A COMPACT MODEL FOR SPHERICAL ROUGH CONTACTS Majid Bahrami M. M. Yovanovich J. R. Culham Microelectronics Heat Transfer Laboratory Department of Mechanical Engineering University of Waterloo Ontario, Canada
OVERVIEW Introduction Objectives Present Model General Pressure Distribution Dimensional Analysis Comparison with Experimental Data Conclusions
INTRODUCTION Contact of Spheres elastic, smooth Hertz (1881) theory of elastic contact of spheres elastic, rough spheres, elastic microcontacts Greenwood and Tripp (1967) plastic, rough surfaces, plastic microcontacts Mikic and Roca (1974) roughness parameter Greenwood et al. (1984) P Hz r/a Hz P,Hz 1 r/a Hz 3
OBJECTIVES develop model to predict the parameters of spherical contact: pressure distribution, elastic deformation, compliance, number of microcontacts, size of the contact area derive simple correlations for determining contact parameters that can be used in other analyses such as thermal contact models 4
COOPER MIKIC YOVANOVICH (CMY) (1969) conforming rough contacts Plastic Model Gaussian surfaces σ σ 1 1 m ω z1 m z mean plane 1 Y σ mean plane ω smooth flat m Y z plane plastically deformed hemispherical asperities a) section through two contacting surfaces equivalent rough b) corresponding section through equivalent rough - smooth flat cross-level theory A r A a 1 erfc Y/ a s 8 m exp erfc n s 1 16 m exp erfc A a 5
MICROHARDNESS Hegazy (1985) 4 Vickers micro-hardness macro-hardness microhardness may not be constant throughout the material as a result of machining process microhardness decreases with increasing depth of indentation until bulk hardness level Hardness H (GPa) 3.5 3.5 1.5 t d V d V /t=7 SS 34 Ni Zr-.5% wt Nb Zr-4 1 1 1 1 1 1 3 Indentation depth t (µm) H v c 1 d v c d v d v /d 6
PRESENT MODEL: ASSUMPTIONS spherical surfaces F Gaussian asperity distribution plastic microcontacts elastic macrocontact flat mean plane dr ρ Y(r) Y(r) dr u O rigid smooth sphere r ω (r) b elastic half-space E' 7
ASSUMPTIONS deformation of each asperity is independent of its neighbors no friction first loading cycle flat mean plane equivalent P(r) distribution rigid sphere fi E' F fi fi fi r elastic half-space discrete point forces plastic zone static contact 8
RELATIONSHIPS FOR MECHANICAL MODEL dimensionless local separation ( σ ) λ() r = Y()/ r effective pressure distribution Pr ( ) = 1 Hmic( r) erfc( λ( r)) elastic deformation of half-space due to applied P(r) force balance local microcontact radius local microhardness Psds () r E = r 4 s ωb() r = sp() s K ds r s π Er > r 4 r PsK () ds r s < π E s r F = π P( r) rdr 8 σ as( r) = exp λ ( r) erfc λ( r) π m ( π s ) H ( r) = c 1 a ( r) mic ( ) ( ) c 1 1 1 1 E = ν E + ν 1 E 9
NUMERICAL ALGORITHM MAIN LOOP Guess u,1 ( cal ) Using P() r calculatef = F F F 1,1 / Start E, ρ, υ, σ 1, 1, 1, m, c, c, F 1, 1 Guess u, Inside Loop Using () calculate F = F F F P r ( cal ), / u, new = Fu Fu 1, 1, F1 F Yes Inside Loop Using P() r calculate F = F F F new ( cal new ), / Acceptable u F = u = new F new u 1 F = u new = F 1 new F new = F F F new : TOL. End F F < new 1 No not acceptable 1
NUMERICAL ALGORITHM INSIDE LOOP From Main Loop u Calculate P(r) Calculate ω bnew, ( r) Pr ( ) = Pnew( r) ω ( r) = ω ( r) b b, new Calculate P ( new r ) Not Acceptable Pnew( r) P( r) : TOL. P ( new) new P() r Acceptable 11
SUCCESSIVE ITERATION F F 1 F F Line passing through points 1, F F = F Calculated Point Calculated Point 1 u,new u,1 u, u 1
OUTPUT PARAMETERS OF MODEL Radius of curvature ρ = 5( mm) Roughness σ = 1.414( µm) Force F = 5( N) Surface slope m =.17 Young s modulus E1 = E = 4( GPa) Poisson s ν1 = ν =.3 Microhardness c1 = 6.7( GPa) Microhardness c =.15 Sample dia. bl = 5( mm) P/P,Hz Non-Dimensional Pressure Distribution 1. 1.8.6.4 a) Hertz Model a s 1 1 8 6 4 Microcontacts Radius (µm) b). a L 1 3 r/a Hz 1 3 r/a Hz 45 Microcontacts Density (m - ) d) Microhardness (GPa) e) 4.5 3 η s H mic 3 15 1.5 1 3 r/a Hz 1 3 r/a Hz 13
GENERAL PRESSURE DISTRIBUTION P/P,H 1. 1.8.6.4. σ =.3 σ =.14 σ =.71 σ =1.41µm σ =5.66 Hertz P(ξ)/P 1.8.6.4. Hertz ( P ' =1) ' P =.95 ' P =.815 ' P =.7 ' P =.41 ' P =.57 ' P =.14 σ =14.14 1 3 4 P r/a H ( ξ ) = P ( 1 ξ ) P γ = 1.5 P P =, H a a F πa ( 1+ γ ) L L H γ 1.5.5.75 1 1.5 Hertzian limit ξ =r/a L ( ) ( r / a ) = P 1 ( r / a ) P γ P H H, H H =.5 1.5F = πa H, H H γ 14
DIMENSIONAL ANALYSIS effective microhardness, H mic = Const. surface slope m is assumed to be a function of surface roughness, Lambert (1995) m.76.5 maximum contact pressure is a function of P P,, E, F, H mic three non-dimensional parameters Parameter Effective elastic modulus, E Force, F Microhardness, H mic Radius of curvature, Roughness, Max. contact pressure, P α = τ = σ ω ρ a H σρ a, H H Dimension ML 1 T MLT ML 1 T M M ML 1 T E E H mic 15
EFFECT OF MICROHARDNESS PARAMETER effect of microhardness parameter on the maximum contact pressure is small and therefore ignored. 1 α =.9 α =.86 1 α =.8 α =.4 P /P,H 1-1 α =4.33 α =8.66 P /P,H 1-1 α =3. α = 16.1 α =17.33 α =3. 1-1 τ = 47 4 6 8 E ' /H mic 1-1 τ = 5333 4 6 8 E ' /H mic 16
CORRELATIONS a L /a H 1 1 1 τ =84.5 τ =47 τ =1149 τ =5333 τ =67467 σ µm τ = 84.5 1-1 -1 1 1 1 1 α τ = 67467 P /P,H 1 1-1 1 - τ =84.5 σ µm τ = 84.5 τ = 47 τ = 1149 τ = 5333 τ = 67467 1-1 -1 1 1 1 1 α τ = 67467 P ' = a L ' = P P a a, H L H 1 = 1+ 1.37α / τ.75 1.65 / P ' = 3.51.51P '.1 P '.47.47 P ' 1 17
COMPARISON WITH GT MODEL Greenwood and Tripp (1967) disadvantages: 1. complex, requires computer programming and numerically intensive solutions b and h s cannot be measured directly, sensitive to the surface measurements constant summit radius b is unrealistic P /P,H 1.8.6 GT, µ =4 present model, τ =1 GT, µ =17 present model, τ = 1.4 1-1 -1 1 α 18
ELASTIC DEFORMATION OF HALF-SPACE using general pressure distribution, relationships are derived for: elastic deformation of half-space compact correlation is derived for compliance ω' b = π E' ω b /4P a L 1.4 1. 1.8.6.4. ω b () / ω b (a L ) ω' b () ω' (a L )..4.6.8 1 P' =P /P,H 4 3.6 3..8.4 ω b () / ω b (a L ) 19
EXPERIMENTAL DATA τ = ρ /a H Kagami, Yamada, and Hatazawa (KYH) 198 16.8 τ 187.4 ρ = 3.15 mm,.8 σ 1.45 µm,.19 F 88 N carbon steel spheres carbon steel and copper flats Greenwood, Johnson, and Matsubara (GJM) 1984 31 τ 17.8 ρ = 1.7 mm,.19 σ. µm, 4.8 F 779 N hard steel balls hard steel flats Tsukada and Anno (TA) 1979 7.8 τ 47.6 ρ = 1.5, 5, 1 mm,.11 σ.1 µm, 3.5 F 1375 N SUJ spheres SK 3 flats 1-3 1-1 -1 1 1 1 1 α = σρ/a H
COMPARISON WITH DATA: CONTACT RADIUS a' L =a L /a H 7 Tsukada and Anno 1979, specimens: SUJ spheres and SK 3 flats KYH1 test TA1 TA TA3 TA4 TA5 TA6 TA7 TA8 KYH- Hz σ µm.11.35.84.11.35.87.1.35 KYH ρ mm 1.5 1.5 1.5 5 5 5 1 1 KYH3 6 KYH4 TA1 test TA9 TA1 TA11 TA1 TA13 TA14 TA15 TA16 TA σ µm.83.4.4.1.19.39.6.3 TA3 5 ρ mm 1 5 5 5 5 5 5 5 TA4 Kagami, Yamada, and Hatazawa 198 TA5 test KYH1 KYH KYH3 KYH4 TA6 MODEL ± 15% TA7 4 σ µm.457.18 1.45.457 TA8 specimens: steel spheres of radius 3.18 mm TA9 KYH 1&: carbon steel (.3% C) flats TA1 KYH 3&4: pure copper (99.9% pure) flats TA11 3 TA1 TA13 MODEL more than 16 points TA14 TA15 TA16 1 Greenwood, Johnson, and Matsubara 1984 TA- Hz test GJM1 GJM GJM3 GJM4 GJM1 σ µm.19.54 1.7. GJM specimens: hard steel balls of radius 1.7 mm GJM3 andhardsteelflats GJM4 RMS difference 6.%. +15% 15%..4.6.8 1 1. P' =P /P,H MODEL 1
COMPARISON WITH DATA: COMPLIANCE κ' =κ / κ H 1 1 8 6 4 more than 4 points Kagami, Yamada, and Hatazawa 198 test KYH5 KYH6 KYH7 KYH8 σ (µm) 1.45.8.457.8 specimens: steel spheres of radius 3.18 mm. KYH 5&6: carbon steel (.3% C) flats KYH 7&8: pure copper (99.9% pure) flats KYH5 KYH6 KYH7 KYH8 MODEL RMS difference 7.7%...4.6.8 1 1. P' =P /P,H
SUMMARY AND CONCLUSIONS a general pressure distribution that encompasses all spherical rough contacts including Hertzian limit is proposed compact correlations for contact radius and compliance are proposed and validated with experimental data It is shown that the non-dimensional maximum contact pressure is the main parameter that controls the solution of spherical rough contacts 3
ACKNOWLEDGMENTS Natural Sciences and Engineering Research Council of Canada (NSERC) The Center for Microelectronics Assembly and Packaging (CMAP) 4