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The Contact Mechanics Challenge Martin Mser Dept. of Materials - - PowerPoint PPT Presentation

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The Contact Mechanics Challenge Martin Mser Dept. of Materials Science and Engineering Saarland University ICTP-COST-MODPHYSFRICT Conference: Trends in Nanotribology > The Tribology Letters Contact Mechanics Challenge Contact Mechanics

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The Contact Mechanics Challenge

Martin Müser

  • Dept. of Materials Science and Engineering

Saarland University

ICTP-COST-MODPHYSFRICT Conference: Trends in Nanotribology

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Contact Mechanics

deformation of solids that touch each other single-asperity contacts nominally rough surfaces quantities of primary interest:

  • area of load or pressure
  • displacement of load or p

quantities of secondary interest:

  • distribution functions of gap, contact patch size, stress

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Why posing a contact-mechanics challenge?

GW model 50+ years

May

new theories keep arising simulations are becoming competitive

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Contact mechanics: Course of action

math. model (PDEs) reality

make (reasonable) approximations

solution

make (reasonable) approximations

  • small-slope approximation
  • linear elasticity, no overlap
  • randomly-rough surfaces
  • short-range adhesion, …

brute-force computing

  • controllable approximations

mapping onto simpler equations

  • uncontrolled approximations

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Building the model

math. model (PDEs) reality

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  • small-slope approximation
  • linear elasticity, no overlap
  • randomly-rough surfaces
  • short-range adhesion, …
  • periodic boundary conditions, hard-wall constraint

Atrue Aapparent ≈ 2 p E*g

σ local =σ 0 exp(−gap / ρ)

adhesion law details don’t matter as long as r is ”small”

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Page Trieste Trends in Nanotribology (2017) > Building the model

Surface height spectra

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H = 0.72 H = 1 H = 0.8

C(q) ~ q−2(1+H )

experimental data compiled in: Persson, Tribol. Lett. (2014)

H = 0.7 roll-off region self-affine region used spectrum

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Page

x (µm) y (µm) z (µm)

Trieste Trends in Nanotribology (2017) > Building the model

Surface height in real space

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H = 0.8 dark areas touch first Lsys = 0.1 mm ; rms-h = 0.7 µm ρ = 2 nm ; g = 50 mJ/m2 ; E* = 25MPa

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

The tasks:

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Compute any measurable and well-defined “observable” (a function or functional of displacement field) Spatially resolved observables (at a reference load ):

  • gap and stress along the reference line

Histograms (at a reference load à 3% contact):

  • gap, stress, and contact patch size

Mean values as function of load:

  • relative contact area and mean gap

Omitted: Stress spectrum (too few submissions)

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

The contestants:

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Austria AC2T research G Vorlaufer, A Vernes France INSA Lyon R Bugnicourt, P Sainsot … TA Lubrecht Germany FZ-Jülich BNJ Persson

Saarland Univ.

MH Müser, WB Dapp Italy Polytech Bari G Carbons, F Bottiglione, L Afferante Iran Isfahan Univ. HA Esfahani, M Kadkhodai, S Akbarzadeh NL Univ. of Groningen S Solhjoo, AI Vakis Taiwan Chang Gung J-J Wu UK Imperial College D Dini, S Medina USA Johns Hopkins J Monti, L Pastewka, MO Robbins

Univ Florida

K Harris, A Bennett … WG Sawyer

Auburn Univ.

KJ Streator, A Rostami

Georgia Tech.

RL Jackson, Y Xu

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Page

redefine problem to new scale “inverse” models

experiment x 1000 all-atom simulations x 0.001

Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

The methods:

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accuracy efficiency

exact (boundary-value) methods (5 times)

  • only controlled approximations

Persson theory (1 time)

  • renormalization group approach

Bearing models (5 times)

  • local constitutive relations
  • no interaction between contact patches

finite-element method (no showing) dimensionality reduction (no showing)

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Page Trieste Trends in Nanotribology (2017) > Methods

exact boundary-value methods

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effectively all minimize total energy or zero forces

  • elastic energy in Fourier space, constraint & adhesion in real space
  • all employ fast Fourier transform (FFT)
  • alter displacements until energy minimized / stress disappears

 σ  q

( ) ≈ qE*

2  u  q

( )+ 

σ ext  q

( )+ 

σ adh  q

( )+...

elastic stress of a semi-infinite solid in q-space

FFT-BVM + conjugate gradient Bugnicourt, Sainsot, Lubrecht BICGSTAB finite-range repulsion Wu BEM+B splining at small scales Vorlaufer, Varnas FFT-IA elastic stress field is varied Dini, Medina GFMD Green’s function molecular dynamics Müser, Dapp, Pastewka, Robbins

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Page Trieste Trends in Nanotribology (2017) > Methods 11 2017-09-11

Persson theory

maps contact mechanics problem onto diffusion process magnification à time, stress à position stress = 0: absorbing boundary roughness at magnification q à diffusion constant starting assumption: Pr(s) = d(s) broaden Pr(s) with each newly resolved h(q) J Chem. Phys. (2001)

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Page Trieste Trends in Nanotribology (2017) > Methods 12 2017-09-11

Bearing-area models

base models (such as Greenwood-Williamson):

  • assume local model for asperity interactions, e.g., Hertz,

JKR, or simple springs (Winkler)

  • assume a distribution of asperity heights and curvatures
  • ignore elastic deformation between asperities

Winkler simple springs Angelini, Sawyer SR-GW spatially resolved GW Esfahani, Kadkhodaei, Akbarzad Archard fly-on-a-fly-on-a-fly… Jackson, Xu, Streator, Rostami SC-GW slightly-corrected Bottiglione, Carbone ICHA interacting & coalescing Hertz asp’s Afferante, Carbone

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Page Trieste Trends in Nanotribology (2017) > Methods 13 2017-09-11

Experiments 3D print surface topography

Harris, Bennett, Schulze, Rohde, Ifju, Sawyer

3D-printed surface glass waveguide molded PDMS waveguide mask diffuse light source blackout fabric real contact

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Page Trieste Trends in Nanotribology (2017) > Methods 14 2017-09-11

All-atom MD scale problem to atomic scale

Solhjoo, Vakis all-atom simulations: EAM potential for calcium E*= 30 GPa à p = 0.3 GPa; pc = 10 GPa rigid substrate with given height profile dislocations

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Page

experiment

Trieste Trends in Nanotribology (2017) > Contact-mechanics challenge: Results

Contact visualization

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Page Trieste Trends in Nanotribology (2017) > Contact-mechanics challenge: Results

Contact visualization

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Page Trieste Trends in Nanotribology (2017) > Contact-mechanics challenge: Results

Contact visualization

all-atom MD

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Page Trieste Trends in Nanotribology (2017) > Contact-mechanics challenge: Results

Contact visualization

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Page

20 40 60 80 100

y (µm)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

g (µm)

reference gap exact methods Experiment Winkler SRGW all-atom MD

Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Results: Spatially resolved quantities Gap across reference line (appr. methods)

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Results: Spatially resolved quantities Gap across reference line (exact methods)

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20 40 60 80 100

y (µm)

0.0 0.5 1.0 1.5 2.0

g (µm)

GFMD FFT-BVM BICGSTAB BEM+B FFT-IA

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Page

38 40 42 44 46 48

y (µm)

  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

σ ( E*/g )

GFMD, FFT-BVM SRGW BICGSTAB BEM+B FFT-IA _

Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Results: Stress across reference line (local zoom-in)

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Page

38 40 42 44 46 48

y (µm)

  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

σ ( E*/g )

GFMD, FFT-BVM SRGW BICGSTAB BEM+B FFT-IA _

Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Results: Stress across reference line (local zoom-in)

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge 23 2017-09-11

Results: Stress distribution

  • 0.5

0.0 0.5 1.0 1.5

σ (E*g) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Pr(σ) (1/E*g)

GFMD GFMD (contact) FFT-BVM BEM+B BICGSTAB FFT-IA GFMD (w/o adhesion) SCGW (w/o adhesion) ICHA (w/o adhesion)

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Page

10

  • 2

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  • 1

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10

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a (µm

2)

0.0 0.2 0.4 0.6 0.8 1.0 CPr (a)

GFMD FFT-BVM Winkler BEM+B BICGSTAB FFT-IA ICHA

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2

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a (nm

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Pr (a) (nm

  • 2)

GFMD FFT-BVM Winkler BEM+B BICGSTAB FFT-IA ICHA

Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Hertz/JKR- like contacts JKR

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Results: Patch-size distribution

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge 25 2017-09-11

Results: Gap distribution

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  • 2

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  • 1

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u (µm) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pr(u) (1/µm)

GFMD FFT-BVM BEM+B BICGSTAB FFT-IA

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  • 2

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  • 1

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u (µm) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pr(u) (1/µm)

GFMD Persson ICHA (only summits)

without adhesion

2 2.5 3

r

1 2 3

g(r) µT = 1/4 µT = 4 z = x

2/2

local gap for long-range and short-range adhesion

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge 26 2017-09-11

Results: Average quantities ar(p)

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p/E*g 10

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  • 1

10 ar

GFMD FFT-BVM Persson experiment Winkler SRGW BICGSTAB BEM+B all-atom MD

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  • 1

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p/E*g 10

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  • 1

10 ar

Archard RL-Archard FFT-IA SCGW ICHA

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  • 4

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  • 2

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  • 1

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p/E*g

0.0 0.5 1.0 1.5 2.0 2.5 3.0

u (µm)

BICGSTAB BEM+B all-atom MD Archard FFT-IA SCGW ICHA

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  • 1

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p/E*g

0.0 0.5 1.0 1.5 2.0 2.5 3.0

u (µm)

GFMD FFT-BVM Persson Winkler SRGW _ _

Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge 27 2017-09-11

Results: mean gap as a function of load

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Conclusions I

  • close agreement between all systematic approaches

differences in quantities that need high resolution GFMD (128k x 128k) and FFT-BVM (32k x 32k) match

  • FFT-BVM (INSA-Lyon) stands out in that

results obtained on a single core (with added RAM) for 32k x 32k and on standard workstation for 4k x 4k (1 hour) à rigorous treatment of measured profiles feasible

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Page Trieste Trends in Nanotribology (2017) > The Tribology Letters Contact Mechanics Challenge

Conclusions II

  • good agreement between all systematic approaches

and inverse models (experiment & all-atom) reverse conclusion: approximations that are commonly made in contact mechanics might be less problematic than believed

  • reasonable agreement with Persson theory
  • n all reported properties

Pr(gap,stress) for small gap but with adhesion missing L

  • bearing models reproduce arel(p) relation

but otherwise do not predict correct trends

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