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

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

> The Tribology Letters Contact Mechanics Challenge Contact Mechanics deformation of solids that touch each other single-asperity quantities of primary interest: contacts - area of load or pressure - displacement of load or p nominally rough surfaces quantities of secondary interest: - distribution functions of gap, contact patch size, stress Page 2017-09-11 Trieste Trends in Nanotribology (2017) 1

> 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 Page 2017-09-11 Trieste Trends in Nanotribology (2017) 2

> The Tribology Letters Contact Mechanics Challenge Contact mechanics: Course of action make (reasonable) make (reasonable) approximations approximations math. reality model solution (PDEs) - small-slope approximation brute-force computing - linear elasticity, no overlap - controllable approximations - randomly-rough surfaces mapping onto simpler equations - uncontrolled approximations - short-range adhesion, … Page 2017-09-11 Trieste Trends in Nanotribology (2017) 3

> The Tribology Letters Contact Mechanics Challenge Building the model adhesion law σ local = σ 0 exp( − gap / ρ ) details don’t matter math. as long as r is ”small” reality model (PDEs) - small-slope approximation A true ≈ 2 p - linear elasticity, no overlap E * g A apparent - randomly-rough surfaces - short-range adhesion, … - periodic boundary conditions, hard-wall constraint Page 2017-09-11 Trieste Trends in Nanotribology (2017) 4

> Building the model roll-off region Surface height spectra self-affine region C ( q ) ~ q − 2(1 + H ) H = 0.72 H = 0.8 used spectrum H = 1 H = 0.7 experimental data compiled in: Persson, Tribol. Lett. (2014) Page 2017-09-11 Trieste Trends in Nanotribology (2017) 5

> Building the model dark areas Surface height in real space touch first H = 0.8 z (µm) y (µm) L sys = 0.1 mm ; rms- h = 0.7 µ m ρ = 2 nm ; g = 50 mJ/m 2 ; E * = 25MPa x (µm) Page 2017-09-11 Trieste Trends in Nanotribology (2017) 6

> The Tribology Letters Contact Mechanics Challenge The tasks: 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) Page 2017-09-11 Trieste Trends in Nanotribology (2017) 7

> The Tribology Letters Contact Mechanics Challenge The contestants: Austria AC2T research G Vorlaufer, A Vernes France INSA Lyon R Bugnicourt, P Sainsot … TA Lubrecht Germany FZ-Jülich BNJ Persson MH Müser, WB Dapp Saarland Univ. 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 K Harris, A Bennett … WG Sawyer Univ Florida KJ Streator, A Rostami Auburn Univ. RL Jackson, Y Xu Georgia Tech. Page 2017-09-11 Trieste Trends in Nanotribology (2017) 8

> The Tribology Letters Contact Mechanics Challenge The methods: exact (boundary-value) methods (5 times) - only controlled approximations redefine problem to new scale finite-element method (no showing) all-atom accuracy simulations Persson theory (1 time) efficiency x 0.001 - renormalization group approach dimensionality reduction (no showing) experiment x 1000 Bearing models (5 times) - local constitutive relations “inverse” - no interaction between contact patches models Page 2017-09-11 Trieste Trends in Nanotribology (2017) 9

> Methods exact boundary-value methods effectively all minimize total energy or zero forces - elastic energy in Fourier space, constraint & adhesion in real space σ  u    ( ) ≈ qE * - all employ fast Fourier transform (FFT)  ( ) +  ( ) +   ( ) + ... q q q q σ ext σ adh 2 elastic stress of a semi-infinite solid in q-space - alter displacements until energy minimized / stress disappears 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 Page 2017-09-11 Trieste Trends in Nanotribology (2017) 10

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

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

> Methods Experiments 3D print surface topography Harris, Bennett, Schulze, Rohde, Ifju, Sawyer waveguide mask diffuse light source real contact 3D-printed surface blackout fabric waveguide glass molded PDMS Page 2017-09-11 Trieste Trends in Nanotribology (2017) 13

> Methods 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; p c = 10 GPa dislocations rigid substrate with given height profile Page 2017-09-11 Trieste Trends in Nanotribology (2017) 14

> Contact-mechanics challenge: Results Contact visualization experiment Page 2017-09-11 Trieste Trends in Nanotribology (2017) 15

> Contact-mechanics challenge: Results Contact visualization Page 2017-09-11 Trieste Trends in Nanotribology (2017) 16

> Contact-mechanics challenge: Results Contact visualization all-atom MD Page 2017-09-11 Trieste Trends in Nanotribology (2017) 17

> Contact-mechanics challenge: Results Contact visualization Page 2017-09-11 Trieste Trends in Nanotribology (2017) 18

> The Tribology Letters Contact Mechanics Challenge Results: Spatially resolved quantities Gap across reference line (appr. methods) 3.0 reference gap exact methods 2.5 Experiment Winkler SRGW 2.0 all-atom MD g ( µ m) 1.5 1.0 0.5 0.0 0 20 40 60 80 100 y ( µ m) Page 2017-09-11 Trieste Trends in Nanotribology (2017) 19

> The Tribology Letters Contact Mechanics Challenge Results: Spatially resolved quantities Gap across reference line (exact methods) GFMD 2.0 FFT-BVM BICGSTAB BEM+B FFT-IA 1.5 g ( µ m) 1.0 0.5 0.0 0 20 40 60 80 100 y ( µ m) Page 2017-09-11 Trieste Trends in Nanotribology (2017) 20

> The Tribology Letters Contact Mechanics Challenge Results: Stress across reference line (local zoom-in) 2.0 GFMD, FFT-BVM 1.5 SRGW BICGSTAB BEM+B 1.0 FFT-IA σ ( E */ g ) _ 0.5 0.0 -0.5 -1.0 38 40 42 44 46 48 y ( µ m) Page 2017-09-11 Trieste Trends in Nanotribology (2017) 21

> The Tribology Letters Contact Mechanics Challenge Results: Stress across reference line (local zoom-in) 2.0 GFMD, FFT-BVM 1.5 SRGW BICGSTAB BEM+B 1.0 FFT-IA σ ( E */ g ) _ 0.5 0.0 -0.5 -1.0 38 40 42 44 46 48 y ( µ m) Page 2017-09-11 Trieste Trends in Nanotribology (2017) 22

> The Tribology Letters Contact Mechanics Challenge Results: Stress distribution 0.07 GFMD GFMD (contact) 0.06 FFT-BVM BEM+B BICGSTAB 0.05 Pr( σ ) (1/ E * g ) FFT-IA GFMD (w/o adhesion) 0.04 SCGW (w/o adhesion) ICHA (w/o adhesion) 0.03 0.02 0.01 0.00 -0.5 0.0 0.5 1.0 1.5 σ ( E * g) Page 2017-09-11 Trieste Trends in Nanotribology (2017) 23

> The Tribology Letters Contact Mechanics Challenge Results: Patch-size distribution Hertz/JKR- like contacts -3 10 -4 10 1.0 JKR GFMD -5 FFT-BVM 10 Winkler 0.8 -2 ) BEM+B Pr ( a ) ( nm -6 BICGSTAB 10 GFMD FFT-IA 0.6 CPr ( a ) FFT-BVM ICHA -7 10 Winkler BEM+B 0.4 -8 BICGSTAB 10 FFT-IA ICHA -9 0.2 10 -10 10 0.0 7 2 3 4 5 6 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 2 ) a ( nm 2 ) a ( µ m Page 2017-09-11 Trieste Trends in Nanotribology (2017) 24