JKR Point loading of an elastic half-space 2 3 Pressure applied to a circular region Boussinesq, n =

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JKR 17 9 15 1 Point loading of an elastic half-space Pressure applied to a circular region 4.1 Boussinesq, n = 1.............................. 4. Hertz, n = 1.................................. 6 4 Hertz : (normal contact of elastic solids) 7 5 (JKR ) 1 5.1............................................. 1 5...................................... 17 5.................................. 19 1

1 R 1, R E 1, E ν 1, ν a F δ Hertz H. Hertz (Hertz, 1881) Hertz (adhesion γ ) JKR Johnson, Kendall, and Roberts (1971) JKR JKR Johnson (1987) Hertz JKR JKR Johnson et al.(1971) Chokshi et al.(199) Point loading of an elastic half-space 1: Johnson(1987) Fig.. ( E ν) xy z O P 1 Hooke Johnson (1987)

ρ = x + y + z r = x + y σ x = σ y = [ P (1 ν) π r [ (1 ν) P π z r {( 1 z ) x y ρ r ) y x {( 1 z ρ r + zy ρ + zx ρ } ] zx ρ 5 } ] zy ρ 5 (.1) (.) σ z = P (.) π ρ [ 5 {( P (1 ν) τ xy = 1 z ) xy π r ρ r xyz } xyz ] (.4) ρ ρ 5 τ xz = P xz (.5) π ρ 5 τ yz = P yz (.6) π ρ 5 { } (1 + ν)p xz x u x = (1 ν) (.7) πe ρ ρ(ρ + z) { } (1 + ν)p yz y u y = (1 ν) (.8) πe ρ ρ(ρ + z) u z = (1 + ν)p πe { z (1 ν) + ρ ρ z (r, θ, z) { ( P 1 σ r = (1 ν) π r z ) } zr ρr ρ 5 σ θ = P ( 1 π (1 ν) r z ρr z ) ρ } (.9) (.1) (.11) σ z = P z (.1) π ρ 5 τ rz = P rz (.1) π ρ 5 { } (1 + ν)p rz z u r = (1 ν)ρ (.14) πe ρ ρr u z = (1 + ν)p πe { z (1 ν) + ρ ρ (z = ) ū r, ū z (.14)(.15) } (.15) (1 + ν)(1 ν) ū r = πe ū z = 1 ν πe P r P r (.16) (.17)

Pressure applied to a circular region C(ξ, η) p C B(x, y) z (.17) ū z (x, y) = 1 ν πe psdsdφ s = 1 ν pdsdφ (.1) πe s B C s (ξ x) + (η y) φ B C : Johnson(1987) Fig..1 S p(x, y) B(x, y) z ū z (x, y) = 1 ν πe S p(s, φ)dsdφ (.) S a ( ) n p(r) = p 1 r (.) a r = x + y p n n = ± 1.1 Boussinesq, n = 1 ( ) 1 p = p 1 r a a S ū z r Uniform normal displacement Boussinesq (.) ū z (.4) 4

: Johnson(1987) Fig..5 (a) B(r, ) ū z (S r < a) C t t = r + s rs cos(π φ) = r + s + rs cos φ (.5) C p(s, φ) p(s, φ) = p a ( a t ) 1 = p a ( a r rs cos φ s ) 1 = p a ( α βs s ) 1 (.6) α a r, β r cos φ p(s, φ) (.) ū z (r) = 1 ν πe p a π dφ s1 ds ( α βs s ) 1 s 1 (a) B B t = a s α βs s = s 1 = β + β + α (.8) dx ax + bx + c = 1 a arcsin ax + b (a < ) (.9) b 4ac (.7) s s1 ds ( α βs s ) 1 = arcsin s β β + α = π arctan ( β α ) s 1 β = arcsin( 1) + arcsin β + α (.7) (.1) 5

β(φ) = r cos φ β(φ + π) = β(φ). arctan ( ) ( β(φ+π) α = arctan β(φ) ) α arctan ( ) β(φ) α φ [, π] (.7) π { ( )} π β(φ) [ π ] π dφ arctan = α φ = π (.11) (.7) ū z ū z r ū z = 1 ν πe p a π = 1 ν E πap (.1) S F F = a ( ) 1 p 1 r πrdr = πa p a (.1) (.1)(.1) p ū z F ū z = 1 ν ae F (.14) B S (b) p(s, φ) = p a ( α + βs s ) 1 (.15) ū z (r > a) ū z (r) = 1 ν πe p a = 1 ν πe p a φ s dφ φ 1 arcsin( a r ) s 1 ds ( α + βs s ) 1 arcsin( a r ) πdφ = (1 ν ) E ( a ) ap arcsin r (.16) r = a (.1) Boussinesq Boussinesq. Hertz, n = 1 ( ) 1 p = p 1 r a (.17) Hertz Boussinesq B(r, ) z ū z ū z (r) = 1 ν πe p a π dφ s1 ds ( α βs s ) 1 (.18) 6

ax + bx + cdx = ax + b 4a ax + bx + c b 4ac 8a dx ax + bx + c (.19) (.9) (.18) s s1 ds ( [ α βs s ) ] 1 s1 s + β = α βs s + β + α { π ( β α)} arctan = 1 αβ + 1 ( α + β ) { π ( β α)} arctan (.) β(φ + π) = β(φ) αβ arctan ( β α) φ [, π] (.18) π π 4 ( α + β ) dφ = π 4 π ( a r + r cos φ ) dφ = π 4 ( a r ) (.1) ū z ū z (r) = 1 ν E πp 4a ( a r ) (.) S F p m F = a ( ) 1 p 1 r πrdr = a πa p (.) p m F πa = p (.4) 4 Hertz : (normal contact of elastic solids) Hertz R 1, R δ 1, δ δ = δ 1 + δ a F δ, a, F xy z 4 Hertz (non-conforming) a R a R a R, a l. l 7

4: Johnson(1987) Fig.4. 8

r = x + y r R 1, z 1 = 1 R 1 r, z = 1 R r (4.1) r z ū z1, ū z R 1 R = 1 R 1 + 1 R ū z1 + ū z = δ (z 1 + z ) (4.) ū z1 + ū z = δ 1 R r (4.) ū z1, ū z (4.) ū z1, ū z Hertz p(r) = p 1 r a (4.4) Landau and Lifshitz (.) (4.) ū z1 + ū z = 1 ν 1 + 1 ν πp E 1 4a E ( a r ) = 1 E πp 4a ( a r ) (4.5) (?) 1 E 1 ν 1 E 1 + 1 ν E (4.) : πp 4aE ( a r ) = δ 1 R r (4.6) r Hertz (4.4) r r πp 4aE = 1 R πap E = δ (4.8) F (.) p F a (4.7)(4.8) p = (4.7) F πa (4.9) ( ) 1 F = 4E R R a a = 4E F (4.1) 9

( δ = a R = p = 9F 16E R ) 1 F πa = ae πr = F = 4E ( 6E F π R (4.1)(4.11)(4.1) Hertz ) 1 Rδ (4.11) (4.1) (4.4) Hertz (4.) p = p (1 r a ) 1 (4.1) Boussiesq p > p < Hertz Boussiesq p = p 1 r a = F 1 r πa a = ae 1 r (4.14) πr a 5 σ r 1

5: Johnson(1987) Fig.4. 11

5 (JKR ) Hertz 5.1 ū z1 + ū z = δ 1 R r (5.1) Hertz Boussinesq ( ) 1 p = p H 1 r a + pb (1 r p B < Boussinesq ū z (.1) (5.1) a ) 1 (5.) πp H 4aE ( a r ) + πap B E = δ 1 R r (5.) Hertz r p H = ae πr (5.4) δ = πa E (p H + p B ) (5.5) F a ( ) F = pπrdr = p H + p B πa (5.6) p H (a) p B (a) p B U E a = a 1 δ = δ 1. Hertz a = a 1 δ = δ 1 U 1. a = a 1 δ = δ Boussinesq (p B < ) Boussinesq a = a 1 δ U 1

1. Hertz U 1 = = δ1 F = πa p H = 4E R a (5.7) Fdδ = 8 E 15 R a 1 δ = a R a1 4E a R a R da = 8 E 15 R a5 1 ( πr E p H1) = 15 (5.8) π a 1 E p H1 (5.9) p H1 p H (a 1 ) = a 1E πr. (.1)(.1) F = F 1 + πa 1 p B1 (5.1) δ = δ 1 + πa 1 E p B1 (5.11) F 1 1. δ = δ 1 p B1 p B (a 1 ) (5.1)(5.11) p B1 F 1 = πa 1 p H1 (5.1) F = a 1 E (δ δ 1 ) + F 1 (5.1) (5.1) U = δ δ 1 Fdδ = F F = F 1 + πa 1 p B1 = U = π a 1 E F 1 F df a 1 E = F F 1 4a 1 E (5.14) ( ) p H1 + p B1 πa 1 (5.15) ( ) p H1 p B1 + p B1 (5.16) U E a 1, p H1, p B1 a, p H, p B U E = U 1 + U = π a ( E 15 p H + ) p H p B + p B (5.17) δ a U E ( ) UE a δ (5.18) 1

δ δ (5.5) p B = E πa δ p H = E πa δ ae πr U E (5.17) (5.4) U E = π a ( ) 1 ae E E ( δ E ) 5 πr π R + δ πa (5.19) (5.) ( ) UE a δ = π a E ( ae πr ) E δ π R + ( E ) δ πa = π a ( ) ae E πr E δ πa = π a E p B (5.1) U T (5.1)(5.) ( UT a ( ) U T a = δ π a p B (< ) U S = γπa (5.) U T = U E + U S (5.) ) δ = π a E p B 4γπa (5.4) E p B = 4γπa (5.5) p B = 4γE πa (5.6) p B p H (5.4) (5.6) F a F = ( ) p H + p B πa = 4E R a πa 4γE πa = 4E R a 16πγE a (5.7) a x = a x a = R { } 4E F + 6πγR + 1πγRF + (6πγR) (5.8) ( 6) γ = Hertz 14

6: Johnson(1987) Fig.5.8 F = a = a (5.8) ( 9πγR a = E ) 1 (5.9) a F a < a F F ( ) F F = F c, (F c > ) F a = F c 1 a = a s F c = πγr (5.) ( ) 1 9πγR a s = = 4 1 4E a =.6a (5.1) F c F c R γ a a s δ δ < a ( 9) a < a s F < F c F c a a s F c 1 (5.8) 1πγRF + (6πγR) F πγr 15

δ (5.5) p B (5.6) p H (5.4) πa δ = E (p H + p B ) = a 4πγa R E = a 4πγR R 1 E a = a R 1 4πγR E 1 a ( a ) a (5.) a (5.9) δ = a R 1 ( a a ) (5.) a = a δ δ δ = a R (5.4) 7 7: Johnson(1987) Fig.5.9 16

5. a, F, δ a, F c, δ (5.7)(5.8)(5.) (5.8) a (5.9) ( ) a 1 = 1πγR a { } F + 6πγR + 1πγRF + (6πγR) = 1 4 F πγr + + 4F πγr + 4 (5.5) F F c ( ) a = 1 F + + 1 + F 4 F c F c a ( F a = 4 F c (5.) (5.4) ( δ a = δ a a ) ( a a ) ) ( ) 1 a a (5.6) (5.7) (5.8) δ δ a a ( 9) δ < a a a δ δ δ c (δ c > ) δ = δ c (5.8) d(δ/δ ) d(a/a ) = a a = ( 1 6 ) (5.9) a c ( ) 1 a c = a =.a (< a s =.6a ) (5.4) 6 a = a s a = a c δ (5.8) a c δ δ = δ c = ( ) 4 ( ) 1 1 1 = 6 6 ( ) 1 1 δ =.85δ = 1 6 δ c δ(a) (5.) ( ) δ = 6 1 a δ c 4 ( a a a = a c (or δ = δ c ) ( ) F 1 = 4 F c 6 ( ) 1 1 =.85 (5.41) 6 a ( ) 1 1 a 6 R ) 1 (5.4) (5.4) ( ) 1 6 = 5 =.556 (5.44) 9 a, F, δ 8, 9 17

8: JKR F F c a a 9: JKR δ F F c a a δ δ δ δ c 18

5. U S U E U T F c δ c U S U T U T F c δ c = U S F c δ c = γπa πγr 1 ( 1 6)1 a R ) ( δ d ( 1 F Fdδ = F c δ c F c ( ) ( ) a a = 4 a a 6 1 ( ) 5 = 6 1 8 a 8 ( a 5 a a ( ) 4 = 6 1 a (5.45) a δ c ) 4 ( a a ) ) 7 ( ) a + a ( a a ) 1 ( ) a d a (5.46) U E U E U T = U S F c δ c F c δ c F c δ c ( = 6 1 8 a 5 a ) 5 8 ( ) 7 ( ) a 4 a + a a (5.47) U E F c δ c, U S F c δ c, U T F c δ c δ δ c 1 (δ = δ c ) U T > U T U T F c δ c = U T δ δc =1 F c δ c = 4 =.9 (5.48) a a =( 1 6) 45 U K U K > 4 45 F cδ c =.9F c δ c (5.49) 4 45 F cδ c a = ( ε ε = ) a =.76a U K 4 45 F cδ c = 1 U K 4 45 F cδ c U K (5.5) 1 ( ) 4 δ = δ = δc = 1.1δ c (5.51) 19

U T U T = 4 5 6 1 Fc δ c = 1.45F c δ c (5.5) 4 5 6 1 Fc δ c + 4 45 F cδ c = (1.45 +.9)F c δ c = 1.54F c δ c (5.5) F c δ c F c δ c = πγr 1 ( ) 1 1 6 ( 9πγR E ) ( ) 4 1 1 R = 9 5 γ 5 R 4 π E = 4.7 γ 5 R 4 E (5.54)

1: U E F c δ c U S F c δ c U T F c δ c δ δ c 1

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