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

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D.M. Netodiogo.neto@dem.uc.pt

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

CMN 2017Technical University of Valencia, Spain

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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

9

• 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

CMN 2017Technical University of Valencia, Spain

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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

CMN 2017Technical University of Valencia, Spain

D.M. Netodiogo.neto@dem.uc.pt

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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D.M. Netodiogo.neto@dem.uc.pt

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Thank you for your attention!