were taken on a steel-blasted 304 stainless steel (0Cr18Ni9) sample with E = 196.5 GPa, v = 0.2855, k = 12.5 W/(mK), and H = 2040 MPa. Table 1 summarizes the equivalent fractal parameters of four contact pairs in vacuum. The interface temperature is 210 K. The contact simulations were performed for surfaces with apparent contact areas Aa = 1 m2 , and the results are limited to F < 8 MPa (i.e. small average size of the microcontact relative to their average spacing in order for interaction effects to be ignored as secondary). A lower limit of A? > 10?4 is used since an extremely small real contact area may result in a very small number of asperity microcontacts, which would be unrealistic. The dependences of both the calculated TCR and the experimental TCR on the applied pressure for different rough surfaces are shown in Fig. 7. The comparison between the theoretical TCR and the experimental TCR at the joint indicates that the present fractal model for TCR is appropriate and the theoretical TCR agrees with the experimental data. Previous studies on TCR (Tien [17], Cooper et al. [3], Mikic [4], Yovanovich [18]) have all concluded that the non-dimensional conductance in vacuum and the load are related as [7] σ hσ = = ξσ k kRc F HAa

x

(25)

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(a) Contact conductance (W/m2k)

350 300 250 200 150 100 50 0 0 1

theory experiment

Contact conductance (W/m2k)

400

(b)

650 600 550 500 450 400 350 300 250 0 1 2 3 4 5 6 7 8 Contact pressure (MPa) theory experiment

2 3 4 5 6 Contact pressure (MPa)

7

8 (d)

(c) Contact conductance (W/m2k) 1600 1400 1200 1000 800 600 400 200 0 1 2 3 4 5 6 7 8 Contact pressure (MPa) theory experiment

Contact conductance (W/m2k)

600 500 400 300 200 100 0 0 1 2 3 4 5 6 7 8 theory experimental

Contact pressure (MPa)

Fig. 7 TCC versus contact pressure for stainless steel samples at 210 K compared with experimental results: (a) 1/1 contact pair, (b) 2/2 contact pair, (c) 3/3 contact pair, and (d) 4/4 contact pair

Table 1 Equivalent parameter values of the contact pairs

Ra (μm) 17.6 6.25 2.83 10.16 D 1.5609 1.6182 1.8567 1.6351 G (× 107 m) 11.980 4.1252 5.1120 8.5514 D 1.1 1.2 1.3 1.4 10?7 0.690 46 0.723 84 0.775 14 0.844 59

Table 3 Load exponents versus various D and G ? values

G 10?9 10?11

Contact pairs 1/1 2/2 3/3 4/4

0.806 12

0.836 63

Table 2

Comparison of load exponents of previous investigations

Load exponent 0.5 0.56 0.66 0.86 0.85 0.99 0.92 0.95

Reference Laming Fletcher Mal’kov Shlykov Tien Cooper Thomas Yovanovich

Since the relation between the fractal dimension D and load exponent has been established, equation (26) shows the origins of the exponent x in equation (25), which lies between 0.5 and 1/(3 ? D); the load exponents x with various D and G ? values are listed in Table 3 and they are in agreement with Table 2. For a lack of surface material properties such as hardness and elastic modulus, a comparison of the present analysis was not made with some experimental results obtained from previous studies. 7 CONCLUSIONS

Several empirical and theoretical studies have reported that different values of load exponent x usually lie between 0.5 and 0.99, as shown in Table 2. In this article, the fractal model of TCR uses a deterministic approach to analyse the machined rough surfaces. According to equations (13), (20), and (21), load P as a function of the conductance was obtained as P ? ? h?2 + mh?(3?D)

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(26)

A new analytical model of TCR in vacuum is obtained based on the classical heat conduction theory and fractal geometry for the surface topography description. An approximate analytical approach was used to identify the dimensionless parameters affecting the TCR. In view of the presented results and discussions, the following conclusions can be obtained.

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1. The TCR decreases with an increase in contact load, fractal dimension D, and a decrease in fractal roughness G. 2. For a given contact load, lower TCR is obtained with a highly reduced elastic modulus-to-yield strength ratio. 3. For the same contact area A? , the value or the TCR depends on the surface topography. 4. Results of this model agree well with experimental data of previous studies, although the theory tends to underpredict the data. ACKNOWLEDGEMENTS The authors acknowledge the support of the National Science Foundation (grant nos 50475073 and 50775036) and Jiangsu Province Science and Technology Project (grant nos BK2002059, BG2006035, and BK2009612). ? Authors 2010 REFERENCES

1 Greenwood, J. A. and Williamson, J. B. P. Contact of nominally ?at surfaces. Proc. R. Soc. Lond. A, 1966, A295, 300–319. 2 Leung, M., Hsieh, C. K., and Goswami, D. Y. Prediction of thermal contact conductance in vacuum by statistical mechanics. Trans. ASME, J. Heat Transf., 1998, 120, 51–57. 3 Cooper, M. G., Mikic, B. B., and Yovanovich, M. M. Thermal contact conductance. Int. J. Heat Mass Transf., 1969, 12, 279–300. 4 Mikic, B. B. Thermal contact conductance: theoretical considerations. Int. J. Heat Mass Transf., 1974, 17, 205–214. 5 Thomas, T. R. and Probert, S. D. Thermal contact resistance: the directional effect and other problems. Int. J. Heat Mass Transf., 1970, 13, 789–807. 6 Bush, A. W., Gibson, R. D., and Thomas, T. R. The elastic contact of rough surface. Wear, 1972, 35, 87–111. 7 Majumdar, A. and Tien, C. L. Fractal network model for contact conductance. Trans. ASME, J. Heat Transf., 1991, 113, 516–525. 8 Nayak, P. R. Random process model of rough surfaces. Trans. ASME, J. Lubr. Technol., 1971, 17, 398–477. 9 Sayles, R. S. and Thomas, T. R. Surface topography as a nonstationary random process. Nature (London), 1978, 271, 431. 10 Yan,W. and Komvopoulos, K. Contact analysis of elasticplastic fractal surfaces. Trans. ASME, J. Appl. Phys., 1998, 84, 3617–3624. 11 Tabor, D. The hardness of metals, 1951 (Oxford University Press, Oxford). 12 Majumdar, A. and Bhushan, B. Fractal model of elasticplastic contact between rough surfaces. Trans. ASME, J. Tribol., 1991, 113(1), 1–11. 13 Fenech, H. and Rohsenow, W. M. Prediction of thermal conductance of metallic surfaces in contact. Trans. ASME, J. Heat Transf., 1963, 85(2), 15–24.

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14 Mandelbrot, B. B. The fractal geometry of nature, 1982 (Freeman, New York). 15 Komvopoulos, K. and Ye, N. Three-dimensional contact analysis of elastic–plastic layered media with fractal surface topographies. Trans. ASME, J. Tribol., 2001, 123, 632–640. 16 Zhao, L. and Xu, L. Experimental research on the heat transfer between the solid interfaces at low temperatures and vacuum. Chin. Space Sci. Technol., 2003, 1, 51–75. 17 Tien, C. L. A correlation for thermal contact conductance of nominally ?at surfaces in vacuum. In Proceedings of the 7th Thermal Conductivity Conference, 1968, pp. 755– 759 (US Bureau of Standards). 18 Yovanovich, M. M. Theory and applications of constriction and spreading concepts for microelectronic thermal management. In Proceedings of the International Symposium on Cooling technology for electronic equipment, Honolulu, HI, 1987.

APPENDIX Notation a ac aL aL? as as? Aa Ar

? Ar Au c D

Ds E/Y E1 , E2 G G? hc ? hc hce hcp

k1 , k2

truncated area or a microcontact (m2 ) critical truncated contact area (m2 ) truncated area of the largest microcontact (m2 ) dimensionless largest truncated microcontact areas smallest truncated microcontact areas (m2 ) dimensionless smallest truncated microcontact areas apparent contact area (m2 ) total real contact area of a fractal domain (m2 ) real-to-apparent ratio total untouched area (m2 ) radius of the contact area (m) fractal dimension of a surface pro?le (1 < D < 2) fractal dimension of a rough surface, D = Ds ? 1 material properties elastic moduli of the two surfaces (Pa) fractal roughness parameter (m) dimensionless fractal parameters thermal contact conductance (W/m2 /K) dimensionless thermal contact conductance thermal contact conductance of a single elastic microcontact (W/m2 /K) thermal contact conductance of a single microcontact in the fully plastic deformation regimes (W/m2 /K) thermal conductivities of the two materials, respectively (W/m/K)

JMES1799

Downloaded from pic.sagepub.com at Southeast University on December 6, 2012

Thermal contact resistance based on the W–M fractal function

967

L m r Rc Rci sL sLu v1 , v2

length of a fractal sample to be characterized (m) magni?cation in equation (26) radius of the contact area a (m) thermal contact resistance (W?1 K) TCR of the ith microcontact (W?1 K) largest island area (m2 ) largest lake area (m2 ) Poisson ratios of the two materials

x Y z γ ξ ψ(c/b)

lateral distance (m) yield strength (Pa) the surface height (m) scaling parameter for determining the spectral density and self-af?ne property (γ > 1) constant in equation (26) interface constriction ratio

JMES1799

Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

Downloaded from pic.sagepub.com at Southeast University on December 6, 2012

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