Numerical study on the elastic-plastic contact
between rough surfaces
D.M. Neto1 • M.C. Oliveira1 • L.F. Menezes1 • J.L. Alves2
1CEMMPRE, Department of Mechanical Engineering, University of Coimbra, Portugal
2CMEMS, Department of Mechanical Engineering, University of Minho, Portugal
University of Coimbra
CMN 2017The Congress on Numerical Methods in Engineering
3 – 5 July 2017
Technical University of Valencia, Spain
CMN 2017Technical University of Valencia, Spain
D.M. [email protected]
2
Introduction
Body-in-white
More than 300 sheet metal
parts:
• Closures
• Structural parts
• Reinforcements
Current challenges in the sheet metal forming industry:
• Adoption of new materials (ultra high strength steels, lightweight alloys, etc.)
• Reduced manufacturing cost and lead times
• Shorter product development cycles
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• Today the numerical simulation is an indispensable tool in the development
of new components manufactured by forming
• Simulation is used to predict formability issues before going into production
Sheet metal forming simulation
Numerical simulation
Friction
Material behavior
Failure criterion
Contact
The numerical solution is
strongly influenced by the
computational models
implemented in the FEM code
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• Frictional contact between the forming tools and the blank
• Typically the friction behavior is modelled by the Coulomb’s law
• The formability predicted by simulation is significantly influenced by the friction
coefficient used in the FE model (difficult to evaluate experimentally)
Objective:
Understand the relationship between the microscopic contact and the
macroscopic friction forces generated during sliding contact.
• Finite element simulation of contact between rough surfaces
Frictional contact
Existing friction laws are inadequate for the
realistic description of local contact conditions
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• All engineering surfaces are rough under certain magnification
• Most of rough surfaces are fractals, i.e. self-repeated patterns at every scale
• Frictionless contact between two linearly elastic half-spaces is equivalent to
contact between an effective elastic rough half-space and a rigid flat
surface
• Sinusoidal rough surface
Surface roughness
Measured surface roughness Self-affine surface
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• Half asperity studied under plane strain conditions
• 2 geometries: asperity height g=1 μm and g=5 μm
• 2 materials: reduced Young modulus E*=43.9 GPa and E*=65.9 GPa
FE model – 1D sinusoidal surface
λ=100 μm
h=
150 μ
m
Deformable body
(linear elastic)
Rigid surface
2g
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• Vertical displacement of the rigid surface until achieving full contact
• 50000 hexahedral finite elements (half asperity)
• Frictionless contact (μ=0)
• DD3IMP in-house finite element code
FE model – 1D sinusoidal surface
250x200x1=50,000 FE
Sym
metr
y c
onditio
ns
Sym
me
try c
onditio
ns
Rigid surface
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• Some definitions:
A’= A/A0 – ration between real contact area (A) and nominal contact area (A0)
– average contact pressure
– mean contact pressure
F – applied force
– average contact pressure at full contact
Contact pressure distribution:
Analytical solution for 1D sinusoidal surface
2
0
* sin ( '/ 2)gE Ap F A
2
m
* sin ( '/ 2)
'
gE Ap F A
A
**
gEp
2 22 * cos( )
( ) sin ( ' 2) sin ( )gE x
p x A x
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
A'=
A/A
0
p'=p̅/p*
Analytical solution
E*=65.9 GPa; g=1 μm
E*=65.9 GPa; g=5 μm
E*=43.9 GPa; g=1 μm
E*=43.9 GPa; g=5 μm
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• Evolution of the real contact area for 2 materials and 2 geometries of asperity
• Numerical results in very good agreement with the analytical solution
Real contact area (analytical vs simulation)
The difference between
analytical and numerical
solution increases for large
values of asperity amplitude
Fullcontact
Lightload
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• Material: E*=65.9 GPa; asperity geometry: g=5 μm (largest amplitude)
• Contact pressure slightly overestimated by the analytical solution
0
4
8
12
16
20
24
0 10 20 30 40 50
Co
nta
ct
pre
ssu
re [
GP
a]
x-coordinate [μm]
A'=1
A'=0.5
A'=0.25
A'=1
A'=0.5
A'=0.25
Contact pressure distribution (analytical vs simulation)
Analytical
Numerical
F/A0=1.5 GPa
F/A0=9.4 GPa
F/A0=5.0 GPa
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• von Mises stress distribution on half asperity, for 3 values of real contact area
• Maximum value of von Mises stress lies in the asperity interior (like in the
Hertz solution)
Stress distribution (E*=65.9 GPa and g=5 μm)
A’=0.25 A’=0.50 A’=1.00
[GPa]
p ̅=F/A0=1.5 GPa p ̅=F/A0=9.4 GPap ̅=F/A0=5.0 GPa
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0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
pmλ/(
gE
*)
A'=A/A0
Analytical solution
E*=65.9 GPa; g=1 μm
E*=65.9 GPa; g=5 μm
E*=43.9 GPa; g=1 μm
E*=43.9 GPa; g=5 μm
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• Evolution of the mean contact pressure for 2 materials and 2 geometries
• The difference between analytical and numerical solution increases for large
values of asperity amplitude
Mean contact pressure evolution (analytical vs simulation)
Decrease of the mean
contact pressure because
the real contact area
increases quickly at the end
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• The results previously presented consider linear elastic material behaviour
• Very good agreement between numerical and analytical solutions
New analysis
• Elastic-perfectly plastic material behaviour
• Elastic and plastic properties: E*=43.9 GPa and σy=1 GPa
• Elastic and plastic properties: E*=65.9 GPa and σy=2 GPa
• 2 geometries: asperity height g=1 μm and g=5 μm
Mechanical material behavior
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• Small increase of the mean contact pressure after onset of plasticity
• Onset of plasticity identified by the deviation of numerical solution from the
analytical solution (linear elastic)
Mean contact pressure evolution (elastic-perfectly plastic)
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
pmλ/
(gE
*)
A'=A/A0
Analytical solution (elastic)
E*=65.9 GPa; σ₀=2 GPa; g=1 μm
E*=65.9 GPa; σ₀=2 GPa; g=5 μm
E*=43.9 GPa; σ₀=1 GPa; g=1 μm
E*=43.9 GPa; σ₀=1 GPa; g=5 μm
Elastic deformation
is predominant
Plastic deformation
is predominant
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• Equivalent plastic strain distribution on half asperity (3 different instants)
• Maximum value arises clearly in the asperity interior
Plastic strain distribution (E*=65.9 GPa and σy=2 GPa)
A’=0.25 A’=0.50 A’=1.00
p ̅=F/A0=1.1 GPa p ̅=F/A0=4.9 GPap ̅=F/A0=3.0 GPa
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0
4
8
12
16
20
24
0 10 20 30 40 50
Conta
ct pre
ssure
[GP
a]
x-coordinate [μm]
A'=1
A'=0.5
A'=0.25
A'=1
A'=0.5
A'=0.25
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• Material: E*=65.9 GPa and σy=2 GPa; asperity geometry: g=5 μm
• Contact pressure approximately constant on the asperity tip
• Slight increase as the applied force rise (real contact area)
Contact pressure distribution
Analytical(elastic)
Numerical(elastic-plastic)
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Next steps – 2D sinusoidal surface
• Analytical solution unavailable for elastic material behaviour (2D wavy surface)
• Vertical displacement of the rigid surface (frictionless contact)
• Linear elastic material behaviour
50x50x50=125,000 FE
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Preliminary results – linear elastic material
• Evolution of the contact area (red) with applied load
• From circular to square-like shape of contact area
Cir
cula
r n
on
-co
nta
ct
area
Cir
cula
r co
nta
ct
area
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Conclusions
• Finite element simulation of frictionless contact between a deformable
sinusoidal asperity and a rigid flat
• Both linear elastic and elastic-perfectly plastic material behaviour
• Roughness described by a sinusoidal function (amplitude and wavelength)
• Very good agreement between numerical and analytical solution considering
elastic material and 1D wavy surface
• The increase of the mean contact pressure stabilizes after onset of plasticity
• Contact pressure on the asperity tip is approximately constant when the
plastic deformation is predominant
• Study of 2D sinusoidal surfaces is essential to approximate real surfaces
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This research work was sponsored by national funds from the Portuguese
Foundation for Science and Technology (FCT) under the projects with reference
P2020-PTDC/EMS-TEC/0702/2014 (POCI-01-0145-FEDER-016779) and P2020-
PTDC/EMS-TEC/6400/2014 (POCI-01-0145-FEDER-016876) by UE/FEDER
through the program COMPETE 2020
Acknowledgements
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Thank you for your attention!
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