rigazzi_phd_defense

The effects of roughness on the area of contact
and on the elastostatic friction
FEM simulation of micro-scale rough contact
and real world applications
PhD dissertation defense of Alessandro Rigazzi
Institute of Computational Science, University of Lugano
November 27th 2014

INTRODUCTION
Effects of roughness on the area of contact and on the elastostatic friction 2

The importance of rough contact
• All real surfaces are –at some scale– rough
– At macro-scale (10-3–10-2 m)
– At micro-scale (10-6 m)
– At nano-scale (10-9 m)
• As a consequence, every contact in the real
world involves rough surfaces
– Contact areas are not smooth and continuous
– They are collections of small fragmented contact
islands, spots, or points
3Effects of roughness on the area of contact and on the elastostatic friction

Physical phenomena related to rough contact
• Several quantities related to contact are
affected by roughness, and they influence
different physical phenomena, e.g.
– Area of contact
• Heat transfer
• Charge conduction
– Contact forces
• Fatigue
• Crack initiation
• Friction

Rough contact: strategies
Homogenization
• Surfaces are smooth
• Quantities like friction
coefficient or heat
conduction are used in
averaged forms, as
parameters of the
simulations
• Lack of local variations
of results
Full scale resolution
• Realistic surfaces
• Physical laws imposed at
microscopic level, and
macroscopic quantities
are derived
• Need to consider several
orders of magnitude
• Two numerical approaches for rough contact

The challenge
• We aim to discretize macroscopic bodies up to
microscopic scale and apply the Finite Element
Method
• The resulting system is huge
– Millions of unknowns only on the contact boundary
• Contact problems are highly non-smooth
• Solution of such systems is feasible only using
an efficient massively parallel contact solver
– Multigrid solver implemented in obslib++/UG

The ultimate goal
• With the chosen method, we want to quantify
the influence of roughness on two relevant
features of rough contact
– Evolution of the real contact area
– Elastostatic friction force
• We want to derive the contribution of selected
roughness and parameters

ROUGHNESS
CHARACTERIZATION

Roughness
• We need to formally define roughness
• Many measures exist covering different aspects of
roughness
– Root mean square roughness
– Mean profile depth
– …
• But we prefer a compact description based on the
Power Spectral Density (PSD) of the surfaces
– Geometric measures can be derived [Nayak 1971]

Mathematical representation of rough surfaces
• We represent a rough surface as a periodic
height function h(x, y) with zero mean
• The PSD of h(x, y), denoted by C(q), is the
Fourier Transform of the auto-correlation
function R(x, y)

Self-affine rough surfaces
• All natural surfaces show self-affinity
– Geometric features repeat themselves at different
length scales
– Similar to fractals, up to some limit scales
– Mostly isotropic
• All specimens of the same material have the
same statistical parameters, i.e. the same PSD
– They are different realization of the same random
process

Typical rough surface

Geometry of our self-affine rough surfaces
• In all our experiments, we employ square surfaces
of side L (3 cm)
• h(x, y) is defined over a square lattice with
constant dO (0.1 mm)
• The full power spectrum is used to generate the
surfaces
• The surface is always considered as a rigid
corrugated half-space

Power Spectral Density of a self-affine rough surface
• We can define the PSD using 5 parameters
– H: Hurst exponent, index of the fractal dimension
– h0: root mean square roughness
– qL: smallest wave vector of the considered PSD
– q0: long distance roll-off wave vector
– q1: short distance cut-off wave vector

Sample size L 0.03m
Spatial resolution dO 10-4m
Grid points N 300x300
rms roughness h0 10-4 –10-3 m
Hurst exponent H 0.5 – 0.9
Roll-off vector q0 5 qL

• We can define the PSD using 5 parameters
– H: Hurst exponent, index of the fractal dimension
– h0: root mean square roughness, height variance
– qL: smallest wave vector of the considered PSD
– q0: long distance roll-off wave vector
– q1: short distance cut-off wave vector

Different in silico surfaces
Full spectrum Half spectrum 1/4 spectrum 1/8 spectrum 1/16 spectrum
H = 0.5 H = 0.6 H = 0.7 H = 0.8 H = 0.9

CONTACT PROBLEMS
IN ELASTICITY

The typical experiment

Frictionless contact problem in elasticity / Signorini Problem
Effects of roughness on the area of contact and on the elastostatic friction
Elasticity Boundary
Value Problem
Contact
Condition
s
s : elasticity stress tensor
u: displacement
d: imposed displacements
W: domain
GD: Dirichlet boundary
GN: Neumann boundary
t, n: tangential, normal comp.
20
g: gap function

Iterative Signorini problem
• The classic Signorini problem linearizes the
non-penetration condition, but this is not
suited for complex obstacles such as our rough
surfaces
• To overcome this problem, we solve a sequence
of Signorini problems, updating the gap
function
• Instead of using a pre-computed Signed
Distance Function, we compute distances on
the fly

Example of Linearized Contact Constraints

NUMERICAL EXPERIMENTS
REAL AREA OF CONTACT

Theory of contact between rough surfaces
• Full contact breaks down in small islands, when
observed closely

Real area of contact
• The real area of contact is only a fraction of the
nominal area of contact between two bodies
• Its study is crucial to understand physical
phenomena
– Heat and charge conduction
– Force transfer across surfaces
• Experimental observations are difficult, when
not impossible

Real area of contact, analytical models
• There exist two prominent theoretical models
for the relation of load and real area of contact
– Bush-Gibson-Thomas (BGT) model [1975]
– Persson’s model [2000]
• BGT model considers the surface asperities as
acting separately on the elastic body
– No merging of contact islands
• Persson’s model assumes that the stress
diffuses over the contact area uniformly
– No isolation of contact clusters

Unification of the two models
• At low pressures, both models predict the same
asymptotic result:
s0: nominal pressure, A0: nominal contact
area
E*: normalized Young’s Modulus
: rms slope of the surface
k : proportionality constant
BGT: Persson:

Numerical setting
• To analyze the contact evolution, we push a
linear elastic cube onto the surface, with an
incremental normal force
• We impose a Poisson’s parameter of 0.45
(almost incompressible solid), Young Modulus
is unitary
• The cube is discretized with tetrahedra, we
employ two different mesh sizes on the contact
boundary: 50 mm and 25 mm
– 600x600 and 1200x1200 contact nodes respectively

Numerical results: variation of h0
• When varying the root mean square roughness, we
obtain different curves

Coefficient of proportionality varying h0

Comparison of two contact evolutions
h0 = 71 mm h0 = 710 mm
• Different rms roughness values dramatically change
the evolution for incremental load

Coefficient of proportionality varying H

Coefficient of proportionality varying q1

Area of contact at low, medium, and large loading
• Persson’s model is supposed to predict the
contact vs area relation at large loads
• There exists a recently developed model, by
Yastrebov, Anciaux, and Molinari, which states
that the contact evolution follows a power law
• Our experimental results are fitted better by a
polynomial of fourth order, for which we
studied the coefficients for different rough
surfaces

Comparison of Polynomial interpolation to other models

Comparison of Polynomial interpolation to other models
• Close ups reveal that our numerical results (red
dots) are fitted better by the polynomial, than by
the YAM power law
• Moreover: coefficients of the polynomials are easier
to derive

Polynomial coefficients varying h0

NUMERICAL EXPERIMENTS:
FRICTION

What is friction?
• Friction is a well-known, frequently observed,
never completely understood phenomenon
• Occurring in every contact, especially when
surfaces are rough
– But also when surfaces are very smooth
• Force opposed to sliding, transferred across
the contact surface
• Its study is crucial for many applications

Causes of friction
• Friction is a macroscopically perceived effect of
micro- and nano-scopic interactions
• We can distinguish between sources of friction
by the scale at which they act
– Microscopic scale – interaction with asperities
• Elasticity
• Plasticity
• Viscosity
– Nanoscopic scale
• Atomic interactions
• Chemical reactions

Experimental setting
• To measure the friction coefficient, we proceed
with a natural extension of our previous
experiment
• After an initial phase of vertical loading, we
displace the top of the cube and let it relax
– We then measure the forces acting on the bottom
surface of the cube at equilibrium
– Friction coefficient is obtained as the ratio of
tangential to normal force

Remarks on shear tests
• The elastic cube stops sliding when the shear force
induced by the displacement of the bottom is
balanced by the contact forces, which are produced
solely by elastic interactions (no friction imposed)
• To reach the final configuration, up to 20 hours of
parallel processing on 2048 cores were needed

The influence of rms roughness on friction
• Behavior of m with respect to h0
44

Qualitative plots: h0 = 710 mm
dC = dO
dC = 1/4 dO
dC = 1/2 dO
dC = 1/8 dO
45

Qualitative plots: h0 = 53 mm
dC = dO
dC = 1/4 dO
dC = 1/2 dO
dC = 1/8 dO
46

VERIFICATION:
TYRE ROAD INTERACTION

Verification: wet tyre-road interactions
• To verify our prediction, we need data for an
experiment with
– A smooth elastic surface in contact with a hard
corrugated substrate
– No adhesive effects
– No plasticity effects
– No viscous effects
• We therefore rely on published experiments of
rubber sliding on wet asphalt
– Mildly wet asphalt: no real aquaplaning
– Studies performed by USA and UK governments

Theory of wet tyre-road contact dynamics
• Persson et al. suggest that the loss of friction of
rubber on wet roads is due to the flooding of
the asphalt, which filters the high-frequency
details of the road texture
• The high-frequency details are responsible for
the hysteretic response of rubber: taking them
off, rubber is less stiff and braking is not
influenced by tangential velocity
• Water is confined in sealed pools, and acts as a
flat rigid obstacle, because of incompressibility

Example of water sealing
• Normally, the tyres
adhere to asphalt, and
small asperities excite
the rubber
• In presence of water,
most of the high
frequency details are
covered by water
Figure taken from “Rubber friction on wet and dry road surfaces: the sealing effect”, Persson et al.

Wet tyre-road interaction: experimental setting
• We covered our surfaces with different amounts of
water and computed the new resulting friction

Wet tyre-road interaction: results
• We found that the
elastostatic friction is
not affected by small
amounts of water (in the
plot, relative m in
percentage)
• Larger water coverage
(where aquaplaning
starts) reduces it
dramatically
• We studied the variation of the friction coefficient
for wet surfaces

Comparison of our prediction to real world data
• We compare the friction
coefficient we obtained
(top) in our previous
experiments with that
measured on wet roads
(American highways,
bottom)
• Good agreement
– Compatible location of the
maximum friction coefficient
– Slightly overestimated friction
coefficient

CONCLUSIONS

Goals achieved by this work
• We successfully applied the FEM to a rough
contact at micro-scale, by employing a massively
parallel MG-based contact solver
– To the best of our knowledge, nobody did it before us
• We validated the results against theoretical
models, and we were able to empirically bridge
two different theories
• We performed a broad parameter study
– area of contact depends on all roughness parameters

Goals achieved by this work (II)
• We computed elastostatic friction
– Never negligible, even for almost smooth surfaces
• We found heuristic bounds on the needed
mesh size, and showed that the wrong mesh
size can result in misleading results
– HPC is a mandatory requirement for future
development

Goals achieved by this work (III)
• We compared our results to measurements
realized in a compatible framework, wet tyre-
road interactions
– With the assumption that viscosity becomes
negligible, our results are in good agreement with
measured data
• We identified a possible explanation of the
aquaplaning regime, based on the water
coverage

Future work
• Now that we have performed a broad
parameter study, we can use more
sophisticated models, targeting a narrower
parameter range to include
– Viscous and plastic effects
– Adhesive forces
– …
• More accurate experiments, with a better
description of the rough surfaces will have to
be used to verify future studies

Algorithm of Iterative Signorini Problem

Validation and verification
• In computational science two steps are usually
required to prove the reliability of a method
– Validation against known theory
– Verification against experimental results
• Validation: we performed validation against
classic benchmarks (e.g. Hertzian contact), and
compared our results to BGT and Persson’s
model.
• Verification: need of experimental data for
friction

The need for experimental data
• Finding good, large, detailed datasets for this
kind of phenomena is hard
– Commercial interests protecting corporate
experiments
– Laboratory equipments expensive
• Our experiments have a limited range of
validity (because of the elastic model, and of
the roughness parameters we employed)

Advantages of our method
• With respect to analytical theories, which
assume that the PSD is uniform over the whole
surface, we can also simulate contact with
anisotropic or inhomogeneous surfaces
• Object do not need to have flat surfaces
• We can identify boundary effects (like the
footprint detachment in the shear tests)

Inhomogeneous anisotropic surface: wood

Scaling results (in press)

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