contact stresses and deformations

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ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-1

Contact Stresses and Deformations

ME EN 7960 – Precision Machine DesignTopic 7


Curved Surfaces in Contact

• The theoretical contact area of two spheres is a point (= 0-dimensional)

• The theoretical contact area of two parallel cylinders is a line (= 1-dimensional)→As a result, the pressure between two curved surfaces should be

infinite→The infinite pressure at the contact should cause immediate

yielding of both surfaces

• In reality, a small contact area is being created through elastic deformation, thereby limiting the stresses considerably

• These contact stresses are called Hertz contact stresses

2


Curved Surfaces in Contact – Examples

Rotary ball bearing

Rotary roller bearing

Linear bearings (ball and rollers)


Curved Surfaces in Contact – Examples (contd.)

Ball screw Gears

3


Spheres in Contact

3

21

2

22

1

21

114

113

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎦

⎤⎢⎣

⎡ −+

−

=

RR

EEF

a

νν

The radius of the contact area is given by:

Where E1 and E2 are the moduli of elasticity for spheres 1 and 2 and ν1and ν2 are the Poisson’s ratios, respectivelyThe maximum contact pressure at the center of the circular contact area is:

2max 23

aFp

π=

R1

R2

E2, 2

E1, 1

F

F

y

z

xSphere 2

Sphere 1

2a

pmax

Circular contact area, resulting in a semi-elliptic

pressure distribution

Z = 0:

x

y

pmax 2a


Spheres in Contact (contd.)

• The equations for two spheres in contact are also valid for:– Sphere on a flat plate (a flat plate is a sphere with an infinitely

large radius)– Sphere in a spherical groove (a spherical groove is a sphere with

a negative radius)

R1E1, 1

y2a

x

F

z

E2, 2Flat plate (R2 = ∞)

Z = 0:

x

y

pmax 2a

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Spheres in Contact – Principal Stresses

The principal stresses σ1, σ2, and σ3 are generated on the z-axis:

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+−====

12

1arctan11

2

2max21

azz

aazpyx νσσσσ

1

2

2

max3 1−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−==

azpzσσ

The principal shear stresses are found as:

231

max21σστττ −

=== 03 =τ


Spheres in Contact – Vertical Stress Distribution at Center of Contact Area

Plot shows material with Poisson’s ratio ν = 0.3

0.5a a 1.5a 2a 2.5a 3a

σz

σX, σy

τmax

0

σ, τ

z

Von Mises

0

0.2

0.4

0.6

0.8

1

Depth below contact area

Rat

i oof

stre

ssto

p max

• The maximum shear and Von Mises stress are reached below the contact area

• This causes pitting where little pieces of material break out of the surface

5


Cylinders in ContactThe half-width b of the rectangular contact area of two parallel cylinders is found as:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎦

⎤⎢⎣

⎡ −+

−

=

21

2

22

1

21

11

114

RRL

EEF

bπ

νν

Where E1 and E2 are the moduli of elasticity for cylinders 1 and 2 and ν1and ν2 are the Poisson’s ratios, respectively. L is the length of contact.The maximum contact pressure along the center line of the rectangular contact area is:

bLFp

π2

max =

x y

z

F

F

2b

E2, ν2

E1, ν1

R2

R1 pmaxL

Rectangular contact area with semi-elliptical pressure distribution


Cylinders in Contact (contd.)• The equations for two cylinders in contact are also valid

for:– Cylinder on a flat plate (a flat plate is a cylinder with an infinitely

large radius)– Cylinder in a cylindrical groove (a cylindrical groove is a cylinder

with a negative radius)

x y

z

F

F

2b

E2, ν2

E1, ν1

R1

pmaxL

Rectangular contact area with semi-

elliptical pressure distribution

Flat plate (R2 = ? ) x

y

zF

F

2b

E2, ν2

E1, ν1

R1 pmax L

Rectangular contact area with semi-elliptical

pressure distribution

Cylindrical groove (R2 = -Rg)

Rg

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Cylinders in Contact – Principal Stresses

The principal stresses σ1, σ2, and σ3 are generated on the z-axis:

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+−==

bz

bzpx 12 2

2

max1 νσσ

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+−−==

−

bz

bz

bzpy 2112 2

21

2

2

max2 σσ

1

2

2

max3 1−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−==

bzpzσσ

2,

2,

221

331

232

1σστσστσστ −

=−

=−

=


Cylinders in Contact – Vertical Stress Distribution along Centerline of Contact Area

• The maximum shear and Von Mises stress are reached below the contact area

• This causes pitting where little pieces of material break out of the surface

Plot shows material with Poisson’s ratio ν = 0.3

0Depth below contact area

Rat

ioof

s tre

ssto

p max

0.5b b 1.5b 2b 2.5b 3b

σx

σy

σz

τ1

Von Mises

z

σ, τ

0

0.2

0.4

0.6

0.8

1

7


Sphere vs. Cylinder – Von Mises Stress

• The Von Mises stress does not increase linearly with the contact force• The point contact of a sphere creates significantly larger stresses than the

line contact of a cylinder

0 20 40 60 80 1000

5 .108

1 .109

1.5 .109

2 .109

2.5 .109

Dia 10 mm sphere (steel) on flat plate (steel)Dia 10 mm x 0.5 mm cylinder on flat plate (s teel)Dia 10 mm sphere (steel) on flat plate (steel)Dia 10 mm x 0.5 mm cylinder on flat plate (s teel)

Sphere vs. Cylinder - Von Mises Stress

Contact Force [N]

Von

Mise

s Stre

ss [P

a]


Effects of Contact Stresses - Fatigue

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Elastic Deformation of Curved Surfaces

31

21

32

2

22

1

21 11

211104.1 ⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−=

RREEFs

ννδ

The displacement of the centers of two spheres is given by:

The displacement of the centers of two cylinders is given by:With ν1 = ν2 = ν, and E1 = E2 = E:

( )⎟⎠⎞

⎜⎝⎛ ++

−=

bR

bR

LEF

c21

2 4ln4ln3212

πνδ

Note that the center displacements are highly nonlinear functions of the load


Sphere vs. Cylinder – Center Displacement

• The point contact of a sphere creates significantly larger center displacements than the line contact of a cylinder

0 20 40 60 80 1000

1 .10 6

2 .10 6

3 .10 6

4 .10 6

5 .10 6

Dia 10 mm sphere (steel) on flat plate (steel)Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel)Dia 10 mm sphere (steel) on flat plate (steel)Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel)

Sphere vs. Cylinder - Center Displacement

Contact Force [N]

Cen

ter D

ispl

acem

ent [

m]

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Sphere vs. Cylinder – Stiffness

• The point contact of a sphere creates significantly lower stiffness than the line contact of a cylinder

0 20 40 60 80 1000

1 .107

2 .107

3 .107

4 .107

Dia 10 mm sphere (steel) on flat plate (steel)Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel)Dia 10 mm sphere (steel) on flat plate (steel)Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel)

Sphere vs. Cylinder - Stiffness

Contact Force [N]

Stiff

ness

[N/m

]


Effects of Material Combinations

• The maximum contact pressure between two curved surfaces depends on:– Type of curvature (sphere vs. cylinder)– Radius of curvature– Magnitude of contact force– Elastic modulus and Poisson’s ratio of contact surfaces

• Through careful material pairing, contact stresses may be lowered

10


Contact Pressure Depending on Material Combination

• Materials with a lower modulus will experience larger deformations, resulting in a lower contact pressure

0 10 20 30 40 50 60 70 80 90 1000

5 .108

1 .109

1.5 .109

2 .109 10 mm Sphere on Flat Plate (Steel)

Contact force [N]

Max

imum

p res

sur e

[Pa]

Tungsten (E = 655 GPa, ν = 0.2)Steel (E = 207 GPa, ν = 0.3)Bronze (E = 117 GPa, ν = 0.35)Titanium (E = 110 GPa, ν = 0.31)Aluminum (E = 71 GPa, ν = 0.33)Acrylic Thermoplastic (E = 2.8 GPa, ν = 0.4)


Center Displacement Depending on Material Combination

0 10 20 30 40 50 60 70 80 90 1000

1 .10 5

2 .10 5

3 .10 5

4 .10 5 10 mm Sphere on Flat Plate (Steel)

Contact force [N]

Cen

t er D

ispl

acem

ent[

m]

Tungsten (E = 655 GPa , ν = 0.2)Steel (E = 207 GPa, ν = 0.3)Bronze (E = 117 GPa, ν = 0.35)Titanium (E = 110 GPa, ν = 0.31)Aluminum (E = 71 GPa, ν = 0.33)Acrylic Thermoplastic (E = 2.8 GPa, ν = 0.4)

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