Contact mechanics I: basics
Georges Cailletaud 1 St´ ephanie Basseville 1,2 Vladislav A. Yastrebov 1
1
Centre des Mat´ eriaux, MINES ParisTech, CNRS UMR 7633
2
Laboratoire d’Ing´ enierie des Syst` emes de Versailles, UVSQ
WEMESURF short course on contact mechanics and tribology
Paris, France, 21-24 June 2010
Table of contents
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
Short historical sketch
Use and opposition to friction
Frictional heat - lighting of fire - more than [40 000 years ago].
Ancient Egypt -lubrication of surfaces with oil [5 000 years ago].
Short historical sketch
First studies on contact and friction
Leonardo da Vinci [1452-1519]
first friction laws and many other trobological topics;
Issak Newton [1687]
Newton’s third law for bodies interaction;
Guillaume Amontons [1699]
rediscovered firction laws;
Leonhard Euler [1707-1783]
roughness theory of friction;
From Leonardo da Vinci’s notebook
Roughness theory of friction
Short historical sketch
First studies on contact and friction
Charles-Augustin de Coulomb [1789]
friction independence on sliding velocity and roughness; the influence of the time of repose.
Heinrich Hertz [1881-1882]
the first study on contact of deformable solids;
Holm [1938],
Ernst and Merchant [1940], Bowden and Tabon [1942]
difference between apparent and real contact areas, adhesion theory.
Photoelasticity analysis of Hertz contact problem (shear stresses)
Apparent and real areas of contact
Practice VS theory
1900: Theory is several steps behind the practice
Theory Practice
Practice VS theory
1940: Theory is behind the practice
Theory Practice
Practice VS theory
1960: Theory catchs up with practice
Practice and Theory
Practice VS theory
1990: The trial-and-error testing becomming more and more difficult. Theory leads practice.
Practice Theory
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
Surface interaction properties
Surface properties:
Coefficient of friction Adhesion
Wear parameters
Surface interaction properties
Surface properties are not fundamental Coefficient of friction /
Adhesion /
Wear parameters /
Surface interaction properties
Surface properties are not fundamental Coefficient of friction /
Adhesion / Wear parameters /
Fundamental properties:
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
Surface:
chemical reactivity;
absorbtion capabilities;
surface energy;
compatibility of
surfaces;
Surface interaction properties
Surface properties are not fundamental Coefficient of friction /
Adhesion / Wear parameters /
Fundamental properties:
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
Surface:
chemical reactivity;
absorbtion capabilities;
surface energy;
compatibility of
surfaces;
Surface interaction properties
Surface properties are not fundamental Coefficient of friction /
Adhesion / Wear parameters /
Fundamental properties are interdependent /
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
Surface:
chemical reactivity;
absorbtion capabilities;
surface energy;
compatibility of
surfaces;
Surface interaction properties
Surface properties are not fundamental Coefficient of friction /
Adhesion / Wear parameters /
More fundamental properties solids are made of atoms;
atoms are linked by bonds;
many of the volume and surface properties are the properties of the bonds.
Fundamental properties are interdependent /
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
Surface:
chemical reactivity;
absorbtion capabilities;
surface energy;
compatibility of
surfaces;
Surface interaction properties
Surface properties are not fundamental Coefficient of friction /
Adhesion / Wear parameters /
More fundamental properties solids are made of atoms;
atoms are linked by bonds;
many of the volume and surface properties are the properties of the bonds.
Fundamental properties are interdependent /
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
Surface:
chemical reactivity;
absorbtion capabilities;
surface energy;
compatibility of
surfaces;
Material properties interdependence
Young’s modulus and yield strength interdependence [Rabinowicz, ]
Material properties interdependence
Penetration hardness and yield stress interdependence
[Rabinowicz, ]
Young’s modulus and melting temperature
interdependence [Rabinowicz, ]
Material properties interdependence
Thermal coefficient of expansion and Young’s modulus interdependence [Rabinowicz, ]
Surface energy and hardness interdependence
[Rabinowicz, ]
Real area of contact
Real area of contact depends on normal load:
real area of contact is proportional to the normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance:
contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials)
real area of contact increases with time;
surface energy:
the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
Real area of contact
Real area of contact depends on normal load:
real area of contact is proportional to the normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance:
contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials)
real area of contact increases with time;
surface energy:
the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
A r ∼ F
A
r- real contact area, F -
applied load
Real area of contact
Real area of contact depends on normal load:
real area of contact is proportional to the normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance:
contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials)
real area of contact increases with time;
surface energy:
the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
A r = F p
A
r- real contact area, F -
applied load; p - hardness.
Real area of contact
Real area of contact depends on normal load:
real area of contact is proportional to the normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance:
contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials)
real area of contact increases with time;
surface energy:
the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
A r = F
p
Real area of contact
Real area of contact depends on normal load:
real area of contact is proportional to the normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance:
contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials)
real area of contact increases with time;
surface energy:
the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
A r = F
p
Real area of contact
Real area of contact depends on normal load:
real area of contact is proportional to the normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance:
contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials)
real area of contact increases with time;
surface energy:
the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
A r = F
p
Engineering friction
First approximations: friction coefficient does not depend on normal load
apparent area of contact velocity
sliding surface roughness time
Friction force direction is opposite to the sliding
Engineering friction
First approximations: friction coefficient does not depend on normal load ,
apparent area of contact , velocity /
sliding surface roughness / / , time / / ,
Friction force direction is opposite to the sliding ,
Real friction :: normal load
First approximation:
friction coefficient does not depend on normal load.
Exceptions:
at micro scale for small slidings (fig. 1);
for very large normal loads (metal forming) friction force is limited;
for very hard (diamond) or very soft (teflon) materials:
generally T = cF
α, α ∈ ˆ
23
; 1 ˜
;
thin hard coating and a softer substrate (fig.2).
Real friction :: normal load
First approximation:
friction coefficient does not depend on normal load.
Fig. 1. For very small sliding, the force of friction is not proportional to the normal force [Rabinowicz, ]
Exceptions:
at micro scale for small slidings (fig. 1);
for very large normal loads (metal forming) friction force is limited;
for very hard (diamond) or very soft (teflon) materials:
generally T = cF
α, α ∈ ˆ
23
; 1 ˜
;
thin hard coating and a softer substrate (fig.2).
Real friction :: normal load
First approximation:
friction coefficient does not depend on normal load.
Fig. 1. For very small sliding, the force of friction is not proportional to the normal force [Rabinowicz, ]
Exceptions:
at micro scale for small slidings (fig. 1);
for very large normal loads (metal forming) friction force is limited;
for very hard (diamond) or very soft (teflon) materials:
generally T = cF
α, α ∈ ˆ
23
; 1 ˜
;
thin hard coating and a softer substrate (fig.2).
Real friction :: normal load
First approximation:
friction coefficient does not depend on normal load.
Fig. 1. For very small sliding, the force of friction is not proportional to the normal force [Rabinowicz, ]
Exceptions:
at micro scale for small slidings (fig. 1);
for very large normal loads (metal forming) friction force is limited;
for very hard (diamond) or very soft (teflon) materials:
generally T = cF
α, α ∈ ˆ
23
; 1 ˜
;
thin hard coating and a softer substrate (fig.2).
Real friction :: normal load
First approximation:
friction coefficient does not depend on normal load.
Fig. 1. For very small sliding, the force of friction is not proportional to the normal force [Rabinowicz, ]
Exceptions:
at micro scale for small slidings (fig. 1);
for very large normal loads (metal forming) friction force is limited;
for very hard (diamond) or very soft (teflon) materials:
generally T = cF
α, α ∈ ˆ
23
; 1 ˜
;
thin hard coating and a softer substrate (fig.2).
Fig. 2. In case of hard surface layer on a softer substrate, at moderate loads friction is determined by the hard surface, higher load brakes the coating and softer material begins
to define the frictional properties[Rabinowicz, ]
Real friction :: normal force
Friction coefficient versus tangential movement; experiments from
[Courtney-Pratt and Eisner, 1957]
Real friction :: friction direction
First approximation:
friction force direction is opposite to the sliding.
Exceptions:
the direction of the friction force remains within [178; 182] degrees to sliding direction (fig. 1);
the difference is higher for oriented surface
roughnesses.
Real friction :: friction direction
First approximation:
friction force direction is opposite to the sliding.
Exceptions:
the direction of the friction force remains within [178; 182] degrees to sliding direction (fig. 1);
the difference is higher for oriented surface roughnesses.
Fig. 1. Change of the direction of friction force with sliding
[Rabinowicz, ]
Real friction :: friction direction
First approximation:
friction force direction is opposite to the sliding.
Exceptions:
the direction of the friction force remains within [178; 182] degrees to sliding direction (fig. 1);
the difference is higher for oriented surface roughnesses.
Fig. 1. Change of the direction of friction force with sliding
[Rabinowicz, ]
Real friction :: apparent area and roughness
First approximation:
Friction coefficient does not depend on the apparent area of contact.
Exceptions:
very smooth and very clean surfaces.
First approximation:
Friction coefficient does not depend on sliding surface roughness.
Exceptions:
very smooth or very rough surfaces
(fig. 1).
Real friction :: apparent area and roughness
First approximation:
Friction coefficient does not depend on the apparent area of contact.
Exceptions:
very smooth and very clean surfaces.
First approximation:
Friction coefficient does not depend on sliding surface roughness.
Exceptions:
very smooth or very rough surfaces
(fig. 1).
Real friction :: apparent area and roughness
First approximation:
Friction coefficient does not depend on the apparent area of contact.
Exceptions:
very smooth and very clean surfaces.
First approximation:
Friction coefficient does not depend on sliding surface roughness.
Exceptions:
very smooth or very rough surfaces (fig. 1).
Fig. 1. Friction roughness influences the coefficient of friction [Rabinowicz, ]
Real friction :: time and velocity
First approximation:
Friction coefficient does not depend on time.
Exceptions:
creeping materials.
First approximation:
Friction coefficient does not depend on sliding velocity.
Exceptions:
if material behaves differently at
different loading rate, then the
friction depends on the sliding
velocity;
Real friction :: time and velocity
First approximation:
Friction coefficient does not depend on time.
Exceptions:
creeping materials.
First approximation:
Friction coefficient does not depend on sliding velocity.
Exceptions:
if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
Static coefficient of friction evolution with time
Real friction :: time and velocity
First approximation:
Friction coefficient does not depend on time.
Exceptions:
creeping materials.
First approximation:
Friction coefficient does not depend on sliding velocity.
Exceptions:
if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
Static coefficient of friction evolution with time Kinetic friction decreases with increasing sliding
velosity
Real friction :: velocity
First approximation:
Friction coefficient does not depend on sliding velocity.
Exceptions:
if material behaves differently at
different loading rate, then the
friction depends on the sliding
velocity;
Real friction :: velocity
Friction coefficient slightly decreses with increasing velocity of sliding, titanium on
titanium [Rabinowicz, ]
First approximation:
Friction coefficient does not depend on sliding velocity.
Exceptions:
if material behaves differently at
different loading rate, then the
friction depends on the sliding
velocity;
Real friction :: velocity
Friction coefficient slightly decreses with increasing velocity of sliding, titanium on
titanium [Rabinowicz, ]
First approximation:
Friction coefficient does not depend on sliding velocity.
Exceptions:
if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
Friction coefficient dependence on velocity of
sliding for lubricated surfaces [Rabinowicz, ]
Real friction :: velocity
Friction coefficient increases and decreases with increasing velocity of sliding, hard on soft (steel
on lead, steel on indium) [Rabinowicz, ]
First approximation:
Friction coefficient does not depend on sliding velocity.
Exceptions:
if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
Friction coefficient dependence on velocity of
sliding for lubricated surfaces [Rabinowicz, ]
Three scales of contact study
Nanoscale:
Study of molecular junctions, van des Waals forces and Casimir effect.
Microscale:
Roughness and microstructure study
Macroscale:
Stress-strain state of contacting
solids
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
Macroscopic contact
Signorini contact law (1933)
r
u n
n
F n ≤ 0 u n ≤ 0 F n u n = 0 Compliance contact law
r
u n
n
[Kragelsky, 1982], [Oden-Martins, 1985]
−F n = C n (u n ) m +
n[Song, Yovanovich, 1987]
−F n = C 1 e c
2u
2nHertz theory (1882)
Geometry of smooth, non-conforming surface in contact
P z
0
S1
δ uz
2
P δ2
δ2
δ2 x−y plane
2 a δ1
uz
1
S2
Expression of the profile of each surface
z
1= 1 2R
10x
12+ 1
2R
100y
12z
2= −
„ 1
2R
20x
12+ 1 2R
200y
22«
where R
0iand R
i00are the principal radii of curvature of the surface i.
Separation between the two surfaces
h = z
1− z
2= Ax
2+ By
2Displacement
u
z1+ u
z2+ h = δ
1+ δ
2[Johnson, 1996]
Hertz theory (1882)
Assumptions in the Hertz theory:
The surface are continuous and non-conforming, a << R The strains are small, a << R
Each solid can be considered as an elastic half-space, a << R
1,2, a << l The surfaces are frictionless, q − x = q
y= 0
Applications
1 Solids of revolution
2 Two-dimensional contact of cylindrical bodies Note
1
E
∗= 1 − ν
1E
1+ 1 − ν
2E
2Hertz theory : Solids of revolution
Simple case of solids of revolution Principal radii of curvature
R
0i= R
i00= R
i, i = 1,2 Boundary conditions for the displacement
u
z1+ u
z2= δ − (1/2R)r
2Pressure distribution
p = p
0{1 − (r/a)
2}
1/2Consequences
Pressure
p
0= 6PE
∗2P
3R
2!
1/3Radius of the contact circle
a =
„ 3PR
4E
∗2«
1/3Displacement
δ = 9P
216RE
∗2!
1/3Hertz theory : Solids of revolution
Distributions of stresses
σr
p0
(x = 0, z) =
σθp0
(x = 0, z)
= −(1 + ν){1 − (z/a)tan
−1(a/z)}
+
12(1 + z
2/a
2)
−1σz
p0
(x = 0,z) = −(1 + z
2/a
2)
−1Maximum shear stress τ
1=
12|σ
r− σ
θ|
(τ
1)
max= 0.31p
0at the deph of 0.48a (for ν = 0.3)
2D contact of cylindrical bodies
O
x
z
a a
p0
p(x)
q(x)
M(x, y) σxx
σzz τxz
1
E∗
=
1−νE 11
+
1−νE 22 1
R
=
R11
+
R12
p(x) = p
0(1 − (x/a)
2)
−1/2a =
q
4PR πE∗p
0= q
PE∗πR
2D contact of cylindrical bodies
Example : cylinder/plate
Distributions of normal pressure (Hertz) and tangential stress
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σ
xx(x = 0, z) = −
pa0{(a
2+ 2z
2)(a
2+ z
2)
−1/2− 2z}
σ
zz(x = 0,z) = −p
0a(a
2+ z
2)
−1/2τ
max(x = 0, z) = p
0a{z − z
2(a
2− z
2)
−1/2}
σ
xz= σ
xy= σ
yz= 0
Macroscopic friction 1/2
Tresca
8
> >
> >
> <
> >
> >
> :
|F
t| ≤ g
If |F
t| < g, then V
slide= 0
If |F
t| = g, ∃λ > 0 such V
slide= −λF
tCoulomb
8
> >
> >
> <
> >
> >
> :
|F
t| ≤ µ|F
n|
If |F
t| < µ|F
n|, then V
slide= 0 stick
If |F
t| = µ|F
n|, ∃λ > 0 such V
slide= −λF
tslip
Macroscopic friction 2/2
Regularized Coulomb [Oden, Pires, 1983], [Raous, 1999]
|F t | = −µφ i (V slide ) |F n | φ 1 = √ V
slideV
slide2+ε
2φ 2 = tanh V
slideε
Variable friction
8
> >
> >
> <
> >
> >
> :
|F
t| ≤ C
t(u
t)
mtIf |F
t| < C
t(u
t)
mt, then V
slide= 0
If |F
t| = C
t(u
t)
mt, ∃λ > 0 such V
slide= −λR
tTransition toward the slip
Definition of sliding
Relative peripheral velocity of the surfaces at their point of contact Sliding of non-conforming elastic bodies
S Fixed slider
V
P Q
Q x
z S2
1
2a
Sliding contact
Question
The tangential traction due to the friction at the contact surface influences the size and shape of the contact area or the distribution of normal pressure ? Evaluation of the elastic stresses and displacements Basic premise of the Hertz theory
Relationship between the tangential traction and the normal pressure
|q(x, y)|
p(x, y) = |Q|
P = µ Coulomb’s law Application
Cylinder sliding perpendicular to its axis
[Johnson, 1996, Goryacheva, 1998]
Cylinder sliding perpendicular to its axis
Distributions of normal pressure (Hertz) and tangential traction p(x ) =
πa2Pq
1 − (
xa)
2q(x ) = ±µp(x ) = ±µ
2Pπaq 1 − (
xa)
2(1)
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Stress components
σxx(x,z= 0) = −p0 nq
1−(x a)2 + 2µx
a o
σzz(x,z= 0) = −p0 q
1−(x a)2 σyy(x,z= 0) = ν(σxx+σzz)
τxz(x,z= 0) = −µp0 q
1−(x a)2 τxy(x,z= 0) = τyz(x,z= 0) = 0
Partial slip
Relation between slip zone (c) and contact zone (a)
−a +a
q1(x) =µ p0
1−x2 a2
1/2
+ =
−a−c +c+a
q2(x) =µc ap0
1−x2
c2 1/2
q(x) =q1(x)−q2(x)
−a−c +c+a glissement
zone coll´ee
c a =
s 1 − Q
µP
If x < c : stick condition. The local contact shear stress is τ
xz= µp
0r 1 − ( x
a )
2− c a µp
0r 1 − ( x
c )
2If c < x < a: slip condition. The local contact shear stress is
τ
xz= µp
0r
1 − ( x
a )
2Friction instability : “stick-slip” 1/4
Definiton of the stick-slip :
Intermittent relative motion between the contact surfaces, alternation of slip and stick.
time F
F
s
d
time F
time F
Phenomenon occurs at various scales:
Macroscopic : discontinuities in the gravity center displacement of contact body and loads.
Microscopic : location of the phenomenon at the interface
Friction instability : “stick-slip” 2/4
The stick-slip is a coupling result :
The dynamic response of the friction system stiffness, damping, inertia
Friction dynamic at the interface
Difference between static (µ
s) and dynamic (µ
d) friction coefficient µ
sand µ
ddependence on the sliding velocity and time
A simple stick-slip model
m k F
X
v
Friction law
F
v F
F
s d
Fig : Plot of frictional force vs. time.
Friction instability : “stick-slip” 3/4
During sliding, the problem is:
m¨ x − F
d= −kx x(0) = F
sk x(0) = ˙ v and the solution is
x(t) = 1
k {(F
s− F
d)cos(ωt) + F
d} + v
ω sin(ωt), ω =
r k
m
or the velocity v is negligible compared to dx/dt:
x(t) ≈ 1
k {(F
s− F
d)cos(ωt) + F
d}
Characteristic time of sliding
T
inertia= 2π
r m
k
The force F oscillates between F
sand 2F
d− F
s.
F
t F
2F − Fd s
Tinertia
s
Friction instability : “stick-slip” 4/4
Fig : Regions of stable and stick-slip motion.
The red curve in the parameters plane at the other parameters being fixed, demarcates the regions
of stable and unstable motion.
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
Examples
(a) Vickers indenta- tion test, palladium glasses
(b) Contact zone under Vickers indenter, zirconium glasses
(c) Sracth resistance of soda-Lime Silica Glasses
Hill’s theory: Elastic-plastic indentation
Cavity model of an
elastic-plastic indentation cone
da Plastic
Elastic dh dc c
da a
da
a β
core r du(r)
Assumptions Within the core:
Hydrostatic component of stress p Outside the core:
Radial symmetry for stresses and displacement At the interface ( between core and plastic zone) Hydrostatic stress (in the core)= radial component of stress (in the external zone)
The radial displacement on r=a during an increment dh
must accommodate the volume of material.
Characteristic
In the plastic zone: a ≤ r ≤ c
σr
Y
= −2ln(c/r) − 2/3
σθ
Y
= −2ln(c/r) + 1/3
In the elastic zone: r ≥ c
σr
Y
= −
23`
cr
´
3σθ
Y
=
13`
cr
´
3where Y denotes the value of the yield strees of material in simple shear and simple compression.
Core pressure
p
Y
= −[
σrY]
r=a=
23+ 2ln(c/a) Radial displacement
du(r)
dc
=
TE{3(1 − ν)(c
2/r
2) − 2(1 − 2ν)(r/c)}
Conservation volume
2πa
2du(a) = πa
2dh = πa
2tan(β)da
9
> >
> >
> >
> >
> >
> >
> >
> =
> >
> >
> >
> >
> >
> >
> >
> ;
Pressure in the core, for an incompressible material
p Y = 2
3
1 + ln
„ Etan(β) 3Y
«ff
Unloading indentation: elastic strain energy
Example: spherical indenter
R R’
a a
R ρ
a a
Before loading Under loading After unloading
Residual depth δ − δ
0: Estimation of the energy dissipated ∆W in one cycle of the load
∆W = α R
Pdδ where α is the hysteresis-loss factor. (α = 0,4% for hard bearing steel) W =
25„
9E∗2 P5 16R
«
4a
3“
1 R
−
1ρ”
=
E3P∗a = “
3P 4E0
”
1/3δ =
aR2= “
9P2 16RE03
”
1/39
> >
> >
> >
> >
> >
> =
> >
> >
> >
> >
> >
> ;
δ
0=
9πPpm16E02
with p
m= 0.38Y in fully plastic state
P
PY
= 0,38(δ
0/δ
Y)
2Sharp indentation
Schematic illustration of a typical P-h reponse of an elasto-plastic material
Characterisation of P-h reponse During the loading, P = Ch
2Kick’s law During the unloading,
dPu dh
˛
˛
˛
hminitial slope
h
rResidual indentation depth after complete unloading Three independent quantities
[?]
Plastic behavior
The power law elasto-plastic stress-strain behavior
Model σ =
8
> <
> :
Eε for σ ≤ σ
yRε
nfor σ ≥ σ
ywith
E Young’s modolus R a strength coefficient n the strain hardening exponent σ
ythe initial yield stress Assumption :
The theory of plasticity with the von Mises effect stress.
Parameters for an elasto-plastic behavior
E, ν, σ
y, n
Dimensional analysis
Objective
Prediction of the P − h reponse from elasto-plastic properties Application of the universal dimensionless functions : the Π theorem
Material parameter set
(E, ν, σ
y, n) or (E, ν, σ
r,n) or (E, ν, σ
y, σ
r)
Load P
P = P(h, E
∗, σ
y, n) or P = P(h, E
∗, σ
r,n) or P = P(E, ν, σ
y, σ
r) with
E
∗= 1 − ν
2E + 1 − ν
2iE
i!
−1Unload
P
u= P
u(h,h
m, E, ν, E
i, ν
i, σ
r,n) or P
u= P
u(h,h
m, E
∗, σ
r, n)
Determination of h m
Application of the dimensional analysis during the load
Load Applying the Π theorem in dimensional analysis P = P(h, E
∗, σ
y,n) C =
Ph2
= σ
rΠ
1“
E∗ σr, n ”
P = P(h,E
∗, σ
r, n) C =
Ph2
= σ
yΠ
A1“
E∗ σy
,
σyσr”
P = P(E, ν, σ
y, σ
r) C =
Ph2
= σ
rΠ
B1“
E∗ σr
,
σσyr
”
with Π are dimensionless functions.
And then h
m!
Determination of h r
Application of the dimensional analysis during the unload
dP
udh = dP
udh (h, h
m,E
∗, σ
r, n) thus
dP
udh = E
∗hΠ
02„ h
mh , σ
rE
∗, n
«
Consequently,
dP
udh
˛
˛
˛
˛
h=hm= E
∗h
mΠ
02„ 1, σ
rE
∗, n
«
= E
∗h
mΠ
2„ σ
rE
∗, n
«
Or
P
u= P
u(h, h
m, E
∗, σ
r, n) = E
∗Π
u„ h
mh , σ
rE
∗, n
«
Finaly,
P
u= 0 implies 0 = Π
u„ h
mh
r, σ
rE
∗,n
«
whether h
rh
m= Π
3„ σ
rE
∗, n
«
and then h
r!
Π
i,i =1,2,3 can be used to relate the indentaion reponse to mechanical properties.
FE Vickers indentation test
Maximum penetration h
maxs3.11 µm
Parameters
Elastic E=81600MPa, ν = 0,36
Elasto-plastic model E=81600MPa, ν = 0,36, σ
y= 1610MPa
Drucker-Prager model E=81600MPa, ν = 0,36, σ
ty= 1600MPa, σ
cy= 1800MPa
[Laniel, 2004]
Computation results
Penetration depth
von Mises max Residual stress
Computation results
Elasto-plastic contact reponse
Computation results
Elasto-plastic contact reponse
Spherical indentation on a single crystal
Hypothesis
Sphere radius : R=100µm
Copper and zinc single crystals : crystal plasticity Silicon substrate : isotropic elastic
Maximum penetration h
maxs: 3.5 µ m
[Casal and Forest, 2009]
Contact reponse
(d)
Elastic anisotropic contact reponse(e)
Elasto-plastic anisotropic contact reponsevon Mises stress fields
Figure: (a) f.c.c and (b) h.c.p crystals. Penetration depth: h
s= 1.25µm
Plastic zone morphology
(a)
f.c.c copper crystals(b)
h.c.p zinc crystalsFigure: Penetration depth: h
s= 3.5µm
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
Bounds for the global coefficient of friction
9
Lubricant µ 1 = 0.2
P P P
i Coating µ 2 = 0.8
µ(x) = q(x)
p(x) µ =
R µ(x)p(x)dS R p(x )dS Uniform stress ≡ X
µ i f i ≤ µ ≤
P µ i c i f i P c i f i
≡ Uniform strain with c i = E
i(1−ν
i2) (1−ν
i)
2(1−2ν
i)
[Dick and Cailletaud, 2006]
Bounds for the global coefficient of friction
Cont A ν E (GPa) C (GPa) µ Cont B ν E (GPa) C (GPa) µ
Comp 1 0.32 8 11.45 0.1 Comp 1 0.15 55 58.08 0.1
Comp 2 0.15 55 58.08 0.5 Comp 2 0.32 08 11.45 0.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 20 40 60 80 100
µ
Component 2 (%) P1=P2 εy1=εy2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 20 40 60 80 100
µ
Component 2 (%) P1=P2 εy1=εy2
Case A: µ 1 = 0.1, E 1 = 11.45GPa Case B: µ 1 = 0.5, E 1 = 58GPa
µ 2 = 0.5, E 2 = 58GPa µ 2 = 0.1, E 2 = 11.45GPa
FE computations VS analytic estimation
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.2 0.4 0.6 0.8 1
COF
Component 2 (%) analytic analytic P1-10 P2-10
-8 -7 -6 -5 -4 -3 -2 -1 0
-0.01 -0.005 0 0.005 0.01
normalized contact pressure
x (mm)
CComp.1 CComp.2 P1-10: pComp.1, pComp.2 P2-10: pComp.1, pComp.2
comp.1
comp.2 comp.2
Different CSL geometries 1/2
Bulk material Component 1 Component 2
E (GPa) 119 8 55
ν 0.29 0.32 0.15
R0 ( MPa) - 200 500
Different CSL geometries 2/2
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.2 0.4 0.6 0.8 1
COF
Component 2 (%) analytic analytic S20_el R20_el L20_el Lxx_el
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.0150.02 0.025
normalized contact pressure
x (mm)
CComp.1 CComp.2 S20_el: pComp.1, pComp.2 R20_el: pComp.1, pComp.2 L20_el: pComp.1, pComp.2 Lxx_el: pComp.1, pComp.2
comp.2 comp.1 comp.2
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.0050.010.0150.02 0.025
normalized contact pressure
x (mm) CComp.1 CComp.2 S20_el: pComp.1, pComp.2 R20_el: pComp.1, pComp.2 L20_el: pComp.1, pComp.2 Lxx_el: pComp.1, pComp.2
comp.2 comp.1 comp.2
Influence of the number of dimples
10% comp 2 70% comp 2
ESD type 2a (µm) lESD(µm) nb.ESD lESD(µm) nb.ESD
R10 360 11.1 32.4 33.3 10.8
R20 360 22.2 16.2 66.6 5.4
R40 360 44.4 8.1 133.3 2.7
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.2 0.4 0.6 0.8 1
COF
Component 2 (%) R10_el R20_el R40_el S20_el S40_el
Influence of plastic deformations
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.2 0.4 0.6 0.8 1
COF
Component 2 (%) analytic analytic R40_el R40_pl S40_el S40_pl
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5
-0.04 -0.02 0 0.02 0.04
normalized contact pressure
x (mm) CComp.1 CComp.2 R40_el: pComp.1, pComp.2 R40_pl: pComp.1, pComp.2
comp.1
comp.2 comp.2
0 0.0038 0.00760.011
0.0150.019 0.0230.027
0.0310.035 0.0380.042
0.0460.05