Modeling of Thermal Joint Resistance for. Rough Sphere-Flat Contact in a Vacuum

Transcription

1 Modeling of Thermal Joint Resistance for Rough Sphere-Flat Contact in a Vacuum by Majid Bahrami A research proposal Presented to University of Waterloo Department of Mechanical Engineering In fulfillment of the Ph.D. comprehensive examination Waterloo, Ontario, 00 Majid Bahrami, 00

2 Abstract The nature of heat transfer across non-conforming rough contacts is complex, and usually is described through the thermal contact resistance concept. This is the resistance that thermal energy should overcome to pass from one body to another at the contact plane. It arises due to imperfect contact. Thermal contact resistance is an important issue in such applications as nuclear reactor cooling, electronics packaging, spacecraft thermal control, gas turbine, and internal combustion engine. A thermal contact resistance problem can be divided into three main parts: i) geometrical, ii) mechanical, and iii) thermal. Thermal contact resistance (TCR) theoretical models have been developed for two limiting cases, i) flat rough, and ii) spherical smooth contacts, which are only, simplified cases of real contacts. However, engineering (real) surfaces have both out-of-flatness and roughness simultaneously. None of the existing theoretical models were developed and validated to cover the above mentioned limiting cases and the case in which both roughness and out-of-flatness are present and their effects on contact resistance are of the same order. The main difficulty of a TCR problem is the mechanical analysis. The most famous existing mechanical model for spherical rough contacts requires parameters that should be estimated through tedious statistical calculations and are sensitive to the surface measurements. It also requires computationally intensive solutions. An analytical model is developed to predict the thermal joint resistance of nonconforming rough surfaces under vacuum conditions. A mechanical model is developed assuming the asperities undergo full plastic deformation and the contacting bodies (bulk) ii

3 deform elastically. The present model relates the deformation of microcontacts (micro scale problem) to the bulk deformations (macro scale problem) through the local separation of two mean planes of the contacting surfaces. A closed set of relationships, which govern the mechanical problem, is solved numerically. The results of the mechanical analysis, the micro along with the macro, are used to calculate the thermal contact resistance. Some of the main features of the model can be summarized as follows: the present model is developed to consider the effect of both out-of-flatness and roughness on TCR output of the numerical simulation at limits collapses to conforming rough and smooth spherical contacts input parameters of the model can be found in references or measured in lab assumptions are realistic, and consistent throughout the modeling process comparison of the model with the existing experimental TCR data shows a reasonably good agreement iii

4 Table of Contents Abstract... ii 1. Introduction Definition of Contact Resistance Problem Statement Literature Review Rough Surface Parameters Hardness and Microhardness Geometrical Model Mechanical Analysis Macrocontact Problem Microcontact Problem....4 Deformation Mode of Asperities Non-Conforming Rough Surfaces Thermal Analysis Constriction Resistance- Flux Tube Solution Clausing and Chao (1965) Lambert (1995) Proposed Model Overview Assumptions of The Present Model Geometrical Analysis Mechanical Analysis iv

5 3.4.1 Numerical Integration Set of Relationships For Mechanical Analysis Numerical Procedure Macrocontact Radius Thermal Resistance Analysis Macrocontact Thermal Resistance Microcontact Resistance Joint Thermal Resistance Results Comparison The Model With Experimental Data Future Work and Research Objectives Analytical Goals Experimental Goals Research Time Line References v

6 List of Figures Figure 1.1:Non-conforming rough surface contact- macro, and microcontact(s) constriction / spreading of heat flux, and temperature distribution in the vicinity of contact... Figure 1.: Resistance network analogy for the thermal joint resistance... 4 Figure 1.3, The thermal contact problem statement flow diagram... 7 Figure.1: Measured hardness and microhardness, Hegazy (1985) Figure.: Relation between contact spot radius and the Vickers s diagonal Figure.3: Flow diagram of the geometrical model Figure.4: Geometry of the elastic contact of two smooth spheres Figure.5: Equivalent contact of conforming rough surfaces Figure.6: Mechanical analysis flow diagram for non-conforming rough surfaces Figure.7: Axisymmetric pressure distribution on a circular area Figure.8: Greenwood and Williamson geometrical model... 5 Figure.9: Numerical procedure in Greenwood and Tripp model Figure.10: The Flux tube geometry and boundary condition Figure.11: Clausing and Chao geometrical model Figure 3.1: The rough non-conforming contact geometry... 4 Figure 3.: The contact geometry after loading Figure 3.3: Plastic zone, discrete point forces, and the equivalent pressure distribution generated in the elastic half-space Figure 3.4: Logarithmic singularity of the displacement integral Figure 3.5: The inside loop flow chart vi

7 Figure 3.6: The main loop flow chart Figure 3.7: The successive iterative method to estimate u 0, new... 5 Figure 3.8: Thermal analysis geometry Figure 3.9: Left: distribution and size of the microcontact spots in the contact area- Right: Thermal resistance network for a surface element Figure 3.10: Microcontact thermal resistance network Figure 3.11: The output parameters of the model Figure 3.1: Effect of roughness on the pressure distribution... 6 Figure 3.13: The effect of roughness level on the maximum contact pressure, and the radius of the macrocontact area Figure 3.14: The effect of roughness level on the micro, macro, and the joint thermal resistance Figure 3.15: Comparison the model with experimental data- Kitscha σ = 0.76( µm) test Figure 3.16: Comparison the model with experimental data- Kitscha σ = 0.17( µm) test vii

8 List of Tables Table.1, Vickers Microhardness Correlations, Hegazy (1985) Table 3.1: Input data for a typical contact problem Table 3.: Geometry and properties for the Kitscha σ = 0.76( µm) test Table 3.3: Geometry and properties for the Kitscha σ = 0.17( µm) test Table 4.1: Geometrical and mechanical parameters of the proposed tests Table 4.: Time schedule for the proposed research viii

9 Nomenclature A = area ( m ) a b = radius of contact area (m) = flux tube radius (m) c1, c = microhardness Vickers correlation coefficient c ( GPa ) 1 d = mean plane separation in Greenwood and Williamson (1966) model (m) d V = Vickers indentation diagonal ( µ m ) * dr s = non-dimensional microcontact thermal resistance of a surface element dr E = increment in radial direction (m) = Young s elastic modulus (GPa) * * E = equivalent elastic modulus = ( ν1 ) 1+ ( ν) E (.) = Elliptical integral of the second kind 1 E 1 / E 1 / E ( GPa) F * F f = applied force (N) = relative error for the calculated pressure distribution in the numerical procedure = discrete point forces acting on microcontacts (N) H, H B = bulk hardness (GPa) H mic = microhardness (GPa) H BGM = geometric mean Brinell hardness (GPa) h = contact thermal conductance ( W / m K ) c K = constant K (.) = Elliptic integral of the first kind k = thermal conductivity (W/mK) ix

10 k = harmonic mean thermal conductivity k k k ( k k ) s s = + (W/mK) 1 1 L m = sampling length (m) = effective mean absolute surface slope m= m + m 1 n s = number of microcontact spots P = applied pressure (Pa) P Q = average (mean) pressure acting on surface elements (Pa) = heat transfer rate (W) q = heat flux ( W / m ) r = radial position (m) * r = non-dimensional radial position r * = r a / Hz R = thermal resistance (K/W) * R = non-dimensional thermal resistance R * = ρ s kr s S Y = dummy variable = yield or flow stress (Pa) T = temperature ( C) u Y = sphere profile in the vicinity of the contact region (m) = mean surface plane separation (m) x

11 Greek Symbols α β = non-dimensional roughness (roughness level) α = σ ρ = surface summits radii of curvature a Hz T = effective temperature drop across the interface ( C) ε = flux tube relative radius ε = a/ b= A / A r a η = microcontacts density ( m ) s λ = non-dimensional separation λ = Y σ θ ν = polar coordinate angle (radians) = Poisson ratio ρ = effective radius of curvature ( ) 1 ρ = 1/ ρ + 1/ ρ ( m) 1 τ σ ω ψ = redistribution parameter = effective r.m.s. surface roughness σ = σ1 + σ ( µm) = deformation (m) = thermal constriction / spreading parameter xi

12 Subscripts 0 = value at the origin, i.e. r = 0 1, = surface 1 and a b c e g Hz j L m r s V = apparent area = bulk = conduction heat transfer, critical = effective = gap heat transfer = Hertz = joint = large (macro scale) = mean (average) value = real area, radiative heat transfer = small (micro scale), microcontact, summits = Vickers xii

13 1. Introduction It has been an accepted fact that the interface formed by two members in contact represents an additional resistance to the flow of heat from one member to another. The study of interface resistance between contacting surfaces had received little attention until late 1940 s, before which time it was either neglected or assigned some arbitrarily or empirically constant value for a given contact. Thermal contact problem, i.e. constriction resistance through a non-conforming contact of two rough surfaces, as a basic problem, occurs in a wide range of applications. Some examples are; i) microelectronic cooling, ii) spacecraft structure, iii) satellite bolted joints, iv) nuclear engineering, v) ball bearings, vi) solar collectors and vii) heat exchangers. Generally, contact between two surfaces occurs only over microscopic contacts. The macroscopic contact region, as shown in Figure 1.1, arises due to radius of curvature / out-of-flatness of bodies; the microcontacts arise due to the microscopic irregularities or asperity characteristics of rough surfaces. The true or real area of contact, A r the total area of all microcontacts, is typically only a small fraction (a few percent or often much less) of the apparent contact area A a. This effect is usually observed through a relatively high temperature drop across the interface (changing up to 100 K or more over a few micrometers). Under these circumstances, the contact spots or the microcontacts may be under high heat fluxes depending upon the heat flow, mechanical and thermal properties of the contacting bodies and load. 1

14 Body 1 F Sphere Insulation Z Temperature profile T1 ρ Q a L Vacuum Q i Q = Q i T T Heat flux lines Microcontacts constriction, spreading resistance Body Q F Flat Macrocontact constriction, spreading resistance Figure 1.1:Non-conforming rough surface contact- macro, and microcontact(s) constriction / spreading of heat flux, and temperature distribution in the vicinity of contact 1.1 Definition of Contact Resistance Thermal contact resistance between contacting surfaces results in a temperature drop T across the interface for a given heat flow rate through the interface. In general, the total heat flow between contacting bodies can occur by three different modes: conduction (through microcontacts, i.e. solid-solid contact) convection / conduction (through the interstitial fluid in the gap between solids) thermal radiation across the gap

15 The total rate of heat flow through the interface is assumed to consist of three separate components: Q = Qc + Qg + Qr (1.1) where, Q, Q, and Q r are the rates of heat transfer by conduction, conduction/convection c g through the interstitial gap, and radiation. The thermal joint resistance is defined as: T Rj = (1.) Q where, R j is the thermal resistances of the joint, and T = T 1 T (see Figure 1.1) is the effective temperature drop across the joint. The radiation heat transfer across the interface remains small as long as the body temperatures are not too high. In typical applications, such as electronic packaging, the heat transfer by radiation is often negligible. However, radiation can be an important factor if the temperatures involved are high. According to McWaid (1990) natural convection does not occur within a fluid when the Grashof number is below 000. The Grashof number can be interpreted as the ratio of buoyancy to viscous forces. In most practical situations concerning thermal contact resistance the nominal gap thickness between the two pressed contacts is quite small ( < 001. mm ). The Grashof number based on the nominal gap thickness is therefore usually less than 000. Consequently, in most instances the heat transfer through the interstitial fluid occurs by conduction, not convection. In this study we are interested in the contact of two non-conforming rough surfaces in a vacuum, i.e.: R g, R. Therefore, heat is constrained to pass through a macroscopic r contact and then, in turn through microscopic contact spots, as shown in Figure 1.1. As a 3

16 result of this assumption two sets of resistances in series may represent the thermal contact resistance for a joint in a vacuum: the large-scale or macroscopic constriction resistance R L and the small-scale or microscopic constriction resistance R s. Figure 1. shows the resistance network analogy for analysis of the joint thermal resistance. Figure 1.: Resistance network analogy for the thermal joint resistance The total joint resistance R j can be written as: Rj = RL, 1+ Rs, 1+ RL, + Rs, (1.3) n 1 s 1 = R s R (1.4) 1, s, i i = 1 1, 4

17 Where, n s is number of microcontact spots and subscripts 1, signify bodies 1,. Contact problems may also be formulated in terms of contact conductance. The thermal contact conductance coefficient is defined in the same manner as the film coefficient in convective heat transfer, and can be derived as: Q = h A T = c a 1 RA c a (1.5) 1. Problem Statement The resistance to heat flow through contacting, rough, non-conforming bodies, in general, depends upon the radius of the macrocontact area, a L, as well as the number, size and distribution of microcontacts within the macrocontact area. As stated in section 1.1, the interstitial fluid is assumed to be absent and the radiation heat transfer in the contact is neglected in the present study. Thermal contact resistance problems basically consist of three different problems: geometrical mechanical thermal Figure 1.3 illustrates the thermal contact resistance problem flow diagram and its components. The mechanical problem includes two parts: 1) macro or large-scale contact, and ) micro or small-scale contact problem. The mechanical analysis determines the macrocontact radius size a L and the pressure distribution for the large-scale problem. While for the microcontact problem the microcontact size, number of microcontacts, the relative microcontact radius and their distribution are calculated. The macro and the micro mechanical problems are strongly coupled. 5

18 The thermal analysis then determines the constriction/spreading resistances based on an isothermal flux tube solution for both macro and micro contacts. The necessary assumptions to carry out the analysis will be discussed later. 6

19 Geometrical Problem Effective Radii of Curvature, Effective Roughness Mechanical Problem Macrocontact Large-Scale Problem Coupled Microcontacts Small-Scale Problem Pr ( ), Pressure Distribution a L, Macrocontact Radius as( r), Profile of Microcontact Radius ns( r), Profile of Number of Microcontacts ( r), Profile of Microcontact Relative Radius ε s Thermal Problem Flux Tube Solution Thermal Macrocontact Resistance R L Thermal Microcontact Resistance R s Thermal Joint Resistance R j Figure 1.3, The thermal contact problem statement flow diagram 7

20 . Literature Review As explained in section 1., thermal contact resistance of non-conforming rough surfaces involves three different problems: 1) geometrical, ) mechanical (macro and micro), and 3) thermal problem. Each problem will be reviewed, also the rough surface characteristics topic is usually categorized separately, section.1 is dedicated to introduce and review the rough surfaces characteristics..1 Rough Surface Parameters All solid surfaces are rough, this roughness, or surface texture, can be thought of as a surface s deviation from its nominal topography. Surface textures can be created by a many different processes. Most man-made surfaces, such as those produced by grinding or machining have a pronounced lay. It is not easy to produce a wholly isotropic roughness. The usual procedure for experimental purposes is to air-blast a metal surface with a cloud of fine particles, in the manner of shot peening, which gives rise to a randomly created surface. In this study, we will focus only on random rough surfaces. Rough surfaces consist of asperities that are scattered over a reference plane. The height of these asperities has a Gaussian distribution about the mean plane. A common measure of average roughness of the surface is the root-mean-square or standard deviation, σ, of the height of the surface from the centerline, i.e.: L 1 σ = L zdz 0 (.1) 8

21 where, z is the height of the surface above the datum, and L is the sampling length. According to Liu, et al. (1999) five types of instruments are currently available for measuring the surface topography, namely: 1) stylus-type surface profilometer ) optical (white-light interference) measurements 3) scanning Electron Microscope (SEM) 4) atomic Force Microscope (AFM) 5) scanning Tunneling Microscope (STM) Among them, the first two instruments are usually used for macro-to-macro asperity measurements, whereas the others may be used for micro or nanometric measurements. Surface texture is most commonly measured by a profilometer, which draws a stylus over a sample length of the surface. A datum or centerline is established by finding the straight line (or circular arc in the case of round components) from which the mean square deviation is a minimum. Usually rough surfaces are described with the standard deviation roughness of the surface σ, and with the mean absolute asperities slope m. Tabor (1951) mentioned that their values in practice depend upon both the sample rate and the sampling interval h used in the measurement. The mean square roughness σ is virtually independent of the sampling interval h provided that h is small compared with the sample length L. The parameter m, however, is sensitive to sampling interval..1.1 Hardness and Microhardness Tabor (1951) in his introductory on the hardness of metals said: hardness like storminess of the seas is easily appreciated but not readily measured. In general, 9

22 hardness implies the resistance to permanent deformation. A wide variety of hardness tests have been classified into groups, the most common of which is the static indentation group. In a static indentation test, a steady load is applied to an indenter which may be a ball, cone or pyramid and the hardness H is calculated from the area or depth of indentation produced. Hegazy (1985) demonstrated through experiments with four alloys (SS304, nickel 00, zirconium-.5% niobium, and Zircaloy-4) that the effective microhardness H mic is significantly greater than the bulk hardness H Brinell, or Meyer macrohardness due to work hardening of metallic surfaces during machining processes, see Figure.1. Figure.1: Measured hardness and microhardness, Hegazy (1985) As shown in Figure.1 H mic decreases with increasing depth of the indenter until, H bulk hardness, is obtained. Hegazy concluded that this increase in the plastic yield stress 10

23 (microhardness) of the metals near the free surface is a result of a local extreme work hardening or some surface strengthening mechanism. Hegazy derived empirical correlations to account for the decrease in contact microhardness of the softer surface with increasing depth of penetration of asperities on the harder surface: H V d V = c1 d0 c (.) where, H V is Vickers microhardness in GPa, and d V is Vickers indentation diagonal in µm, d0 = 1( µ m), and c, c are correlation coefficients determined from experimental 1 measurements. Table.1 shows c1, c for some materials: Table.1, Vickers Microhardness Correlations, Hegazy (1985) Material c1 ( GPa) c Zr Zr-.5 wt % Nb Ni SS dv AS av av AV AV =AS as a) Vicker's indentation print b) Circular contact spot Figure.: Relation between contact spot radius and the Vickers s diagonal 11

24 As shown in Figure., if the print area in a Vickers test is assumed to be equal to the microcontact area, i.e. A V = A a relation between the Vickers s diagonal d V and the S microcontact spot radius a s can be derived: d V = π a (.3) s Therefore, effective microhardness can be computed: ( s) H = c 1 π a (.4) mic where, a s is the radius of the microcontact spot. In general microhardness depends on several parameters: mean surface roughness, mean absolute slope of asperities, method of surface preparation, and applied pressure. Sridhar (1994) suggested empirical relations to estimate microhardness coefficients, c1,andc using the bulk hardness of the material. Two least-square-cubic fit expressions were reported: c 1 3 H B H B H B = HBGM (.5) HBGM HBGM HBGM c c 1 H B 1 H B 1 H = H 4. H H BGM BGM 3 B BGM (.6) where, H B is the Brinell hardness of the bulk material, and H = 3178 BGM. ( GPa). The above correlations are valid for a Brinell hardness range of 13. to 76. ( GPa ), and the r.m.s percent difference between data and calculated values are 5.3% and 0.8% for c 1,andc, respectively. 1

25 . Geometrical Model It is necessary to consider the effect of both surface roughness and non-conformity or out-of-flatness upon the contact of two non-conforming rough surfaces. The simple geometry of a sphere-flat is chosen because of its mathematical and geometrical simplicity. A flow diagram of the geometrical model used in this study, is shown schematically in Figure.3. The actual contact between two curved bodies is modeled as a contact between two truncated spherical segments, which is mathematically equivalent to the contact of a flat with a sphere. b L ρ 1 ρ a) Contact of non-conforming rough surfaces ρ σ σ 1 c) Rough sphere-rough flat contact, effective radius of curvature b) Contact of two rough spherical segments ρ σ b L roughness, and effective radius of curvature Figure.3: Flow diagram of the geometrical model Radii of curvature ρ can be calculated from the following geometrical relation: b L ρ = (.7) δ 13

26 where, δ is out-of-flatness, Clausing and Chao (1965). According to Johnson (1985) in static frictionless contact of solids, the contact stresses depend only upon the relative profile of their two surfaces, i.e. upon the shape of the interstitial gap between them before loading. The actual system geometry then may be replaced, without loss of generality, as illustrated in Figure.4 by a flat surface and a profile, which results in the same undeformed gap between the surfaces. For convenience, all elastic deformations are considered to occur in one body, which has an effective elastic modulus E * and the other body is assumed to be rigid: E * = ν E + ν (.8) 1 E = + (.9) ρ ρ1 ρ F Body 1 F ρ E 1 1, ν 1 ρ Rigid sphere Contact plane ρ a E, ν Elastic half-space E * a F Body Figure.4: Geometry of the elastic contact of two smooth spheres 14

27 Assuming a Gaussian distribution of surface profile heights and slopes for rough surfaces, allows a simple approximation that contacts formed by the two Gaussian surfaces are mathematically equivalent to one formed by a single Gaussian surface, having combined surface characteristics, placed in contact with a perfectly smooth flat surface. Figure.5 shows the surface geometry model for conforming rough surfaces. The effective roughnessσ, and slope m are defined as follows: σ = σ + σ 1 m= m + m 1 (.10) (.11) Mean plane 1 ω Smooth flat σ m 1 1 m σ ω Z 1 Z Y σ m Y Z Mean plane Mean plane Equivalent rough Figure.5: Equivalent contact of conforming rough surfaces 15

28 .3 Mechanical Analysis The contact of two smooth non-conforming surfaces (macro or large-scale contact problem) plays an important role in the mechanical analysis. Therefore, existing theories for the contact of two smooth curved bodies will be reviewed. As the other part of the mechanical analysis (micro or small-scale contact problem), rough surface models also will be reviewed. Figure.6 illustrates the mechanical analysis overview for contact of non-conforming rough surfaces. Contact of non-conforming smooth surfaces (Macrocontact) Contact of conforming rough surfaces (Microcontact) F ω Smooth flat ρ Contact plane O a Hz Elastic half-space σ m Y Z Mean plane -Elasticity theory -Elastoplastic deformation -Fully plastic deformation -Plastic models -Elastic models -Elastoplastic models Contact of non-conforming rough surfaces F ρ Contact plane O a P(r) σ Elastic half-space Figure.6: Mechanical analysis flow diagram for non-conforming rough surfaces 16

29 .3.1 Macrocontact Problem In this section different modes of deformation will be reviewed for smooth sphere-flat contacts with increase the external loading. As the first step, normal deformation due to a concentrated load on a half space, Boussinesq problem, will be studied. Then, by using superposition, an expression for normal deformation due to a general normal pressure distribution over a circular region on a half-space will be derived. The Hertz (1881) contact theory can be derived directly from the resulting general equation by assuming a parabolic pressure distribution over a circular contact area. The onset of plastic deformation, plastic, and elastoplastic modes of deformation will be reviewed for completeness. Concentrated Normal Force Acting On A Half-Space Timoshenko (1970) showed that the elastic displacements on the surface of a solid ( z = 0 ), produced by a concentrated point force F acting normally on a surface at the origin is: F ωb( r) z = = 0 πer (.1) From the above equation it can be seen that the profile of the deformed surface is a rectangular hyperboloid, which is asymptotic to the undeformed surface at a large distance from the origin and exhibits a theoretically infinite deflection at the origin. 17

30 Axisymmetric Loading Over A Circular Region Elastic deflections produced by a normal pressure distribution over an area of the surface now can be found by superposition using the result of the concentrated normal force, and the fact that the displacement due to an axisymmetric pressure distribution will also be axisymmetric. P(r) O C s θ r ds B sdθ O a r a) A general pressure distribution over a circular region b) Calculating normal deformation of point B, due to P(r), using superposition Figure.7: Axisymmetric pressure distribution on a circular area Gladwell (1980) using superposition showed that the normal displacement of an axisymmetric normal pressure distribution Pr ( )over a half-space (Figure.7) is: a π 1 1 ωb( r) = sp( s) ds ( r + s rscos θ) / dθ πe 0 0 (.13) The integral is an elliptic type, which can be simplified to the following: 18

31 a ω ( b r ) = 4 s PsKκ ds πe s+ r ( ) ( ) 0 (.14) where, K(.) is the complete elliptic integral of the first kind with the argument κ = rs ( r+ t ), a is the radius of the loaded area, and s is a dummy variable in the range 0 to a. Gladwell showed that the displacement in elastic half-space due to Pr ( )applied on the surface is equal to the following: ω b a ( r) = sp( s) B( r, s) ds 0 (.15) where; 4 Brs (,) = Es K ( r ) π s 4 Brs (,) = Er K ( s ) π r r r < s (.16) > s (.17) Hertz (1881) Theory of Elastic Contact When two non-conforming smooth solids are brought into contact, they touch initially at a single point (spheres) or along a line (cylinders). Hertz first made the hypothesis that the contact area is, in general, elliptical, and then introduced the simplification that, each body can be regarded as elastic half-space loaded over a small elliptical region of its plane surface. Figure.4 shows the sphere-flat geometry in which, a, is the radius of the contact area, and ρ is the effective radius of curvature. Assumptions in Hertz theory can be summarized as follows: surfaces are continuous and non-conforming strains are small, i.e. a << ρ 19

32 each solid can be considered as an elastic half-space, i.e. a << ρ surfaces are frictionless According to the Hertz theory, for the elastic contact of a sphere-flat the following expressions can be written: F a = Hz 3 ρ * 4E 13 / (.18) a Hz ω0 = (.19) ρ F FE Po = = * 3 a 6 π π ρ 13 / (.0) where, ω 0 is the maximum deformation, and P o is the maximum pressure in the Hertz solution. Plastic Yield According to Johnson (1985), the load at which plastic yield begins in the complex stress of two solids in contact is related to the yield point of the softer material in simple tension or shear test through an appropriate yield criterion. The yield of most ductile materials is usually taken to be governed either by Tresca s maximum shear stress criterion, or Von Mises shear strain-energy criterion. However, the difference in the predictions of the two criteria is not large and is hardly significant. When the plastic deformation is severe so that the plastic strains are large compared with the elastic strains, the elastic deformation may be neglected. Provided the material does not strain-harden to a large extent, it may be idealized as a rigid-perfectly plastic solid, which flows plastically at a constant stress S Y in simple tension or compression. A 0

33 loaded body of rigid-plastic material consists of regions in which, plastic flow takes place and regions in where there is no deformation due to the assumption of rigidity. The elasticity of real materials plays an important part in the plastic indentation process. When the yield point is first exceeded the plastic zone is small and fully contained by material, which remains elastic so that the plastic strains are as the same order of magnitude as the surrounding elastic strains. In these circumstances the material displaced by the indenter is accommodated by an elastic expansion of the surrounding solid. As the indentation becomes more severe, either by increasing the load on a curved indenter or by using a more acute-angled wedge or cone, an increasing pressure is required beneath the indenter to produce the necessary expansion. Eventually the plastic zone breaks out to the free surface and displaced material is free to escape by plastic flow to the sides of the indenter. This is the uncontained mode of deformation, which should be analyzed by the theory of rigid-plastic solids. We would expect the plastic zone to break out to the surface and the uncontained mode to become possible when the pressure beneath the indenter reaches the value given by rigid-plastic theory. Based on Johnson s theory this pressure can be written as: P m = cs (.1) where, c has a value about 3.0 depending on the geometry of the indenter and friction at the interface. Equation (.1) also gives the first yield s pressure, where the constant c has a value about unity. There is a transitional range of contact pressures, lying between S Y and 3S Y, where the plastic flow is contained by elastic material and the mode of deformation is one of roughly radial expansion. The three ranges of loading: purely Y 1

34 elastic, elastic-plastic (contained) and fully plastic (uncontained) are a common feature of the most engineering structures..3. Microcontact Problem There are many analytical models for the contact of conforming rough surfaces. Based upon the deformation mode of asperities, existing models can be categorized into three main groups: plastic, elastic and elastoplastic models. In this section some of the existing models for contact of conforming rough surfaces will be discussed. Plastic Models Abott and Firestone (1933) developed the most widely used model for a fully plastic contact, known as the surface microgeometry model. This model assumes that the deformation of a rough surface against a rigid smooth surface is equivalent to the truncation of the undeformed rough surface at its intersection with the flat. With the concept of having only one rough surface (effective roughness) in contact with a smooth one, model implies that the asperities are flattened or equivalently penetrated into the smooth surface without any change in shape of the part of surfaces not yet in contact. Therefore, bringing the two surfaces together within a distance Y is equivalent to slicing off the top of the asperities at a height Y above the mean plane, see Fig..4. Since surfaces are rough and the true area of contact, which is much smaller than the apparent area in contact, must support pressures so large that they are comparable with the strength of the materials of the contacting bodies.

35 Bowden and Tabor (1954) suggested that these contact pressures are equal to the flow pressure of the softer of the two contacting materials and the normal load is then supported by plastic flow of its asperities, so that the mean contact pressure will be equal to the hardness and effectively independent of load and the contact geometry. The true area of contact is then proportional to the load. Cooper, Mikic, and Yovanovich, CMY Plastic Model (1969) Cooper, Mikic, and Yovanovich (1969) developed a model for conforming nominally flat rough surfaces that assumes all the contact asperities undergo plastic deformation. The model is based on the following assumptions: rough surfaces are isotropic deformation of each asperity is independent of its neighbors the surface heights form a Gaussian distribution, and the slope (m) of surface is independent of height. Based on the surface microgeometry model one can write: P H m e A A r = (.) a where, P = F / A is the nominal pressure, and H e is the effective microhardness of the m a softer surface. The main relations of the CMY are as follows: Ar A a ( ) = 1 erfc λ (.3) 1 m exp( λ ) ηs = 16 σ erfc( λ ) (.4) 3

36 as = 8 σ exp( λ ) erfc ( λ ) (.5) π m h c ks m = π σ exp( λ ) 1 erfc( ) [ λ ] (.6) λ = erfc 1 P m H e (.7) 1 kk k = s k + k 1 (.8) where, η, a, h, λ,and k are density of microcontacts, radius of microcontacts, joint s s c s conductance, dimensionless separation, and harmonic mean of thermal conductivies, respectively. Elastic Models Archard (1957) pointed out that for applications such as lubrication, or moving machine parts in which the contacting surfaces meet many times, the asperities may flow plastically at first but they must reach a steady state in which the load is supported elastically. He then offered a model in which each asperity is covered with micro asperities, and each micro asperity with micro-micro asperities, gave successive closer approximations to the law A r F as more stages were considered. Greenwood and Williamson (1966) Elastic Model (GW) Greenwood and Williamson (1966) developed an elastic model for contact of flat rough surfaces, based on the deformation of an average size asperity. They assumed that all summits have the same radius of curvature at their top, and possess a Gaussian distribution about a mean reference plane, see Figure.8. As shown in Figure.8 in the 4

37 GW model, separation is defined as the distance between the mean average summit height and the smooth mean plane. Rigid smooth flat β Separation Y Mean summit height Zs Z Mean surface height Elastic rough surface Figure.8: Greenwood and Williamson geometrical model The GW model is based on the following assumptions: the rough surface is isotropic and has Gaussian height distribution with a standard deviation, σ the distribution of summit heights is the same as the Gaussian standard deviation, i.e.: σ = σ s the deformation of each asperity is independent of its neighbors the asperity summits are spherical with a constant radius β 5

38 the deformation is entirely within the elastic limit and the Hertz equation is applied for each individual summit there is no bulk deformation, i.e. only asperities deform during contact They stated that, the contact between flat surfaces could be determined either by plastic or elastic conditions; plastic flow may be expected for very rough surfaces, while for very smooth ones, contact will be entirely elastic. They reported that for a given nominal area the area of contact depends on the load and not on the nominal pressure. The similarity with the behavior calculated by assuming ideal plastic flow, where the mean pressure is equal to the hardness, led them to suggest that there is an elastic contact hardness, the mean elastic contact pressure, which controls the area of contact under an elastic condition. They estimated the elastic contact hardness for a Gaussian surface as: P e = 05. E σ s β (.9) Mikic (1974) Elastic Model Mikic (1974) offered his elastic model, based on the CMY (1969) model and the fact that at the same separation between contacting surfaces, the elastic contact area is half of the plastic contact area A Elastic A Plastic = 1. According to the Mikic analysis, the average contact pressure, and the ratio of real contact area to apparent area in elastic deformation are: P e = * E m (.30) Ar A a ( ) = 1 erfc λ (.31) 4 6

39 where, m is the slope of the surface. The other relations in the Mikic elastic model are the same as the CMY (1969). Mikic also showed that the contact area is proportional to the load, and it is not sensitive to the nominal pressure. Elastoplastic Models Chang et al. (1987) presented a model (we may call it CEB) based on volume conservation of an asperity control volume during plastic deformation. They started with the GW (1966) model assumptions, and then by using the work of Tabor (1951), the initial yield was assumed to occur when Po = 06. H, where P o is the maximum pressure at each asperity, and H is the hardness of the softer material. The critical interference, ω c, at the inception of plastic deformation is: πkh ωc = β E (.3) They developed an approximate elastoplastic model based on volume conservation of an assumed plastically deformed region of the asperity with the following assumptions: the asperity deformation is localized mainly in the vicinity of the contact. Hence, beyond a certain depth under the contact, the asperity remains undeformed the deformed asperity is modeled by a truncated spherical segment the radius β is assumed to be the same as of the unreformed asperity for all plastically deformed asperities the average pressure over the contact area assumed to be KH, which is constant throughout the elastic-plastic deformation. In the CEB model, there is a discontinuity in the contact pressure at the critical point of the initial yielding. At this point, the average contact pressure is allowed to jump from / 3KH in the elastic regime to KH in the plastic regime. 7

40 Zhao et al. (000), using the CEB (1987) model, developed an elastic-plastic microcontact model for nominally flat rough surfaces. The transition from elastic deformation to fully plastic flow of the contacting asperities was curve-fitted. A cubic polynomial, smoothly joining the expressions for elastic and plastic area of contacts spans the elastoplastic region based on two extremes of the CEB (1987) model..4 Deformation Mode of Asperities Tabor (1951) stated: when two metal surfaces are first placed in contact, they will be supported on the tips of their asperities. It might be assumed that one surface is harder than the other and that the asperities on the harder surface are hemispherical / pyramidal at their tips. At first the deformation is elastic, but for asperities of the order of 10 4 cm radius the minutest loads will produce plastic deformation. Indeed full plasticity may occur even for the hardest steels at a load of the order of a few milligrams. He concluded that in most practical cases, for a wide variety of shapes and types of surface irregularities the real area of contact would be very nearly proportional to the applied load. It will also be inversely proportional to the effective hardness of the surface asperities. The real area bears no direct relation to the actual size of the surface. Greenwood (1967) described the contact of two surfaces as: surfaces touch at a larger number of contacts, and these contacts will be in all the states from fully elastic, to fully plastic. The fully elastic ones are a negligible fraction of the total; the effective flow pressure will be intermediate between plastic and elastic values. Greenwood and Williamson (1966) and Greenwood and Tripp (1971) reported that even in the elastic mode of deformation the real contact area is proportional to the load. 8

41 Greenwood and Williamson (1966) introduced a plasticity index, as criteria for plastic flow: γ = E H σ s (.33) β where, β is the radius of the summits. They also reported that the load has a little effect on the deformation regime. When the index is less than 0.6, plastic contact could be caused only if the surfaces were forced together under very large nominal pressure. When γ > 1.0 plastic flow will occur even at trivial nominal pressures. Based on the plasticity index, they concluded that; most of surfaces have plasticity indices larger than 1.0, and thus, except for especially smooth surfaces, the asperities will flow plastically under the lightest loads, as has been frequently postulated. Chang et al. (1987) with the same assumptions as GW (1966), set the criteria for the deformation mode based on the deformation of an average asperity. For compliances less than the critical complianceω c, ω c is the inception of plastic deformation based on experimental work, the contact is elastic and Hertzian theory can be applied, for ω ω c, they offered an elastoplastic model which finally collapse to the plastic model. According to Johnson (1985), the GW (1966) and Chang et al. (1987) assumptions outlined above are not strictly correct because the summit curvature β is not independent of the summit height. In reality the summit curvatures will have a random variation, but they assumed a constant curvature for all asperities. In addition, if actual surfaces were characterized as having variable asperity radii of curvature, it would be difficult to use the above criteria to predict plastic flow of asperities. 9

42 Mikic (1974) proposed an alternative plasticity index, based on the fact that, when two rough surfaces are in contact the degree of plastic deformation of asperities is proportional to the microhardness, elastic properties, and the slope of the asperities: γ = Hmic (.34) Em This definition avoids the difficulty of measuring two statistical quantities ( β,σ s ) in the GW plasticity index. Mikic also reported that the mode of deformation, as stated in GW model, depends only on the material properties and the shape of the asperities, and it is not sensitive to the pressure level. Mikic performed an analysis to determine the contact pressure over the contact area based on the fact that all contact spots do not have the same contact pressure, although the average contact pressure would remain constant. He concluded that for γ 3, 90% of the actual area will have the elastic contact pressure, therefore the contact will be predominantly elastic, and for γ 033., 90% of the actual area will have the plastic contact pressure, therefore the contact will be predominantly plastic. It might be concluded that there is an average contact pressure acting at the top of microcontact spots. Based upon the Mikic plasticity index, the deformation of asperities and consequently this average contact pressure could be elastic, plastic or elastoplastic. Not to confuse this pressure with macro pressure distribution, we will call this the average asperity pressure. For most engineering surfaces the asperity deformation mode is plastic, therefore the average asperity pressure is microhardness, H mic. As an example, stainless steel 304, with E = 110( GPa), Hmic = 30. ( GPa)and m = 01. (an average value for blasted surfaces) has the plasticity index, γ = 07., which lies in the plastic range. 30

43 .5 Non-Conforming Rough Surfaces There are very few models for the contact of non-conforming rough surfaces in the literature. Greenwood and Tripp (1967) performed the first in-depth analytical study of the effect of roughness on the pressure distribution and deformation of contacting elastic spherical bodies. They developed a model for predicting the continuously varying pressure distribution and deformation. Assumptions were almost the same as the assumptions of Greenwood and Williamson (1966) model. The contacting rough surfaces were modeled as a smooth sphere and a rough flat. The microscopic summits were modeled as hemi spheres with a Gaussian distribution of heights above the mean surface. Like the Greenwood and Williamson (1966) model, this model was based on the assumption that; the top of the asperities were spherical, all with the same radius, and that they deformed elastically according to the Hertz theory. They derived a relation connecting the local separation, between the mean surfaces, with the pressure created by compressing the asperities. Then, having the pressure distribution, the elastic deformation of bodies could be found from elasticity theory; this gave a complementary relation between separation and pressure. Figure.9 shows the iterative cycle, which was used in the Greenwood and Tripp (1967) model to solve these equations numerically. Elastic Deformation of Asperities Local Separation Pressure Elastic Deformation of Sphere-Flat Figure.9: Numerical procedure in Greenwood and Tripp model 31

44 Although their model was based on the total elastic deformation of the summits, they also mentioned that the higher summits must deform plastically. They concluded the essential behavior of the rough surfaces is determined by the statistical distribution of asperity heights and secondarily by their mode of deformation. The most important trends in their model were that increases in roughness: decrease the axial (maximum) contact pressure, P o, compared with the axial (maximum) Hertzian contact pressure, P ohertz, enlarge the effective macroscopic contact radius, a L, beyond the Hertzian contact radius.6 Thermal Analysis In this section first the flux tube solution will be studied as a general solution for both macro and micro contact resistances. Then some existing thermal contact resistance models for non-conforming rough surfaces will be reviewed..6.1 Constriction Resistance- Flux Tube Solution The spreading resistance in general is defined as the difference between the average temperature of the contact area and the average temperature of the heat sink, which is far from the contact area, divided by the total heat transfer rate Q. Solving Laplace s 3

45 equation for steady state heat conduction, Yovanovich (001) showed that the spreading resistance for an isothermal circular contact area on a half-space is: RSpreading = 1 4ka (.35) where, k is the thermal conductivity, and a is the radius of the contact area. The spreading resistance depends on the boundary condition over the contact area; he also showed that the isoflux resistance is approximately 8.1% greater than the isothermal spreading resistance. The difference between constriction and spreading alleviation factors ψ is also neglected, i.e.: ψ ψ = ψ (.36) Spreading Constriction Figure.10 represents the geometry, and boundary conditions of the flux tube. The flux tube consists of a circular heat source area of radius a, which is in perfect contact with a tube with radius b. The constriction resistance for the flux tube is: ( ) ψε ( ) R = ψε flux tube 4ka + 4ka 1 (.37) where, ψ ( ε ) is the constriction/spreading alleviation factor, andε is the relative contact radius ε = ab. Yovanovich (001) proposed the dimensionless spreading resistance based on the ratio of real area to apparent area. The advantage of this definition is that the real shape of the contact spots would be a second order effect. a ε = = b Ar A a (.38) 33

46 Many researchers worked on the flux tube constriction factor and suggested various expressions for ψ most of these expressions are limited to a specific range of ε. Cooper, Mikic and Yovanovich (1969) developed an accurate expression for ψ ( ε ): ( ) = ( 1 ε) ψε Their expression is accurate for all values ofε. 3 (.39) T Q k ( ) 1or = z orz z πb T = 0( z = 0, a < r < b) z T = 0 T = 0( r = b) r T = T ( z = 0,0 < r a) 0 Figure.10: The Flux tube geometry and boundary condition.6. Clausing and Chao (1965) Clausing and Chao (1965) considered the contact of rough non-flat surfaces. They developed a model for determining the macroscopic and microscopic constriction 34

47 resistance for relatively smooth, spherical surfaces in contact in a vacuum, based on the following assumptions: the size of macroscopic contact radius a L is obtained from the Hertz theory for elastic spheres, which assumes that the spheres are smooth, i.e. no roughness the asperities deform plastically, and material hardness which is multiplied by an empirical correction factor introduced by Holm (1958), is used to estimate A r : A r F = = nsπ a s (.40) H e the microscopic contact spots are assumed to be uniformly distributed over the macrocontact area, see Figure.11 the average size of the microscopic contacts a s is independent of load the average size of the microscopic contacts a s (all of which are contained within the Hertzian macroscopic contact region) is of the same order of magnitude as the surface roughness, i.e. a s σ surfaces are clean, i.e. no film effects They used the ratio of the Hertzian macroscopic contact radius a L to the specimen radius b to calculate the macroscopic constriction resistance. As shown in Figure.11, by assuming that the microscopic contact spots are distributed in a triangular array with a distance of b s between centers, the microscopic constriction resistance using a flux tube solution can be calculated. Clausing and Chao s (1965) model agreed quite well with the experiments for smooth SS-303, brass 360, magnesium AZ31B, and moderately rough aluminum 04-T4. However, their measured resistances for smooth aluminum 04-T4 were as much as an order of magnitude greater than their theory. 35