Contact Stresses and Deformations
ME EN 7960 – Precision Machine Design
Topic 7
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-1
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
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-2
1
Curved Surfaces in Contact – Examples
Linear bearings (ball and rollers)
Rotary ball bearing
Rotary roller bearing
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-3
Curved Surfaces in Contact – Examples (contd.)
Ball screw
Gears
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-4
2
Spheres in Contact
The radius of the contact area is
given by:
z
⎡1 −ν 12 1 −ν 22 ⎤
+
3F ⎢
E1
E2 ⎥⎦
⎣
a=3
⎛1
1 ⎞
4⎜⎜ + ⎟⎟
⎝ R1 R2 ⎠
F
y
Z = 0:
x
pmax
Sphere 1
E 1,
2a
R1
1
2a
Circular contact area,
resulting in a semi-elliptic
pressure distribution
y
pmax
E 2,
R2
2
Sphere 2
x
Where E1 and E2 are the moduli of
elasticity for spheres 1 and 2 and ν1
and ν2 are the Poisson’s ratios,
respectively
The maximum contact pressure at
the center of the circular contact
area is:
pmax =
F
3F
2πa 2
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-5
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)
z
y
Z = 0:
F
x
pmax
2a
E1,
R1
1
2a
y
E2,
2
Flat plate
(R2 = ∞)
x
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-6
3
Spheres in Contact – Principal Stresses
The principal stresses σ1, σ2, and σ3 are generated on the z-axis:
⎡
⎤
⎢
⎥
⎛
1
z
a⎞
⎥
σ 1 = σ 2 = σ x = σ y = − pmax ⎢(1 +ν )⎜⎜1 − arctan ⎟⎟ −
2
⎢
a
z
⎛
⎞
z
⎝
⎠ 2⎜ + 1⎟ ⎥
⎢
⎜ a2 ⎟ ⎥
⎝
⎠⎦
⎣
⎛ z2 ⎞
σ 3 = σ z = − pmax ⎜⎜ 2 + 1⎟⎟
⎠
⎝a
−1
The principal shear stresses are found as:
τ 1 = τ 2 = τ max =
σ1 − σ 3
τ3 = 0
2
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-7
Spheres in Contact – Vertical Stress Distribution at
Center of Contact Area
σ, τ
1
Von Mises
Ratio of stress to pmax
0.8
σz
Plot shows
material with
Poisson’s
ratio ν = 0.3
0.6
σX, σy
0.4
τmax
0.2
0
z
0
0.5a
a
1.5a
2a
2.5a
3a
Depth below 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
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-8
4
Cylinders in Contact
F
The half-width b of the rectangular
contact area of two parallel cylinders is
found as:
E1, ν 1
z
⎡1 −ν 12 1 −ν 22 ⎤
4F ⎢
+
E1
E2 ⎥⎦
⎣
b=
⎛1
1 ⎞
πL⎜⎜ + ⎟⎟
R
R
2 ⎠
⎝ 1
E2, ν 2
pmax
R1
L
2b
x
Where E1 and E2 are the moduli of
elasticity for cylinders 1 and 2 and ν1
and 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:
y
R2
F
Rectangular contact area
with semi-elliptical pressure
distribution
pmax =
2F
πbL
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-9
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)
Rectangular contact
area with semi- z
elliptical pressure
distribution
F
E1, ν1
Rectangular contact
area with semi-elliptical
pressure distribution
E1, ν1
F
z
R1
E2, ν 2
pmax
L
Rg
E2, ν2
R1
2b
x
y
Flat plate
(R2 = ? )
F
L
pmax
2b
x
F
Cylindrical groove
(R2 = -Rg)
y
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-10
5
Cylinders in Contact – Principal Stresses
The principal stresses σ1, σ2, and σ3 are generated on the z-axis:
⎡ z2
z⎤
+1 − ⎥
2
b ⎦⎥
⎣⎢ b
σ 1 = σ x = −2νpmax ⎢
⎡⎛ ⎛ z 2 ⎞ −1 ⎞ z 2
z⎤
σ 2 = σ y = − pmax ⎢⎜ 2 − ⎜⎜ 2 + 1⎟⎟ ⎟ 2 + 1 − 2 ⎥
b⎥
⎢⎜⎝ ⎝ b
⎠ ⎟⎠ b
⎦
⎣
⎡ z2 ⎤
σ 3 = σ z = − pmax ⎢ 2 + 1⎥
⎢⎣ b
⎥⎦
τ1 =
σ 2 −σ3
2
, τ2 =
−1
σ1 − σ 3
2
, τ3 =
σ1 − σ 2
2
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-11
Cylinders in Contact – Vertical Stress Distribution
along Centerline of Contact Area
σ, τ
1
Von Mises
σy
Ratio of stress to pm ax
0.8
σx
0.4
0.2
0
•
•
Plot shows
material with
Poisson’s ratio
ν = 0.3
σz
0.6
τ1
0
0.5b
b
1.5b
2b
Depth below contact area
2.5b
3b
z
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
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-12
6
Sphere vs. Cylinder – Von Mises Stress
Sphere vs. Cylinder - Von Mises Stress
9
2.5 .10
Dia 10 mm sphere (steel) on flat plate (steel)
Dia 10 mm x 0.5 mm cylinder on flat plate (s teel)
9
Von Mises Stress [Pa]
2 .10
9
1.5 .10
9
1 .10
8
5 .10
0
0
20
40
60
80
100
Contact F orce [N]
•
•
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
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-13
Effects of Contact Stresses - Fatigue
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-14
7
Elastic Deformation of Curved Surfaces
The displacement of the centers of two spheres is given by:
2
⎡ ⎛ 1 −ν 12 1 −ν 22 ⎞⎤ 3 ⎡ 1 ⎛ 1
1 ⎞⎤
⎟⎟⎥ ⎢ ⎜⎜ + ⎟⎟⎥
δ s = 1.04⎢ F ⎜⎜
+
E2 ⎠⎦ ⎣ 2 ⎝ R1 R2 ⎠⎦
⎣ ⎝ E1
1
3
The displacement of the centers of two cylinders is given by:
With ν1 = ν2 = ν, and E1 = E2 = E:
δc =
(
)
2 F 1 −ν 2 ⎛ 2
4R
4R ⎞
⎜ + ln 1 + ln 2 ⎟
b
b ⎠
πLE ⎝ 3
Note that the center displacements are highly nonlinear functions of the load
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-15
Sphere vs. Cylinder – Center Displacement
5 .10
Sphere vs. Cylinder - Center Displacement
6
Center Displacement [m]
Dia 10 mm sphere (steel) on flat plate (steel)
Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel)
4 .10
6
3 .10
6
2 .10
6
1 .10
6
0
0
20
40
60
80
100
Contact F orce [N]
• The point contact of a sphere creates significantly larger
center displacements than the line contact of a cylinder
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-16
8
Stiffness [N/m]
Sphere vs. Cylinder – Stiffness
4 .10
7
3 .10
7
2 .10
7
1 .10
7
Sphere vs. Cylinder - Stiffness
Dia 10 mm sphere (steel) on flat plate (steel)
Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel)
0
0
20
40
60
80
100
Contact F orce [N]
• The point contact of a sphere creates significantly lower
stiffness than the line contact of a cylinder
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-17
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
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-18
9
Contact Pressure Depending on Material
Combination
10 mm Sphere on Flat Plate(Steel)
9
2 .10
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)
9
Maximum pressure [Pa]
1.5 .10
9
1 .10
8
5 .10
0
0
10
20
30
40
50
60
Contact force [N]
70
80
90
100
• Materials with a lower modulus will experience larger
deformations, resulting in a lower contact pressure
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
7-19
Center Displacement Depending on Material
Combination
Center Displacement [m]
4 . 10
10 mm Sphere on Flat Plate (Steel)
5
3 . 10
5
2 . 10
5
1 . 10
5
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)
0
0
10
20
30
40
50
60
Contact force [N]
70
80
90
ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations
100
7-20
10