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
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
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
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
ME 383S Bryant February 17, 2006 5 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
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
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
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 1 + 1 R 2 }
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
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
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 } 1 1 + 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
Cylinder on Cylinder (line contact): 12 p(x) = p o 1" x 2 a = " 4(k + k )R R % 1 2 1 2 $ ' # 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 3 1 2 1 * / 01 ) + (k 1 + k 2 ) # 1 + 1 &, 2 % (- 4 $ R 1 R 2 '. 34 l: length of contact along axis of cylinders
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 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 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 Brinell Hardness For steels S = S(H) Ultimate strength S u = K B H B Yield Strength S y! 1.05S u - 30ksi
Micro Hardness Measurements 17 Micro hardness tester: Indents specimen Use on thin films, e.g. hard drive coatings
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
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 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 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)