BOOKCOMP, Inc. — John Wiley & Sons / Page 321 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 321 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [321], (61) Lines: 2472 to 2532 ——— 6.71815pt PgVar ——— Normal Page PgEnds: T E X [321], (61) B A = (1/k 2 )E(k ) − K(k ) K(k ) − E(k ) (4.151) where K(k ) and E(k ) are complete elliptic integrals of the first and second kind of modulus k . The Hertz solution requires the calculation of k, the ellipticity, K(k ), and E(k ). This requires a numerical solution of the transcendental equation that relates k, K(k ), and E(k ) to the local geometry of the contacting solids through the geometric pa- rameters A and B. This is usually accomplished by an iterative numerical procedure. To this end, additional geometric parameters have been defined (Timoshenko and Goodier, 1970): cos τ = B − A B + A and ω = A B ≤ 1 (4.152) Computed values of m and n,orm/n and n, are presented with τ or ω as the independent parameter. Table 4.15 shows how k, m, and n depend on the parameter ω over a range of values that should cover most practical contact problems. The parameter k may be computed accurately and efficiently by means of the Newton– Raphson iteration method applied to the following relationships (Yovanovich, 1986): k new = k + N(k ) D(k ) (4.153) TABLE 4.15 Hertz Contact Parameters and Elastoconstriction Parameter ω kmnψ ∗ 0.001 0.0147 14.316 0.2109 0.2492 0.002 0.0218 11.036 0.2403 0.3008 0.004 0.0323 8.483 0.2743 0.3616 0.006 0.0408 7.262 0.2966 0.4020 0.008 0.0483 6.499 0.3137 0.4329 0.010 0.0550 5.961 0.3277 0.4581 0.020 0.0828 4.544 0.3765 0.5438 0.040 0.1259 3.452 0.4345 0.6397 0.060 0.1615 2.935 0.4740 0.6994 0.080 0.1932 2.615 0.5051 0.7426 0.100 0.2223 2.391 0.5313 0.7761 0.200 0.3460 1.813 0.6273 0.8757 0.300 0.4504 1.547 0.6969 0.9261 0.400 0.5441 1.386 0.7544 0.9557 0.500 0.6306 1.276 0.8045 0.9741 0.600 0.7117 1.1939 0.8497 0.9857 0.700 0.7885 1.1301 0.8911 0.9930 0.800 0.8618 1.0787 0.9296 0.9972 0.900 0.9322 1.0361 0.9658 0.9994 1.000 1.0000 1.0000 1.0000 1.0000 BOOKCOMP, Inc. — John Wiley & Sons / Page 322 / 2nd Proofs / Heat Transfer Handbook / Bejan 322 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [322], (62) Lines: 2532 to 2584 ——— -0.0958pt PgVar ——— Normal Page * PgEnds: Eject [322], (62) where N(k ) = k 2 E(k ) K(k ) k 2 + A B − k 4 1 + A B (4.154) D(k ) = E(k ) K(k ) k k 2 − 2k A B + A B k 2 k (4.155) If the initial guess for k is based on the following correlation of the results given in Table 4.15, the convergence will occur within two to three iterations: k = 1 − 0.9446 A B 0.6135 2 1/2 (4.156) Polynomial approximations of the complete elliptic integrals (Abramowitz and Ste- gun, 1965) may be used to evaluate them with an absolute error less than 10 −7 over the full range of k . 4.15.2 Local Gap Thickness The local gap thickness is required for the elastogap resistance model developed by Yovanovich (1986). The general relationship for the gap thickness can be determined by means of the following surface displacements (Johnson, 1985; Timoshenko and Goodier, 1970): δ(x,y) = δ 0 + w(x,y) − w 0 (m) (4.157) where δ 0 (x,y) is the local gap thickness under zero load conditions, w(x,y) is the total local displacement of the surfaces of the bodies outside the loaded area, and w 0 is the approach of the contact bodies due to loading. The total local displacement of the two bodies is given by 3F∆ 2π ∞ µ 1 − x 2 a 2 + t − y 2 b 2 + t dt [(a 2 + t)(b 2 + t)t] 1/2 (4.158) where µ is the positive root of the equation x 2 a 2 + µ + y 2 b 2 + µ = 1 (4.159) When µ > 0, the point of interest lies outside the elliptical contact area: x 2 a 2 + y 2 b 2 = 1 (4.160) When µ = 0, the point of interest lies inside the contact area, and when x = y = 0,w(0, 0) = w 0 , the total approach of the contacting bodies is BOOKCOMP, Inc. — John Wiley & Sons / Page 323 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 323 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [323], (63) Lines: 2584 to 2627 ——— 2.00821pt PgVar ——— Normal Page * PgEnds: Eject [323], (63) w 0 = 3F∆ 2π ∞ 0 dt [(a 2 + t)(b 2 + t)t] 1/2 = 3F∆ πa K(k ) (m) (4.161) The relationships for the semiaxes and the local gap thickness are used in the following subsections to develop the general relationships for the contact and gap resistances. 4.15.3 Contact Resistance of Isothermal Elliptical Contact Areas The general spreading–constriction resistance model, as proposed by Yovanovich (1971, 1986), is based on the assumption that both bodies forming an elliptical contact area can be taken to be a conducting half-space. This approximation of actual bodies is reasonable because the dimensions of the contact area are very small relative to the characteristic dimensions of the contacting bodies. If the free (noncontacting) surfaces of the contacting bodies are adiabatic, the total ellipsoidal spreading–constriction resistance of an isothermal elliptical contact area with a ≥ b is (Yovanovich, 1971, 1986) R c = ψ 2k s a (K/W) (4.162) where a is the semimajor axis, k s is the harmonic mean thermal conductivity of the joint, k s = 2k 1 k 2 k 2 + k 2 (W/m · K) (4.163) and ψ is the spreading/constriction parameter of the isothermal elliptical contact area developed in the section for spreading resistance of an isothermal elliptical area on an isotropic half-space: ψ = 2 π K(k ) (4.164) in which K(k ) is the complete elliptic integral of the first kind of modulus k and is related to the semiaxes k = 1 − b a 2 1/2 The complete elliptic integral can be computed accurately by means of accurate polynomial approximations and by computer algebra systems. This important special function can also be approximated by means of the following simple relationships: BOOKCOMP, Inc. — John Wiley & Sons / Page 324 / 2nd Proofs / Heat Transfer Handbook / Bejan 324 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [324], (64) Lines: 2627 to 2667 ——— 0.29227pt PgVar ——— Short Page * PgEnds: Eject [324], (64) K(k ) = ln 4a b 0 ≤ k<0.1736 2π (1 + √ b/a) 2 0.1736 <k≤ 1 (4.165) These approximations have a maximum error less than 0.8%, which occurs at k = 0.1736. The ellipsoidal spreading–constriction parameter approaches the value of 1 when a = b, the circular contact area. When the results of the Hertz elastic deformation analysis are substituted into the results of the ellipsoidal constriction analysis, one obtains the elastoconstriction resistance relationship developed by Yovanovich (1971, 1986): k s (24F ∆ρ ∗ ) 1/3 R c = 2 π K(k ) m ≡ ψ ∗ (4.166) where the effective radius of the ellipsoidal contact is defined as ρ ∗ = [2(A +B)] −1 . The left-hand side is a dimensionless group consisting of the known total mechanical load F , the effective thermal conductivity k s of the joint, the physical parameter ∆, and the isothermal elliptical spreading/constriction resistance R c . The right-hand side is defined to be ψ ∗ , which is called the thermal elastoconstriction parameter (Yovanovich, 1971, 1986). Typical values of ψ ∗ for a range of values of ω are given in Table 4.15. The elastoconstriction parameter ψ ∗ → 1 when k = b/a = 1, the case of the circular contact area. 4.15.4 Elastogap Resistance Model The thermal resistance of the gas-filled gap depends on three local quantities: the local gap thickness, thermal conductivity of the gas, and temperature difference between the bounding solid surfaces. The gap model is based on the subdivision of the gap into elemental heat flow channels (flux tubes) having isothermal upper and lower boundaries and adiabatic sides (Yovanovich and Kitscha, 1974). The heat flow lines in each channel (tube) are assumed to be straight and perpendicular to the plane of contact. If the local effective gas conductivity k g (x,y) in each elemental channel is assumed to be uniform across the local gap thickness δ(x,y), the differential gap heat flow rate is dQ g = k g (x,y) ∆T g (x,y) δ(x,y) dx dy (W) (4.167) The total gap heat flow rate is given by the double integral Q g = A g dQ g , where the integration is performed over the entire effective gap area A g . The thermal resistance of the gap, R g , is defined in terms of the overall joint temperature drop ∆T j (Yovanovich and Kitscha, 1974): BOOKCOMP, Inc. — John Wiley & Sons / Page 325 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 325 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [325], (65) Lines: 2667 to 2713 ——— 6.5355pt PgVar ——— Short Page PgEnds: T E X [325], (65) 1 R g = Q g ∆T j = A g k g (x,y) ∆T g (x,y) δ(x,y) ∆T j dA g (W/K) (4.168) The local gap thickness in the general case of two bodies in elastic contact forming an elliptical contact area is given above. The local effective gas conductivity is based on a model for the effective thermal conductivity of a gaseous layer bounded by two infinite isothermal parallel plates. Therefore, for each heat flow channel (tube) the effective thermal conductivity is approximated by the relation (Yovanovich and Kitscha, 1974) k g (x,y) = k g,∞ 1 + αβΛ/δ(x,y) (W/m · K) (4.169) where k g,∞ is the gas conductivity under continuum conditions at STP. The accom- modation parameter α is defined as α = 2 − α 1 α 1 + 2 − α 2 α 2 (4.170) where α 1 and α 2 are the accommodation coefficients at the solid–gas interfaces (Wied- mann and Trumpler, 1946; Hartnett, 1961; Wachman, 1962; Thomas, 1967; Kitscha and Yovanovich, 1975; Madhusudana, 1975, 1996; Semyonov et al., 1984; Wesley and Yovanovich, 1986; Song and Yovanovich, 1987; Song, 1988; Song et al., 1992a, 1993b). The fluid property parameter β is defined by β = 2γ (γ + 1)/Pr (4.171) where γ is the ratio of the specific heats and Pr is the Prandtl number. The mean free path Λ of the gas molecules is given in terms of Λ g,∞ , the mean free path at STP, as follows: Λ = Λ g,∞ T g T g,∞ P g,∞ P g (m) (4.172) Two models for determining the local temperature difference, ∆T g (x,y), are proposed (Yovanovich and Kitscha, 1974). In the first model it is assumed that the bounding solid surfaces are isothermal at their respective contact temperatures; hence ∆T g (x,y) = ∆T j (K) (4.173) This is called the thermally decoupled model (Yovanovich and Kitscha, 1974), since it assumes that the surface temperature at the solid–gas interface is independent of the temperature field within each solid. BOOKCOMP, Inc. — John Wiley & Sons / Page 326 / 2nd Proofs / Heat Transfer Handbook / Bejan 326 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [326], (66) Lines: 2713 to 2741 ——— -0.96779pt PgVar ——— Normal Page * PgEnds: Eject [326], (66) In the second model (Yovanovich and Kitscha, 1974), it is assumed that the tem- perature distribution of the solid–gas interface is induced by conduction through the solid–solid contact, under vacuum conditions. This temperature distribution is ap- proximated by the temperature distribution immediately below the surface of an in- sulated half-space that receives heat from an isothermal elliptical contact. Solving for this temperature distribution, using ellipsoidal coordinates it was found that ∆T g (x,y) ∆T j = 1 − F(k , ψ) K(k ) (4.174) where F(k , ψ) is the incomplete elliptic integral of the first kind of modulus k and amplitude angle ψ (Abramowitz and Stegun, 1965; Byrd and Friedman, 1971). The modulus k is given above and the amplitude angle is ψ = sin −1 a 2 a 2 + µ 1/2 (4.175) where the parameter µ is defined above. It ranges between µ = 0, the edge of the elliptical contact area, to µ =∞, the distant points within the half-space. Since the solid–gas interface temperature is coupled to the interior temperature distribution, it is called the coupled half-space model temperature drop. The general elastogap model has not been solved. Two special cases of the general model have been examined. They are the sphere-flat contact, studied by Yovanovich and Kitscha (1974) and Kitscha and Yovanovich (1975), and the cylinder-flat contact, studied by McGee et al. (1985). The two special cases are discussed below. 4.15.5 Joint Radiative Resistance If the joint is in a vacuum, or the gap is filled with a transparent substance such as dry air, there is heat transfer across the gap by radiation. It is difficult to develop a general relationship that would be applicable for all point contact problems because radiation heat transfer occurs in a complex enclosure that consists of at least three nonisother- mal convex surfaces. The two contacting surfaces are usually metallic, and the third surface forming the enclosure is frequently a reradiating surface such as insulation. Yovanovich and Kitscha (1974) and Kitscha and Yovanovich (1975) examined an enclosure that was formed by the contact of a metallic hemisphere and a metallic circular disk of diameter D. The third boundary of the enclosure was a nonmetallic circular cylinder of diameter D and height D/2. The metallic surfaces were assumed to be isothermal at temperatures T 1 and T 2 with T 1 >T 2 . These temperatures corre- spond to the extrapolated temperatures from temperatures measured on both sides of the joint. The joint temperature was defined as T j = (T 1 +T 2 )/2. The dimensionless radiation resistance was found to have the relationship Dk s R r = k s πσDT 3 j 1 − 2 2 + 1 − 1 2 1 + 1.103 (4.176) BOOKCOMP, Inc. — John Wiley & Sons / Page 327 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 327 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [327], (67) Lines: 2741 to 2803 ——— -2.31227pt PgVar ——— Normal Page PgEnds: T E X [327], (67) where σ = 5.67×10 −8 W/m 2 ·K 4 is the Stefan–Boltzmann constant, 1 and 2 are the emissivities of the hemisphere and disk, respectively, and k s is the effective thermal conductivity of the joint. 4.15.6 Joint Resistance of Sphere–Flat Contact The contact, gap, radiative, and joint resistances of the sphere–flat contact shown in Fig. 4.17 are presented here. The contact radius a is much smaller than the sphere diameter D. Assuming an isothermal contact area, the general elastoconstriction resistance model (Yovanovich, 1971, 1986; Yovanovich and Kitscha, 1974), becomes R c = 1 2k s a (K/W) (4.177) where k s = 2k 1 k 2 /(k 1 + k 2 ) is the harmonic mean thermal conductivity of the contact, and the contact radius is obtained from the Hertz elastic model (Timoshenko and Goodier, 1970): 2a D = 3F∆ D 2 1/3 (4.178) where F is the mechanical load at the contact and ∆ is the joint physical parameter defined above. The general-coupled elastogap resistance model for point contacts reduces, for the sphere–flat contact, to (Yovanovich and Kitscha, 1974; Kitscha and Yovanovich 1975; Yovanovich, 1986): 1 R g = D L k g,0 I g,p (W/K) (4.179) where L = D/2a is the relative contact size. The gap integral for point contacts proposed by Yovanovich and Kitscha (1974) and Yovanovich (1975) is defined as I g,p = L 1 2x tan −1 √ x 2 − 1 2δ/D + 2M/D dx (4.180) The local gap thickness δ is obtained from the relationship 2δ D = 1 − 1 − x L 2 1/2 + 1 πL 2 (2 − x 2 ) sin −1 1 x + x 2 − 1 − 1 L 2 (4.181) where x = r/a and 1 ≤ x ≤ L. The gap gas rarefaction parameter is defined as M = αβΛ (m) (4.182) where the gas parameters α, β, and Γ are as defined above. BOOKCOMP, Inc. — John Wiley & Sons / Page 328 / 2nd Proofs / Heat Transfer Handbook / Bejan 328 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [328], (68) Lines: 2803 to 2859 ——— 5.01927pt PgVar ——— Normal Page PgEnds: T E X [328], (68) Contacts in a Vacuum The joint resistance for a sphere–flat contact in a vacuum is (Yovanovich and Kitscha, 1974; Kitscha and Yovanovich, 1975) 1 R j = 1 R c + 1 R r (W/K) (4.183) The models proposed were verified by experiments conducted by Kitscha (1982). The test conditions were: sphere diameter D = 25.4 mm; vacuum pressure P g = 10 −6 torr; mean interface temperature range 316 ≤ T m ≤ 321 K; harmonic mean thermal conductivity of sphere-flat contact k s = 51.5 W/(m · K); emissivities of very smooth sphere and lapped flat (rms roughness is σ = 0.13 µm) 1 = 0.2 and 2 = 0.8, respec- tively; elastic properties of sphere and flat E 1 = E 2 = 206 GPa and ν 1 = ν 2 = 0.3. The dimensionless joint resistance is given by the relationship 1 R ∗ j = 1 R ∗ c + 1 R ∗ r (4.184) where R ∗ j = Dk s R j R ∗ c = Dk s R c = LR ∗ r = 1415 300 T m 3 (4.185) The model and vacuum data are compared for a load range in Table 4.16. The agree- ment between the joint resistance model and the data is excellent over the full range of tests. Effect of Gas Pressure on Joint Resistance According to the Model of Yovanovich and Kitscha (1974), the dimensionless joint resistance with a gas in the gapisgivenby 1 R ∗ j = 1 R ∗ c + 1 R ∗ r + 1 R ∗ g (4.186) TABLE 4.16 Dimensionless Load, Constriction, Radiative, and Joint Resistances FLT m R ∗ r R ∗ j R ∗ j (N) D/2a (K) Model Model Test 16.0 115.1 321 1155 104.7 107.0 22.2 103.2 321 1155 94.7 99.4 55.6 76.0 321 1155 71.3 70.9 87.2 65.4 320 1164 61.9 61.9 195.7 50.0 319 1177 48.0 48.8 266.9 45.1 318 1188 43.4 42.6 467.0 37.4 316 1211 36.4 35.4 Source: Kitscha (1982). BOOKCOMP, Inc. — John Wiley & Sons / Page 329 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 329 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [329], (69) Lines: 2859 to 2903 ——— -0.06282pt PgVar ——— Normal Page PgEnds: T E X [329], (69) where R ∗ g = Dk s R g = k s L 2 k g,∞ I g,p (4.187) The joint model for the sphere–flat contact is compared against data obtained for the following test conditions: sphere diameter D = 25.4 mm; load is 16 N; dimensionless load L = 115.1; mean interface temperature range 309 ≤ T m ≤ 321 K; harmonic mean thermal conductivity of sphere–flat contact k s = 51.5 W/m · K); emissivities of smooth sphere and lapped flat are 1 = 0.2 and 2 = 0.8, respectively. The load was fixed such that L = 115.1 for all tests, while the air pressure was varied from 400 mmHg down to a vacuum. The dimensionless resistances are given in Table 4.17. It can be seen that the dimensionless radiative resistance was relatively large with respect to the dimensionless gap and contact resistances. The dimension- less gap resistance values varied greatly with the gas pressure. The agreement be- tween the joint resistance model and the data is very good for all test points. 4.15.7 Joint Resistance of a Sphere and a Layered Substrate Figure 4.18 shows three joints: contact between a hemisphere and a substrate, contact between a hemisphere and a layer of finite thickness bounded to a substrate, and contact between a hemisphere and a very thick layer where t/a 1. In the general case, contact is between an elastic hemisphere of radius ρ and elastic properties: E 3 , ν 3 and an elastic layer of thickness t and elastic properties: E 1 , ν 1 , which is bonded to an elastic substrate of elastic properties: E 2 , ν 2 . The axial load is F . It is assumed that E 1 <E 2 for layers that are less rigid than the substrate. The contact radius a is much smaller than the dimensions of the hemisphere and the substrate. The solution for arbitrary layer thickness is complex because the contact radius depends on several parameters [i.e., a = f(F,ρ,t,E i , ν i ), i = 1, 2, 3]. The contact radius lies in the range a S ≤ a ≤ a L , where a S corresponds to the very thin TABLE 4.17 Effect of Gas Pressure on Gap and Joint Resistances for Air T m P g R ∗ g R ∗ r R ∗ j R ∗ j (K) (mmHg) Model Model Model Test 309 400.0 77.0 1295 44.5 46.8 310 100.0 87.6 1282 47.9 49.6 311 40.0 97.4 1270 50.7 52.3 316 4.4 138.3 1211 59.7 59.0 318 1.8 168.9 1188 64.7 65.7 321 0.6 231.3 1155 72.1 73.1 322 0.5 245.9 1144 73.4 74.3 325 0.2 352.8 1113 80.5 80.3 321 vacuum ∞ 1155 104.7 107.0 Source: Kitscha (1982). BOOKCOMP, Inc. — John Wiley & Sons / Page 330 / 2nd Proofs / Heat Transfer Handbook / Bejan 330 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [330], (70) Lines: 2903 to 2927 ——— -1.94696pt PgVar ——— Normal Page PgEnds: T E X [330], (70) Figure 4.18 Contact between a hemisphere and a layer on a substrate: (a) hemisphere and substrate; (b) hemisphere and layer of finite thickness; (c) hemisphere and very thick layer. (From Stevanovi ´ c et al., 2001.) layer limit, t/a → 0 (Fig. 4.18a) and a L corresponds to the very thick layer limit, t/a →∞(Fig. 4.18c). For the general case, a contact in a vacuum, and if there is negligible radiation heat transfer across the gap, the joint resistance is equal to the contact resistance, which is equal to the sum of the spreading–constriction resistances in the hemisphere and layer–substrate, respectively. The joint resistance is given by Fisher (1985), Fisher and Yovanovich (1989), and Stevanovi ´ c et al. (2001, 2002) R j = R c = 1 4k 3 a + ψ 12 4k 2 a (K/W) (4.188) where a is the contact radius. The first term on the right-hand side represents the con- striction resistance in the hemisphere, and ψ 12 is the spreading resistance parameter in the layer–substrate. The thermal conductivities of the hemisphere and the substrate appear in the first and second terms, respectively. The layer–substrate spreading resis- tance parameter depends on two dimensionless parameters: τ = t/a and κ = k 1 /k 2 . This parameter was presented above under spreading resistance in a layer on a half- space. To calculate the joint resistance the contact radius must be found. A special case arises when the rigidity of the layer is much smaller than the rigidity of the hemisphere and the layer. This corresponds to “soft” metallic layers such as indium, lead, and tin; or nonmetallic layers such as rubber or elastomers. In this case, since E 1 E 2 and E 1 E 3 , the hemisphere and substrate may be modeled as perfectly rigid while the layer deforms elastically. The dimensionless numerical values for a/a L obtained from the elastic contact model of Chen and Engel (1972) according to Stevanovi ´ c et al. (2001) are plotted in Fig. 4.19 for a wide range of relative layer thickness τ = t/a and for a range of values of the layer Young’s modulus E 1 . The contact model, which is represented by the correlation equation of the numerical values, is (Stevanovi ´ c et al., 2002) . Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 321 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [321],. 2nd Proofs / Heat Transfer Handbook / Bejan 322 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [322],. Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 323 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [323],