Contact Mechanics of theory of elasticity.ppt

Contact stresses

1. Contact Mechanics
Maria Persson Gulda
Kathleen DiSanto

2. Outline
What is Contact Mechanics?
The two different kind of contacts.
Boussinesq and Cerruti Potential Functions
The specific case of an Applied Normal Force
Hertz Equations- Derivation, Assumptions
Rigid Sphere Contacting a Deformable Plate
Deformable Sphere Contacting a Rigid Plate

3. What is Contact Mechanics?
“[The theory of contact mechanics] is
concerned with the stresses and
deformation which arise when the
surfaces of two solid bodies are
brought into contact.”
Professor Johnsson

4. Two kinds of contact
Conforming contacts
The two surfaces fit exactly or closely
together without deformation
Non-conforming contact
The surfaces, or one of the two
surfaces, deforms when there is a
contact area in between them.

5. Derivation: Boussinesq and Cerruti
Potential Functions
Here are the potential functions:
Each satisfy Laplace’s equation:

€F
1=q
xξ,η()⋅Ω⋅d
S
∫∫ ξ⋅dη
G
1=q
yξ,η()⋅Ω⋅d
S
∫∫ ξ⋅dη
H
1=pξ,η()⋅Ω⋅d
S
∫∫ ξ⋅dη

€
F=
∂F
1
∂z
=q
x
ξ,η()⋅lnρ+z()⋅d
S
∫∫ ξ⋅dη
G=
∂G
1
∂z
=q
y
ξ,η()⋅lnρ+z()⋅d
S
∫∫ ξ⋅dη
H=
∂H
1
∂z
=pξ,η()⋅lnρ+z()⋅d
S
∫∫ ξ⋅dη
€
ρ=ξ−x()
2
+η−y()
2
+z
2
{ }
1
2
€
Ω=z⋅lnρ+z()−ρ

€
∇
2
ϕ=0
QuickTime™ and a
decompressor
are needed to see this picture.

6. Special Case: Applied Pressure Only
The potential functions are reduced as follows:
Displacement equations:
By Hooke’s Law, the stresses are:

€
Ψ=
∂H
∂z
=
∂Ψ
1
∂z
=pξ,η()
S
∫∫ ⋅
1
ρ
⋅dξ⋅dη

€u
x=−
1
4πG
1−2υ( )
∂ψ
1
∂x
+z
∂ψ
∂x

⎧

⎨

⎩

⎫

⎬

⎭
u
y
=−
1
4πG
1−2υ( )
∂ψ
1
∂y
+z
∂ψ
∂y

⎧

⎨

⎩

⎫

⎬

⎭

€
σ
x=
1
2π
2υ
∂ψ
∂z
−z
∂
2
ψ
∂x
2
−(1−2υ)
∂
2
ψ
1
∂x
2

⎧

⎨

⎩

⎫

⎬

⎭
σ
y
=
1
2π
2υ
∂ψ
∂z
−z
∂
2
ψ
∂y
2
−(1−2υ)
∂
2
ψ
1
∂y
2

⎧

⎨

⎩

⎫

⎬

⎭
σ
z
=
1
2π
∂ψ
∂z
−z
∂
2
ψ
∂z
2

⎧

⎨

⎩

⎫

⎬

⎭

€
F=F
1
=G=G
1
=0

€
u
z=
1
4πG
21−υ()ψ+z
∂ψ
∂z

⎧

⎨

⎩

⎫

⎬

⎭

€
τ
xy=−
1
2π
(1−2υ)
∂
2
ψ
1
∂x∂y
+2
∂
2
ψ
∂x∂y

⎧

⎨

⎩

⎫

⎬

⎭
τ
yz
=−
1
2π
z
∂
2
ψ
∂y∂z
τ
zx=−
1
2π
z
∂
2
ψ
∂x∂z

7. Concentrated Normal Force on an Elastic
Half Space
The displacements are:
The stresses in polar coordinates:
QuickTime™ and a
decompressor
are needed to see this picture.
€
ρ=x
2
+y
2
+z
2
{ }
1
2
€
pξ,η()⋅d
S
∫∫ ξ⋅dη=P

€
Ψ
1
=
∂H
1
∂z
=H=P⋅ln(ρ+z)

€
u
x
=
P
4πG
xz
ρ
−1−2υ( )
x
ρρ+z()

⎧

⎨

⎩

⎫

⎬

⎭
u
y
=
P
4πG
yz
ρ
3
−1−2υ( )
y
ρρ+z()

⎧

⎨

⎩

⎫

⎬

⎭

€
u
z
=
P
4πG
z
2
ρ
3
+
21+ν()
ρ

⎧

⎨

⎩

⎫

⎬

⎭

€
σ
r=
P
2π
1−2υ( )
1
r
2
−
z
ρr
2

⎛

⎝

⎜

⎞

⎠

⎟
−
3zr
2
ρ
5

⎧

⎨

⎩

⎫

⎬

⎭
σ
θ=−
P
2π
1−2υ( )
1
r
2
−
z
ρr
2
−
z
ρ
3

⎛

⎝
⎜

⎞

⎠
⎟

€
σ
z=−
3P
2π
z
3
ρ
5
τ
rz
=−
3P
2π
rz
2
ρ
5

€
Ψ=
∂H
∂z
=
P
ρ

8. Concentrated Force Cont.
Now looking only at the surface, z=0
The displacements in polar coordinates become:
For a general pressure distribution, the displacement for
any surface point in S, by Green’s function method,
becomes:
€
u
rz=0
=−
1−2ν( )
4πG
P
r

€
u
z
z=0
=
1−ν()
2πG
P
r

€
u
z
z=0
=
1−ν
2
πE
ps,φ()⋅ds⋅dφ
S
∫∫
QuickTime™ and a
decompressor
are needed to see this picture.

9. Hertz Pressure
The Pressure distribution is:
Equation for determining surface displacement:

The Hertz displacement equation:
€
p(r)=
p
o
a
2
−r
2
( )
1/2
a

€
u
z
z=0
=
1−ν
2
πE
ps,φ()⋅ds⋅dφ
S
∫∫
where a is the radius of the contact area

10. Hertz Theory of Elastic Contact
Assumptions:
The radii of curvature of the contacting bodies
are large compared with the radius of the
circle of contact.
The dimensions of each body are large
compared to the radius of the circle of
contact.
The contacting bodies are in frictionless
contact.
The surfaces in contact are continuous and
nonconforming.

11. Examples
Focus on two examples:
1.Rigid spherical indenter pushing to deformable flat
surface.
2.Deformable sphere contacting rigid plate.
(2)
QuickTime™ and a
decompressor
are needed to see this picture.
(1)

12. Equations to be Used
(2) – where δ is the vertical distance the point where the load is
applied moves and a is the contact area radius determined
by the equation:
(1)– where R’ is the radius of the rigid sphere and
R
S
is the radius of the deformable plate
(3) – h is the original distance between a point on the rigid sphere
and the deformable plate before load application.
(4) – These are the equations of
displacement derived
previously
(5) – This states that the translation of the point of load
application equals the surface displacement of the
plate and sphere plus the original distance
between the surfaces.

€
a=3⋅P⋅R
4⋅E
*( )
1/3

13. Rigid Sphere Contacting Deformable Flat
Surface with Abaqus
Theoretical Contact Radius: 11.995 mm
Abaqus Contact Radius: 11.6 mm
Error: 3%

14. Deformable Sphere Contacting Rigid
Plate with Abaqus
Theoretical Contact Radius: 9.288 mm
Abaqus Contact Radius: 8.5 mm
Error: 6%

15. Conclusion
Contact problems in general are very
complicated to model numerically and
theoretically
Other factors
Friction - rough surfaces
Blunt edges, sharp corners
Sliding and rolling contact
Dynamic impact

16. A Special Thank You To:
Dr. Ashkan Vaziri
Professor James Rice

17. References
Johnson, K. L. Contact Mechanics, Cambridge:
Cambridge University Press; 1985
Fisher-Cripps, A. C. The Hertzian contact surface. J.
Materials Science. 1999;34:129-137
Kogut, L., Etsion, I. Elastic-Plastic Contact Analysis
of a Sphere and a Rigid Flat. J. of Applied
Mechanics. 2002;69:657-662
Johnson, K. L., Greenwood, J. A. An Adhesion Map
for the contact of elastic Spheres. J. of Colloid and
Interface Science. 1997;192:326-333
Barber, J. R.,Clavarella, M. Contact mechanics.
Inter. J. of Solids and Structures. 2000;37:29-43

18. Any Questions?