Contact
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)
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)
