ME 383S Bryant February 17, 2006 1

CONTACT • 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 2

NATURE OF CONTACT 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 3

CONTINENTAL ANALOGY OF CONTACT

• Earth's surface rough: mountains & valleys

• Place 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 4

CONTACT MECHANICS FUNDAMENTALS

• 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)

• Point force P, normal to semi-infinite elastic space • 2D: plane strain or plain stress

P

x

y

(x, y)

v

u

• Elastic deformations: (u, v) along (x, y)

!

u = "P

4#µ($ "1)% "

2xy

r2

& ' (

) * + ,

!

v = "P

4#µ($ +1)log r "

2y 2

r2

% & '

( ) *

• Stresses:

!

" xx = #2P

$

y

r2#y3

r4

% & '

( ) *

’

!

" yy = #2P

$

y3

r4

% & '

( ) *

’

!

" xy = #2P

$

xy2

r4

% & '

( ) *

µ: elastic shear modulus, ν: Poisson's ratio Elastic modulus: E = 2(1 + ν)µ Dundars constant: κ = 3 – 4ν, plane strain, = (3 – ν)/(1 + ν), plane stress

!

r = x2

+ y2

, tan θ = x/y

ME 383S Bryant February 17, 2006 6

SOLUTION (2D) • Point force P, tangential to semi-infinite elastic space • 2D: plane strain or plain stress

P

x

y

(x, y)

v

u

• Elastic deformations: (u, v) along (x, y)

!

u = "P

4#µ($ +1)log r +

2y 2

r2

% & '

( ) * ,

!

v =P

4"µ(# $1)% +

2xy

r2

& ' (

) * + ,

• Stresses:

!

" xx =2P

#$x

r2

+xy

2

r4

% & '

( ) * ’

!

" yy = #2P

$

xy2

r4

% & '

( ) *

’

!

" xy =2P

#$y

r2

+y3

r4

% & '

( ) *

µ: elastic shear modulus, ν: Poisson's ratio κ: Dundars constant = 3 – 4ν, plane strain; = (3 – ν)/(1 + ν), plane stress

!

r = x2

+ y2

, tan θ = x/y

ME 383S Bryant February 17, 2006 7

BOUSSINESQ SOLUTION (3D) • Point force P on semi-infinite elastic space

P

x

z

y(x, y, z)

w

u

• Elastic deformations: (u, v, w) along (x, y, z)

u = P4πµ { x z

r3 - (1 - 2 ν) xr (r + z) }

v = P4πµ { y z

r3 - (1 - 2 ν) yr (r + z) }

w = P4πµ { z

2

r3 + 2(1 - ν)

r } µ: elastic shear modulus, ν: Poisson's ratio r = x2+ y2 + z2

ME 383S Bryant February 17, 2006 8

HERTZIAN CONTACT THEORY • Contact between elastic curved bodies • Initial Contact

• bodies (spheres) touch at

origin • curvatures R1, R2 • elastic parameters (E1,ν1) (E2,ν2) • initial separation

(parabolas)

Zo(x, y)= x2 + y2

2{ 1R1

+ 1

R2

}

E1 !1

E2 !2

xZo(x, y)

R2

R1

initial contact

ME 383S Bryant February 17, 2006 9

HERTZIAN CONTACT THEORY • Compressive normal load P • Induces contact pressures p = p(x, y)

2a

P

P

Zf(x, y)x

p(x,y)

• normal surface

deformations w1, w2 wi = wi[p(x, y); Ej, νj, Rj] (from Boussinesq) • final separation Zf(x, y) = Zo(x, y) - α + w1 + w2 • contact diameter 2a, defines contact area • normal approach α, amount centers of bodies come together

10

HERTZIAN CONTACT THEORY Problem Statement

2a

P

P

Z f(x, y)x

p(x,y)

Unknowns • a, α (eigenvalues) • Contact pressures p(x, y) (eigenfunction) Physics: static equilibrium Boundary Conditions • over contact (x2+ y2 < a2) Zf(x, y) = 0, p(x, y) ≥ 0 • outside (x2+ y2 ≥ a2) p = 0 • P = ⌡⌠

contact areap(x, y) dx dy

11

HERTZIAN CONTACT THEORY Solution

2a

P

P

Z f(x, y)x

p(x,y)

Sphere on Sphere (point contact):

• p(x, y) = po

!

1"x2

a2"y 2

a2

po = 3P2π a2

• a = {

!

3"P(k1

+ k2)R1R2

4(R1

+ R2)

}13

ki = 1- νi2

πEi

• α = {

!

9"2P2R1

+ R2

( ) k1

+ k2

( )16R

1R2

2}13

12

Cylinder on Cylinder (line contact):

•

!

p(x) = po 1"x2

a2 ,

!

po =2P

"al

•

!

a =4P(k

1+ k

2)R

1R2

l(R1+ R

2)

"

# $

%

& '

1/ 2

, ki = 1- νi2

πEi

•

!

" =P

l(k1 + k2) 1+ ln

4l3

(k1 + k2)P

1

R1

+1

R2

#

$ %

&

' (

) * +

, - .

/

0 1 1

2

3 4 4

• l: length of contact along axis of cylinders

13

• PLASTIC CONTACT THEORY • Indentation (Meyer) hardness

Bodies in contact Load P > elastic limit ⇒ plastic deformations

• Contact pressures p(x, y)

approximately uniform • Hardness pressure

(indentation hardness)

Η ≡ p ≈ PδA

H ≈ 3 x Yield stress • Use: estimate contact

area, given H and P

!A

P

P

p

14

OVERALL CONTACT MODEL

2a

P

P

p(x,y)

P

P

α2

α1

• Spheres • Increasing normal load P 0 ≤ P < Pe ; Elastic (Hertzian) contact model

α = α1+α2 = {9π2P2(k1+k2)2(R1+R2)16R1R2

}13

P ≥ Pe ; Elastic-Plastic contact model

α = α1+α2 > {9π2Pe2(k1+k2)2(R1+R2)16R1R2

}13

• Similar formulations, tangential loads & deformations P >> Pe, 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

– HB = F/(! D t) [N/m2 ] (units of

stress)

– t " {D - (D 2 - d2 )1/2 }/2

• Other hardness tests– Vickers: Diamond point indentation

– Rockwell: measured like Brinell or Vickers

– scratch test

Hard St eel Ball

(d iamete r D)

s pecim en

F

dt

16

Brinell Hardness

• For steels S = S(H)

• Ultimate strength

Su = KB HB

• Yield Strength

Sy ! 1.05Su - 30ksi

17

Micro Hardness Measurements

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

18

CONTACT QUIRKS • Nonlinear contact stiffness P = P(α)

P = C α3/2 , C = 4

3π(k1+k2)R1R2

R1+R2

• ⇒ nonlinear contact vibrations • Tangential motions: slip/stick with friction high pitched "squeal" / fingernail on blackboard

19

ROUGH CONTACT MODELS • Greenwood & Williamson

d

P

P

• Rough surfaces contact: current separation = d

d

P

P • Asperities contact interference α = (z1 + z2) - d z1 , z2 surface heights, upper & lower

20

• Model asperities as spheres

doz1

z2

r1

r2

h

• Relate contact quantities to surface heights z = z1 + z2 (random variable) • Expected values ⇒ Macroscopic Contact Parameters

E[H(z)] = N

!

H (z)d

"

# F(z)dz 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: Pi(z) = C α3/2 = C (z - d)3/2 (Hertz)

Total force: H(z) = Pi(z)

P = E[Pi(z)] = N

!

Pi(z)

d

"

# F( z)dz

• Real contact area Asperity area: H(z) = Ai(z) = π a2

From Hertz: a = C1

!

"1/ 2

!

= C1

!

z " d( )1/ 2

!

• Contact conductance

Asperity conductance: Gi(z) = ρ/2a (Holm)