Contact

Published on May 2016 | Categories: Documents | Downloads: 75 | Comments: 0 | Views: 1451
of x
Download PDF   Embed   Report

Comments

Content

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 u y v

(x, y)

• Elastic deformations: (u, v) along (x, y) P & 2xy ) P % 2y 2 ( u=" '($ "1)% " 2 * , v = " &($ + 1)log r " 2 ) 4#µ ' r * 4#µ ( r + • Stresses:

!

2P % y y 3 ( 2P % y 3 ( 2P % xy 2 ( " xx = # & 2 # 4 ) ’ " yy = # & 4 ) ’ " xy = # & 4 ) $ 'r r *! $ 'r * $ 'r *

!

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 = x 2 + y , tan θ = x/y

2

!

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) y v u x

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

P % 2y 2 ( P & 2xy ) u=" &($ + 1)log r + 2 ) , v = '(# $1)% + 2 * , 4#µ ' r * 4"µ ( r +
• Stresses: 2P % x xy 2 ( 2P % y y3 ( 2P % xy 2 ( " xx = &$ + ) " = # & 4 ) ’ " xy = &$ + ) # ' r 2 r 4 * ’! yy # ' r2 r4 * $ 'r * µ: elastic shear modulus, ν: Poisson's ratio κ: Dundars constant ! = 3 – 4ν, plane strain; = (3 – ! (1 + ν), plane stress ν)/

!

!

r = x 2 + y , tan θ = x/y

2

!

ME 383S

Bryant

February 17, 2006 7

BOUSSINESQ SOLUTION (3D)
• Point force P on semi-infinite elastic space P
y z w x u

(x, y, z)

• Elastic deformations: (u, v, w) along (x, y, z) P u = 4πµ P v = 4πµ P w = 4πµ xz { r3 yz { r3 z2 { r3 x - (1 - 2 ν) r (r + z) y - (1 - 2 ν) r (r + z)
+

} }

2(1 - ν) r

}

µ: elastic shear modulus, ν: Poisson's ratio r = x 2+ y 2 + z 2

ME 383S

Bryant

February 17, 2006 8

HERTZIAN CONTACT THEORY
• Contact between elastic curved bodies • Initial Contact
E1 !1 R1 Z o (x, y) R2 E2 !2 initial contact x

• bodies (spheres) touch at origin • curvatures R1, R2 • elastic parameters (E1,ν1) (E2,ν2) • initial separation (parabolas) Zo(x, y)=
x2 + y2 2

{R +R }
1 2

1

1

ME 383S

Bryant

February 17, 2006

9

HERTZIAN CONTACT THEORY • Compressive normal load P • Induces contact pressures p = p(x, y) • normal surface deformations w1, w2 wi = wi[p(x, y); Ej, νj, Rj] (from Boussinesq) • final separation
x 2a

P

p(x,y)
Z f (x, y)

Zf(x, y) = Zo(x, y) - α + w1 + w2 • contact diameter 2a, defines contact area • normal approach α, amount centers of bodies come together

P

10

HERTZIAN CONTACT THEORY Problem Statement

P
p(x,y)

Unknowns • a, α (eigenvalues) • Contact pressures p(x, y) (eigenfunction) Physics: static equilibrium
x

Z f(x, y) 2a

Boundary Conditions • over contact (x2+ y2 < a2) Zf(x, y) = 0, p(x, y) ≥ 0 • outside (x2+ y2 ≥ a2) p=0 • P=
⌠p(x, ⌡ contact area

P

y) dx dy

11

HERTZIAN CONTACT THEORY Solution Sphere on Sphere (point contact):
P
p(x,y)
x2 y 2 " a 2 a2

• p(x, y) = po 3P po = 2π a2
x 2a

1"

Z f (x, y)

• a=

{

3"P(k1 + k2 )R1 R2 4(R1 + R2 )

}

1 3

P

1- νi2 ki = πE i •α=

{

9" P ( R1 + R2 ) ( k1 + k2 ) 16R1 R2

2

2

2

}

1 3

!

!

12

Cylinder on Cylinder (line contact):

x2 • p(x) = po 1" a 2 ,
" 4P(k1 + k 2 )R1R2 % • a = $ l(R + R ) ' , # & 1 2 !
1/ 2

po =

2P "al

!

1- νi2 ki = πE i

!


/ ) 4l 3 #1 P 1 &,2 " = (k1 + k 2 )11+ ln* % + (-4 l 1 + (k1 + k 2 )P $ R1 R2 '.4 0 3
l: length of contact along axis of cylinders



!

13 •

PLASTIC CONTACT THEORY

• Indentation (Meyer) hardness •
P p

Contact pressures p(x, y) approximately uniform Hardness pressure (indentation hardness) P Η ≡ p ≈ δA H ≈ 3 x Yield stress


!A P

Bodies in contact Load P > elastic limit ⇒ plastic deformations •

Use: estimate contact area, given H and P

14

OVERALL CONTACT MODEL
P

α

1

P p(x,y)

α
P

2

2a P

• Spheres • Increasing normal load P 0 ≤ P < Pe ; α = α1+α2 = P ≥ Pe ; α = α1+α2 > Elastic (Hertzian) contact model
1 3

{ {

9π2P2(k1+k2)2(R1+R2) 16R1R2

}

Elastic-Plastic contact model 9π2Pe2(k1+k2)2(R1+R2) 16R1R2

}

1 3

• 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
F
Hard St eel Ball (d ia mete r D)

– HB = F/(! D t) [N/m 2 ] (units of
stress) – t " {D - (D 2 - d 2 )1/2 }/2
t

s pecim en

d

• Other hardness tests
– Vickers: Diamond point indentation – Rockwell: measured like or Vickers – scratch test Brinell

16

Brinell Hardness
• For steels S = S(H) • Ultimate strength
Su = KB HB

• Yield Strength
Sy ! 1.05S u - 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=Cα
3/2

4 , C = 3π(k +k ) 1 2

R1R2 R1+R2

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

19

ROUGH CONTACT MODELS
• Greenwood & Williamson
P

d

P

• Rough surfaces contact: current separation = d

P

d

P
• Asperities contact interference α = (z1 + z2) - d

z1 , z2 surface heights, upper & lower

20

• Model asperities as spheres
r1

h

do

z1 z2 r2

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

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: Pi(z) = C α Total force: H(z) = Pi(z)
"

3/2

= C (z - d)

3/2

(Hertz)

P = E[Pi(z)] = N

# P (z)F( z)dz
i d

• 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)

Sponsor Documents

Or use your account on DocShare.tips

Hide

Forgot your password?

Or register your new account on DocShare.tips

Hide

Lost your password? Please enter your email address. You will receive a link to create a new password.

Back to log-in

Close