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Hw2A.pdf
Homework 2
Contact Mechanics
1
Use the equations obtained by Hertz for the stresses resulting from the elastic deformation
produced by the contact of two identical smooth spheres. Compute and plot the dimensionless stresses obtained by dividing by pmean and use multiples of the contact radius a as the
dimensionless distance scale.
Answer: See Maple and COMSOL files.
2
Use the COMSOL system to create a finite element model approximating a two dimensional
semi-infinite body linear elastic body loaded by a rigid flat punch x ∈ [−a, a] at its smooth,
flat surface. Select as linear elastic material first pure Aluminum, then pure Copper and
validate your model by comparing its predictions about the normal displacement at the
surface just outside the punch, against those obtained from the exact formulae.
Answer: See Maple and COMSOL files.
3
Repeat the above problem but use instead the material properties (linear elastic and plastic)
of pure Aluminum and pure Copper. Compare your results with those obtained for the linear
elastic model.
Answer: See COMSOL file.
1
4
Use the COMSOL system to create a finite element model approximating a three dimensional
semi-infinite linear elastic body loaded by a concentrated normal force at the origin on its
smooth, flat surface. Select as linear elastic material first pure Aluminum, then pure Copper
and validate your model by comparing its predictions against those obtained from the exact
formulae.
Answer: See Maple and COMSOL files.
5
Repeat the above problem but use instead the material properties (linear elastic and plastic)
of pure Aluminum and pure Copper. Compare your results with those obtained for the linear
ealstic model.
Answer: See COMSOL file.
6
Use the Abaqus (or Ansys or COMSOL) systems to create a finite element model of the
effect of a Hertz pressure distribution in x ∈ [−a, a] on the stress field inside a semi-infinite
body considering elasto-plastic deformation behavior according to the following stress strain
relationship (approximate for an Al 7075-T6 alloy; assume also E = 5×1010 Pa and ν = 0.3.)
Strain (-)
0.0
0.01
0.1
Stress (Pa)
0.0
500 × 106 = σY
600 × 106 = σU
Answer: See COMSOL files.
7
The surface of a sample of Al-7075 alloy has been investigated using profilometry. The
standard deviation of the asperity heights was measured as s ≈ 1.4µm and the average tip
2
radius of curvature was R = 5µm. Compute the value of the plasticity index ψ and show that
the asperities are expected to deform plastically. How small would the standard deviation
of asperity heights would have to be in order for the asperities to deform elastically?
Answer:
The plasticity index is defined by
ψ=
E σ 1/2
( )
H R
For the given alloy, E = 5 × 1010 , σY = 500 MPa and H = 3σY = 1500 × 106 Pa, so that
ψ = 17.63, therefore the asperities will undergo plastic deformation.
To get ψ = 1, the standard deviation of asperity heights must not exceed
R(
H 2
) = 5(0.15/5)2 = 0.0045
E
micrometers.
8
The contact angle of water on platinum has been measured to be θ = 40 degrees, while that
on paraffin is θ = 110 degrees. Estimate the work of adhesion of water on platinum and
water on paraffin and comment.
Answer:
Dupre’s equation is
WSL = γLV (1 + cos θ)
where WSL is the work of adhesion of liquid to solid and γLV is the surface tension liquid-air
(γLV = 0.0728 J/m2 .
Therefore for water over platinum WSL = 0.128 J/m2 .
And for water over paraffin WSL = 0.047 J/m2 .
Thus, the adhesion of water to platinum is almost three times stronger than to paraffin.
9
Construct a finite element model using COMSOL to simulate the combined effect of a flat
punch and a Hertzian pressure distribution. Compare against the results obtained for the
Hertzian pressure alone.
3
Answer: See COMSOL file.
10
A cylindrical water bridge (radius r = 0.01 m, thickness h = 0.0001 m) forms between two
flat platinum surfaces. Estimate the adhesive force due to the bridge.
Answer:
The calculated adhesive force is F = 0.35 N.
4
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