ME 383S Bryant February 17, 2006 CONTACT. Mechanical interaction of bodies via surfaces

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1 ME 383S Bryant February 17, 2006 CONTACT 1 Mechanical interaction of bodies via surfaces Surfaces must touch Forces press bodies together Size (area) of contact dependent on forces, materials, geometry, temperature, etc. Contact mechanics

2 ME 383S Bryant February 17, 2006 NATURE OF CONTACT 2 Rough surfaces contact Highest peaks on highest peaks Contact area = discrete islands Real contact area small fraction of apparent area Consequence contact stresses higher heating (friction) more intense electrical contacts: local constriction resistance

3 ME 383S Bryant February 17, 2006 CONTINENTAL ANALOGY OF CONTACT 3 Earth's surface rough: mountains & valleys lace South America on North America Contact: highest peaks against highest peaks - Andes/Appalachia - Highlands/Rockies Apparent contact area large, real contact area small Contact between bodies similar

4 ME 383S Bryant February 17, 2006 CONTACT MECHANICS FUNDAMENTALS 4 Fundamental Solutions: Boussinesq: 3D Elastic deformations from point force normal to semi-infinite space Flamant: 2D Elastic deformations from point force normal to semi-infinite space Contact between spheres elastic: Hertzian contact plastic: Indentation (Meyer) hardness overall quirks

5 ME 383S Bryant February 17, FLAMANT SOLUTION (2D) oint force, normal to semi-infinite elastic space 2D: plane strain or plain stress y (x, y) v u x Elastic deformations: (u, v) along (x, y) u = " & 2xy ) '($ "1)% " * 4#µ ( r 2 +, v = " % 2 2y ( &($ +1)log r " ) 4#µ ' r 2 * Stresses: " xx = # 2 % y $ r # y 3 ( & ) ' 2 r 4 * " = # 2 % y 3 ( yy & ) $ ' r 4 * " = # 2 % xy 2 ( xy & ) $ ' r 4 * µ: elastic shear modulus, ν: oisson's ratio Elastic modulus: E = 2(1 + ν)µ Dundars constant: κ = 3 4ν, plane strain, = (3 ν)/(1 + ν), plane stress r = x 2 + y 2, tan θ = x/y

6 ME 383S Bryant February 17, 2006 SOLUTION (2D) 6 oint force, tangential to semi-infinite elastic space 2D: plane strain or plain stress y (x, y) v u x Elastic deformations: (u, v) along (x, y) u = " % 2 2y ( &($ +1)log r + ) 4#µ ' r 2 *, v = & 2xy ) '(# $1)% + * 4"µ ( r 2 +, Stresses: " xx = 2 % # $ x r + xy 2 ( & ) ' 2 r 4 * " yy = # 2 % xy 2 ( & ) $ ' r 4 * " xy = 2 % # $ y r + y 3 ( & ) ' 2 r 4 * µ: elastic shear modulus, ν: oisson's ratio κ: Dundars constant = 3 4ν, plane strain; = (3 ν)/(1 + ν), plane stress r = x 2 + y 2, tan θ = x/y

7 ME 383S Bryant February 17, 2006 BOUSSINESQ SOLUTION (3D) 7 oint force on semi-infinite elastic space y z (x, y, z) w u x Elastic deformations: (u, v, w) along (x, y, z) u = v = 4πµ 4πµ { x z r 3 - (1-2 ν) x r (r + z) { y z r 3 - (1-2 ν) y r (r + z) } } w = 4πµ { z2 2(1 - ν) r 3 + r } µ: elastic shear modulus, ν: oisson's ratio r = x 2 + y 2 + z 2

8 ME 383S Bryant February 17, 2006 HERTZIAN CONTACT THEORY 8 Contact between elastic curved bodies Initial Contact E 1! 1 R 1 bodies (spheres) touch at origin Z o (x, y) R 2 E 2! 2 x curvatures R 1, R 2 elastic parameters (E 1,ν 1 ) (E 2,ν 2 ) initial contact initial separation (parabolas) Zo(x, y)= x2 + y 2 2 { 1 R R 2 }

9 ME 383S Bryant February 17, 2006 HERTZIAN CONTACT THEORY 9 Compressive normal load Induces contact pressures p = p(x, y) Z f (x, y) 2a p(x,y) x normal surface deformations w 1, w 2 wi = wi[p(x, y); E j, ν j, R j ] (from Boussinesq) final separation Z f (x, y) = Zo(x, y) - α + w 1 + w 2 contact diameter 2a, defines contact area normal approach α, amount centers of bodies come together

10 HERTZIAN CONTACT THEORY 10 roblem Statement p(x,y) Unknowns a, α (eigenvalues) Contact pressures p(x, y) (eigenfunction) hysics: static equilibrium Z f(x, y) 2a x Boundary Conditions over contact (x 2 + y 2 < a 2 ) Z f (x, y) = 0, p(x, y) 0 outside (x 2 + y 2 a 2 ) p = 0 = contact area p(x, y) dx dy

11 HERTZIAN CONTACT THEORY 11 Solution Sphere on Sphere (point contact): p(x, y) = po 1" x2 a " y 2 2 a 2 Z f(x, y) 2a p(x,y) x po = 3 2π a 2 a = { 3"(k } k 2 )R 1 R 3 2 4(R 1 + R 2 ) k i = 1- ν i 2 πe i α = { 9"2 2 R 1 + R 2 ( )( k 1 + k 2 ) 16R 1 R 2 } 1 2 3

12 Cylinder on Cylinder (line contact): 12 p(x) = p o 1" x 2 a = " 4(k + k )R R % $ ' # l(r 1 + R 2 ) & a 2, p o = 2 "al 1/ 2, k i = 1- ν i 2 πe i " = l (k + k ) 1+ ln 4l * / 01 ) + (k 1 + k 2 ) # &, 2 % (- 4 $ R 1 R 2 '. 34 l: length of contact along axis of cylinders

13 LASTIC CONTACT THEORY 13 Indentation (Meyer) hardness p!a Contact pressures p(x, y) approximately uniform Hardness pressure (indentation hardness) Η p δa Bodies in contact Load > elastic limit plastic deformations H 3 x Yield stress Use: estimate contact area, given H and

14 14 OVERALL CONTACT MODEL α 1 p(x,y) α 2 2a Spheres Increasing normal load 0 < e ; Elastic (Hertzian) contact model α = α 1 +α 2 = { 9π2 2 (k 1 +k 2 ) 2 (R 1 +R 2 ) 16R 1 R 2 } 1 3 e ; Elastic-lastic contact model α = α 1 +α 2 > { 9π2 e 2 (k 1 +k 2 ) 2 (R 1 +R 2 ) 16R 1 R 2 } 1 3 Similar formulations, tangential loads & deformations >> e, Meyer Hardness problem

15 15 Hardness (Indentation) Test Brinell hardness H B Hard steel ball (diameter D, load F) indent for 30 sec. measure permanent (pla stic) set H B = F/(! D t) [N/m 2 ] (units of stress) t " {D - (D 2 - d 2 ) 1/2 }/2 Other hardness tests Vickers: Diamond point indentation Rockwell: measured like Brinell or Vickers scratch test t F Hard St eel Ball (dia mete r D) s pecim en d

16 16 Brinell Hardness For steels S = S(H) Ultimate strength S u = K B H B Yield Strength S y! 1.05S u - 30ksi

17 Micro Hardness Measurements 17 Micro hardness tester: Indents specimen Use on thin films, e.g. hard drive coatings

18 CONTACT QUIRKS 18 Nonlinear contact stiffness = (α) = C α 3/2, C = 4 3π(k 1 +k 2 ) R 1 R 2 R 1 +R 2 nonlinear contact vibrations Tangential motions: slip/stick with friction high pitched "squeal" / fingernail on blackboard

19 ROUGH CONTACT MODELS 19 Greenwood & Williamson d Rough surfaces contact: current separation = d d Asperities contact interference α = (z 1 + z 2 ) - d z 1, z 2 surface heights, upper & lower

20 20 Model asperities as spheres r 1 h do z2 z1 r 2 Relate contact quantities to surface heights z = z 1 + z 2 (random variable) Expected values Macroscopic Contact arameters E[H(z)] = N " # H(z) F(z)dz d N: total number asperities H(z): physical quantity, dependent on heights z Lower limit: heights z d for asperities to touch Microscopic understanding Important practical engineering parameters

21 21 Contact force (elastic) on ith asperity Asperity force: i (z) = C α 3/2 = C (z - d) 3/2 (Hertz) Total force: H(z) = i (z) " = E[ i (z)] = N # i (z) F( z)dz Real contact area Asperity area: H(z) = A i (z) = π a 2 d From Hertz: a = C 1 " 1/ 2 = C 1 ( z " d) 1/ 2 Contact conductance Asperity conductance: G i (z) = ρ/2a (Holm)