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Cailletaud2010-Contact_mechanics_I.pdf
Contact mechanics I: basics
Georges Cailletaud1
Stéphanie Basseville1,2
Vladislav A. Yastrebov1
1 Centre des Matériaux, MINES ParisTech,
2 Laboratoire d’Ingénierie des Systèmes de
CNRS UMR 7633
Versailles, UVSQ
WEMESURF short course on contact mechanics and tribology
Paris, France, 21-24 June 2010
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
2/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
3/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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].
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
4/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Short historical sketch
First studies on contact and friction
Leonardo da Vinci [1452-1519]
first friction laws and many
other trobological topics;
From Leonardo da Vinci’s notebook
Issak Newton [1687]
Newton’s third law for bodies
interaction;
Guillaume Amontons [1699]
rediscovered firction laws;
Leonhard Euler [1707-1783]
roughness theory of friction;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Roughness theory of friction
Contact mechanics I
Paris, 21-24 June 2010
5/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Photoelasticity analysis of Hertz
contact problem (shear stresses)
Apparent and real areas of contact
Contact mechanics I
Paris, 21-24 June 2010
6/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Practice VS theory
1900: Theory is several steps behind the practice
Theory
Practice
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
7/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Practice VS theory
1940: Theory is behind the practice
Theory
Practice
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
7/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Practice VS theory
1960: Theory catchs up with practice
Practice and Theory
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
7/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Practice VS theory
1990: The trial-and-error testing becomming more and
more difficult. Theory leads practice.
Practice
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Theory
Paris, 21-24 June 2010
7/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
8/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Surface interaction properties
Surface properties:
Coefficient of friction
Adhesion
Wear parameters
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
9/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Surface interaction properties
Surface properties are not fundamental
Coefficient of friction /
Adhesion /
Wear parameters /
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
9/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Surface interaction properties
Surface properties are not fundamental
Coefficient of friction /
Adhesion /
Fundamental properties:
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
Wear parameters /
Surface:
chemical reactivity;
absorbtion
capabilities;
surface energy;
compatibility of
surfaces;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
9/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Surface interaction properties
Surface properties are not fundamental
Coefficient of friction /
Adhesion /
Fundamental properties:
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
Wear parameters /
Surface:
chemical reactivity;
absorbtion
capabilities;
surface energy;
compatibility of
surfaces;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
9/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Surface interaction properties
Surface properties are not fundamental
Coefficient of friction /
Adhesion /
Fundamental properties
are interdependent /
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
Wear parameters /
Surface:
chemical reactivity;
absorbtion
capabilities;
surface energy;
compatibility of
surfaces;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
9/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Surface interaction properties
Surface properties are not fundamental
Coefficient of friction /
Adhesion /
Fundamental properties
are interdependent /
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
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.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Surface:
chemical reactivity;
absorbtion
capabilities;
surface energy;
compatibility of
surfaces;
Contact mechanics I
Paris, 21-24 June 2010
9/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Surface interaction properties
Surface properties are not fundamental
Coefficient of friction /
Adhesion /
Fundamental properties
are interdependent /
Volume:
Young’s modulus;
Poisson’s ratio;
shear modulus;
yield stress;
elastic energy;
thermal properties.
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.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Surface:
chemical reactivity;
absorbtion
capabilities;
surface energy;
compatibility of
surfaces;
Contact mechanics I
Paris, 21-24 June 2010
9/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Material properties interdependence
Young’s modulus and yield strength interdependence [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
10/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Material properties interdependence
Penetration hardness and yield
stress interdependence
[Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Young’s modulus and melting temperature
interdependence [Rabinowicz, ]
Contact mechanics I
Paris, 21-24 June 2010
11/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Material properties interdependence
Thermal coefficient of expansion
and Young’s modulus
interdependence [Rabinowicz, ]
Surface energy and hardness interdependence
[Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
12/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
13/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Real area of contact
Real area of contact depends on
normal load:
Ar ∼ F
real area of contact is proportional to the
normal load; coefficient of proportionality is
inverse of the material hardness;
Ar - real contact area, F applied load
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]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
13/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Real area of contact
Real area of contact depends on
normal load:
Ar =
real area of contact is proportional to the
normal load; coefficient of proportionality is
inverse of the material hardness;
F
p
Ar - real contact area, F applied load; p - 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]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
13/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Real area of contact
Real area of contact depends on
normal load:
Ar =
real area of contact is proportional to the
normal load; coefficient of proportionality is
inverse of the material hardness;
F
p
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]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
13/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Real area of contact
Real area of contact depends on
normal load:
Ar =
real area of contact is proportional to the
normal load; coefficient of proportionality is
inverse of the material hardness;
F
p
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]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
13/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Real area of contact
Real area of contact depends on
normal load:
Ar =
real area of contact is proportional to the
normal load; coefficient of proportionality is
inverse of the material hardness;
F
p
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]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
13/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
14/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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 ,
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
14/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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:
ˆ
˜
generally T = cF α , α ∈ 23 ; 1 ;
thin hard coating and a softer substrate (fig.2).
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
15/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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:
ˆ
˜
generally T = cF α , α ∈ 23 ; 1 ;
thin hard coating and a softer substrate (fig.2).
Fig. 1. For very small sliding, the
force of friction is not proportional
to the normal force [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
15/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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:
ˆ
˜
generally T = cF α , α ∈ 23 ; 1 ;
thin hard coating and a softer substrate (fig.2).
Fig. 1. For very small sliding, the
force of friction is not proportional
to the normal force [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
15/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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:
ˆ
˜
generally T = cF α , α ∈ 23 ; 1 ;
thin hard coating and a softer substrate (fig.2).
Fig. 1. For very small sliding, the
force of friction is not proportional
to the normal force [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
15/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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:
ˆ
˜
generally T = cF α , α ∈ 23 ; 1 ;
thin hard coating and a softer substrate (fig.2).
Fig. 1. For very small sliding, the
force of friction is not proportional
to the normal force [Rabinowicz, ]
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, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
15/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Real friction :: normal force
Friction coefficient versus tangential movement; experiments from
[Courtney-Pratt and Eisner, 1957]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
16/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
17/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
17/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
17/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
First approximation:
Friction coefficient does not depend
on sliding surface roughness.
Exceptions:
very smooth or very rough surfaces
(fig. 1).
Contact mechanics I
Paris, 21-24 June 2010
18/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
First approximation:
Friction coefficient does not depend
on sliding surface roughness.
Exceptions:
very smooth or very rough surfaces
(fig. 1).
Contact mechanics I
Paris, 21-24 June 2010
18/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
18/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Real friction :: time and velocity
First approximation:
Friction coefficient does not depend
on time.
Exceptions:
creeping materials.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
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;
Contact mechanics I
Paris, 21-24 June 2010
19/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
19/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Real friction :: time and velocity
First approximation:
Friction coefficient does not depend
on time.
Exceptions:
creeping materials.
Static coefficient of friction evolution with time
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
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;
Kinetic friction decreases with increasing sliding
velosity
Contact mechanics I
Paris, 21-24 June 2010
19/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
20/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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;
Friction coefficient slightly decreses with
increasing velocity of sliding, titanium on
titanium [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
20/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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;
Friction coefficient slightly decreses with
increasing velocity of sliding, titanium on
titanium [Rabinowicz, ]
Friction coefficient dependence on velocity of
sliding for lubricated surfaces [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
20/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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;
Friction coefficient increases and decreases with
increasing velocity of sliding, hard on soft (steel
on lead, steel on indium) [Rabinowicz, ]
Friction coefficient dependence on velocity of
sliding for lubricated surfaces [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
20/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
21/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
22/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Macroscopic contact
Signorini contact law (1933)
r
n
un

Fn ≤ 0





un ≤ 0





Fn un = 0
Compliance contact law
r
n
[Kragelsky, 1982], [Oden-Martins, 1985]
n
−Fn = Cn (un )m
+
un
[Song, Yovanovich, 1987]
2
−Fn = C1 e c2 un
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
23/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Hertz theory (1882)
Geometry of smooth, non-conforming surface in contact
Expression of the profile of each surface
P
z1 =
δ2
1
1 2
2
x +
y
2R10 1
2R100 1
„
z2 = −
S2
uz
δ2
δ
1
0
δ2
2 a
z
S1
δ1
x−y plane
uz2
P
1 2
1
2
x +
y
2R20 1
2R200 2
«
where Ri0 and Ri00 are the principal radii of curvature
of the surface i.
Separation between the two surfaces
2
h = z1 − z2 = Ax + By
2
Displacement
uz1 + uz2 + h = δ1 + δ2
[Johnson, 1996]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
24/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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 << R1,2 , a << l
The surfaces are frictionless, q − x = qy = 0
Applications
1 Solids of revolution
2 Two-dimensional contact of cylindrical bodies
Note
1
1 − ν1
1 − ν2
+
=
E∗
E1
E2
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
25/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Hertz theory : Solids of revolution
Simple case of solids of revolution
Principal radii of curvature
00
0
Ri = Ri = Ri ,
i = 1, 2
Boundary conditions for the displacement
uz1 + uz2 = δ − (1/2R)r
Pressure distribution
2
2 1/2
p = p0 {1 − (r /a) }
Consequences
Pressure
6PE ∗2
P3R2
p0 =
Radius of the contact circle
„
a=
3PR
4E ∗2
!1/3
«1/3
Displacement
δ=
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
9P 2
16RE ∗2
!1/3
Contact mechanics I
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26/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Hertz theory : Solids of revolution
Distributions of stresses
σr
p0
(x = 0, z)
=
σθ
p0
=
−(1 + ν){1 − (z/a)tan−1 (a/z)}
(x = 0, z)
+ 12 (1 + z 2 /a2 )−1
σz
p0
(x = 0, z)
=
−(1 + z 2 /a2 )−1
Maximum shear stress τ1 = 21 |σr − σθ |
(τ1 )max = 0.31p0 at the deph of 0.48a (for ν = 0.3)
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Paris, 21-24 June 2010
27/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
2D contact of cylindrical bodies
p0
a
a
1
E∗
=
1−ν1
E1
+
1
R
=
1
R1
1
R2
p(x)
=
p0 (1 − (x/a)2 )−1/2
a
=
p0
=
1−ν2
E2
p(x)
O
x
q(x)
M (x, y)
σxx
+
q
4PR
πE ∗
q
PE ∗
πR
τxz
σzz
z
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Paris, 21-24 June 2010
28/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
2D contact of cylindrical bodies
Example : cylinder/plate
Distributions of normal pressure (Hertz) and tangential stress
!# %" $ +( *
3 15476
!# "%$ &')( *
PRQTSVU
8
, $- $ +( *
01 2
@DAE @CD5;B F <=;?>
G?KLHJIMOGN
W XY Z
ced fhg
, ./ . +( *
[\=] ]
^`_ aJb
p0
a
01 2
9:1 2
{(a2 + 2z 2 )(a2 + z 2 )−1/2 − 2z}
σxx (x = 0, z)
=
−
σzz (x = 0, z)
=
−p0 a(a2 + z 2 )−1/2
τmax (x = 0, z)
=
p0 a{z − z 2 (a2 − z 2 )−1/2 }
σxz
=
σxy = σyz = 0
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Contact mechanics I
Paris, 21-24 June 2010
29/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Macroscopic friction 1/2
Tresca
8
|Ft | ≤ g
>
>
>
>
>
<
If |Ft | < g , then Vslide = 0
>
>
>
>
>
:
If |Ft | = g , ∃λ > 0 such Vslide = −λFt
Coulomb
8
|Ft | ≤ µ|Fn |
>
>
>
>
>
<
If |Ft | < µ|Fn |, then Vslide = 0 stick
>
>
>
>
>
:
If |Ft | = µ|Fn |, ∃λ > 0 such Vslide = −λFt
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
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slip
30/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Macroscopic friction 2/2
Regularized Coulomb [Oden, Pires, 1983], [Raous, 1999]
|Ft | = −µφi (Vslide ) |Fn |
φ1 = √ V2slide 2
Vslide +ε
φ2 = tanh Vslide
ε
Variable friction
8
|Ft | ≤ Ct (ut )mt
>
>
>
>
>
<
If |Ft | < Ct (ut )mt , then Vslide = 0
>
>
>
>
>
:
If |Ft | = Ct (ut )mt , ∃λ > 0 such Vslide = −λRt
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Transition toward the slip
Definition of sliding
Relative peripheral velocity of the surfaces at their point of contact
Sliding of non-conforming elastic bodies
Question
P
Fixed slider
The tangential traction due to the friction at the
S2
contact surface influences the size and shape of the
Q
contact area or the distribution of normal pressure ?
S1
V
Q
x
2a
z
Sliding contact
Evaluation of the elastic stresses and displacements
Basic premise of the Hertz theory
Relationship between the tangential traction and the
normal pressure
|Q|
|q(x, y )|
=
=µ
p(x, y )
P
Coulomb’s law
Application
Cylinder sliding perpendicular to its axis
[Johnson, 1996, Goryacheva, 1998]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
Paris, 21-24 June 2010
32/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Cylinder sliding perpendicular to its axis
Distributions of normal pressure (Hertz) and tangential traction
q
2P
p(x) =
1 − ( xa )2
πa
(1)
q(x)
"$# !
MON P Q
8:9 8; <)=
[email protected] ?:B <'>
!
2P
±µp(x) = ±µ πa
=
WYX[Z]\
H$JLI K
TVU
RS
h ij k
q rs
C D EGF
tvuVw w
. +0/21
67+(" &%' (!
*,+(-
5 43 x
!
67+(" &% !
yz y[{
|} |~
€‚
¸¯´¯¹±µg­¯º³¶0®±»· °³²
…‡†
ƒ„
ˆ ‰Š ‹
€
*,+(-
*,+(-
σxx (x, z = 0)
=
−p0
nq
σzz (x, z = 0)
=
−p0
q
σyy (x, z = 0)
=
ν(σxx + σzz )
τxz (x, z = 0)
=
−µp0
τxy (x, z = 0)
=
τyz (x, z = 0) = 0
1 − ( x )2 + 2µ x
a
a
o
1 − ( x )2
a
3 «¬
¨ª©
£ ¤¦ § ¥
ž ¡
£ ¢
Ž‘“” • ’ –˜ ™ — šœ  › žŸ
Œ0
3 " %)%' (!
ab
^`_
lm npo
q
1 − ( x )2
a
3 " # !
. +0/21
Stress components
q
3 " %)%' (!
egf
cd
1 − ( xa )2
3 q
5 43 *,+(-
3 G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Partial slip
Relation between slip zone (c) and contact zone (a)
+c +a
−a −c
−a
glissement
=
+
+a
+c +a
−a −c
zone collée
1/2
x2
q1 (x) = µ p0 1 − 2
a
q2 (x) = µ
c
=
a
1/2
x2
c
p0 1 − 2
a
c
s
1−
q(x) = q1 (x) − q2 (x)
Q
µP
If x < c : stick condition. The local contact shear stress is
r
τxz = µp0
c
x
1 − ( )2 − µp0
a
a
r
x
1 − ( )2
c
If c < x < a: slip condition. The local contact shear stress is
r
τxz = µp0
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
x
1 − ( )2
a
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Friction instability : “stick-slip” 1/4
Definiton of the stick-slip :
Intermittent relative motion between the contact surfaces, alternation of slip and stick.
F
Fs
F
Fd
time
time
time
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
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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 (µs ) and dynamic (µd ) friction coefficient
µs and µd dependence on the sliding velocity and time
A simple stick-slip model
Friction law
F
F
Fs
k
X
Fd
m
v
v
Fig : Plot of frictional force vs. time.
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Friction instability : “stick-slip” 3/4
During sliding, the problem is:
mẍ − Fd = −kx
x(0) =
Fs
k
ẋ(0) = v
and the solution is
x(t) =
r
1
v
{(Fs − Fd )cos(ωt) + Fd } + sin(ωt),
k
ω
ω=
k
m
or the velocity v is negligible compared to dx/dt:
x(t) ≈
1
{(Fs − Fd )cos(ωt) + Fd }
k
F
Fs
Characteristic time of sliding
r
Tinertia = 2π
Tinertia
m
k
The force F oscillates between Fs and 2Fd − Fs .
2Fd − Fs
t
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
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Contact mechanics I
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Examples
(a) Vickers indenta- (b) Contact zone under
tion test, palladium Vickers indenter, zirconium
glasses
glasses
(c) Sracth resistance of soda-Lime Silica Glasses
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Hill’s theory: Elastic-plastic indentation
Cavity model of an
elastic-plastic indentation cone
da
da
a
dc
c
Assumptions
Within the core:
Hydrostatic component of stress p
a
core r
da
Plastic
β
dh
du(r)
Elastic
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.
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Characteristic
In the plastic zone: a ≤ r ≤ c
σr
= −2ln(c/r ) − 2/3
Y
σθ
Y
In the elastic zone:
` r´ ≥ c
σr
= − 23 cr 3
Y
−2ln(c/r ) + 1/3
=
σθ
Y
=
1
3
` c ´3
r
where Y denotes the value of the yield strees of material in simple shear and simple compression.
Core pressure
p
Y
= −[ σYr ]r =a =
2
3
+ 2ln(c/a)
Radial displacement
du(r )
dc
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
{3(1 − ν)(c 2 /r 2 ) − 2(1 − 2ν)(r /c)} >
>
>
>
>
>
>
>
Conservation volume
>
>
>
>
>
;
2πa2 du(a) = πa2 dh = πa2 tan(β)da
=
T
E
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Pressure in the core,
for an incompressible material

„
«ff
p
2
Etan(β)
=
1 + ln
Y
3
3Y
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Unloading indentation: elastic strain energy
Example: spherical indenter
R
R
R’
Before loading
a
a
Under loading
ρ
a
a
After unloading
Residual depth
δ − δ 0 : Estimation of the energy dissipated ∆W in one cycle of the load
R
∆W = α „ Pdδ where
« α is9the hysteresis-loss factor. (α = 0, 4% for hard bearing steel)
∗2 5
>
>
W = 52 9E16RP
>
>
>
>
>
“
”
>
>
3
1
1
3P
>
>
= E∗
4a
9πPpm
=
R − ρ
with pm = 0.38Y in fully plastic state
δ0
=
2
16E 0
”1/3
“
P
>
=
0,
38(δ 0 /δY )2
>
3P
P
>
Y
a = 4E 0
>
>
>
>
>
”1/3 >
“
>
2
2
>
a
9P
;
δ=
=
R
3
16RE 0
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Sharp indentation
Characterisation of P-h reponse
During the loading,
P = Ch2
Kick’s law
During the unloading,
˛
dPu ˛
initial slope
dh ˛
hm
hr
Schematic illustration of a typical P-h
reponse of an elasto-plastic material
Residual indentation depth
after complete unloading
Three independent quantities
[?]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Plastic behavior
Model
8
>
< Eε
σ=
>
: Rεn
with
The power law elasto-plastic stress-strain
behavior
E
R
n
σy
for
σ ≤ σy
for
σ ≥ σy
Young’s modolus
a strength coefficient
the strain hardening exponent
the initial yield stress
Assumption :
The theory of plasticity with the von Mises effect stress.
Parameters for an elasto-plastic behavior
E, ν, σy , n
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
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, σy , n)
or
(E , ν, σr , n)
or
(E , ν, σy , σr )
or
P = P(h, E , σr , n)
Load P
∗
P = P(h, E , σy , n)
∗
or
P = P(E , ν, σy , σr )
with
E
∗
=
1 − ν2
1 − νi2
+
E
Ei
!−1
Unload
Pu = Pu (h, hm , E , ν, Ei , νi , σr , n)
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
or
∗
Pu = Pu (h, hm , E , σr , n)
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Determination of hm
Application of the dimensional analysis during the load
Load
P = P(h, E ∗ , σy , n)
Applying the Π theorem in dimensional
“ ∗ ” analysis
C = P2 = σr Π1 Eσr , n
h
P = P(h, E ∗ , σr , n)
C =
P
h2
= σy ΠA
1
“
E∗
σy
,
σr
σy
”
P = P(E , ν, σy , σr )
C =
P
h2
= σr ΠB
1
“
E∗
σr
,
σy
σr
”
with Π are dimensionless functions.
And then hm !
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Determination of hr
Application of the dimensional analysis during the unload
dPu
dPu
∗
=
(h, hm , E , σr , n)
dh
dh
thus
dPu
∗
0
= E hΠ2
dh
„
hm σr
,
,n
h E∗
«
Consequently,
˛
«
„
«
„
dPu ˛˛
σr
σr
∗
∗
0
,
n
= E hm Π2 1, ∗ , n = E hm Π2
˛
dh h=hm
E
E∗
Or
∗
∗
Pu = Pu (h, hm , E , σr , n) = E Πu
„
hm σr
,
,n
h E∗
«
Finaly,
„
Pu = 0
implies
0 = Πu
hm σr
,
,n
hr E ∗
«
whether
hr
= Π3
hm
„
σr
,n
E∗
«
and then hr !
Πi ,i =1,2,3 can be used to relate the indentaion reponse to mechanical properties.
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
FE Vickers indentation test
Maximum penetration hsmax
3.11 µm
Parameters
Elastic
E=81600MPa, ν = 0, 36
Elasto-plastic model
E=81600MPa, ν = 0, 36,
σy = 1610MPa
Drucker-Prager model
E=81600MPa, ν = 0, 36,
σyt = 1600MPa, σyc = 1800MPa
[Laniel, 2004]
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Computation results
Penetration depth
von Misesmax
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Residual stress
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Computation results
Elasto-plastic contact reponse
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Computation results
Elasto-plastic contact reponse
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Spherical indentation on a single crystal
Hypothesis
Sphere radius : R=100µm
Copper and zinc single crystals : crystal plasticity
Silicon substrate : isotropic elastic
Maximum penetration hsmax : 3.5 µ m
[Casal and Forest, 2009]
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Contact reponse
(d)
Elastic anisotropic contact reponse
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
(e)
Elasto-plastic anisotropic contact reponse
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
von Mises stress fields
Figure: (a) f.c.c and (b) h.c.p crystals. Penetration depth: hs = 1.25µm
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Plastic zone morphology
(a)
f.c.c copper crystals
(b)
h.c.p zinc crystals
Figure: Penetration depth: hs = 3.5µm
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Bounds for the global coefficient of friction
Lubricant µ1 = 0.2
9
P
i
PP Coating µ2 = 0.8
q(x)
µ(x) =
p(x)
Uniform stress ≡
with ci =
X
R
µ=
µ(x)p(x)dS
R
p(x)dS
P
µi ci fi
µi fi ≤ µ ≤ P
≡ Uniform strain
ci fi
(1−νi )2
Ei
(1−νi2 ) (1−2νi )
[Dick and Cailletaud, 2006]
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Bounds for the global coefficient of friction
ν
0.32
0.15
E (GPa)
8
55
C (GPa)
11.45
58.08
µ
0.1
0.5
Cont B
Comp 1
Comp 2
0.5
0.5
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
µ
µ
Cont A
Comp 1
Comp 2
0.2
0.2
0.15
0.15
0.1
E (GPa)
55
08
C (GPa)
58.08
11.45
µ
0.1
0.5
0.1
P1=P2
εy1=εy2
0.05
0
ν
0.15
0.32
0
20
40
60
Component 2 (%)
80
P1=P2
εy1=εy2
0.05
100
Case A: µ1 = 0.1, E1 = 11.45GPa
µ2 = 0.5, E2 = 58GPa
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
0
0
20
40
60
Component 2 (%)
80
100
Case B: µ1 = 0.5, E1 = 58GPa
µ2 = 0.1, E2 = 11.45GPa
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
FE computations VS analytic estimation
0
-1
normalized contact pressure
0.5
0.45
0.4
COF
0.35
0.3
0.25
0.2
analytic
analytic
P1-10
P2-10
0.15
0.1
0
0.2
0.4
0.6
Component 2 (%)
0.8
-2
-3
-4
comp.2
comp.1
-6
CComp.1
CComp.2
P1-10: pComp.1, pComp.2
P2-10: pComp.1, pComp.2
-7
-8
1
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
comp.2
-5
-0.01
-0.005
Contact mechanics I
0
x (mm)
0.005
0.01
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Different CSL geometries 1/2
E (GPa)
ν
R0 ( MPa)
Bulk material
119
0.29
-
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Component 1
8
0.32
200
Component 2
55
0.15
500
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Different CSL geometries 2/2
0.5
0.45
0.4
COF
0.35
0.3
0.25
analytic
analytic
S20_el
R20_el
L20_el
Lxx_el
0.2
0.15
0.1
0
0.2
0.4
0.6
0.8
1
0
0
-1
-0.5
-2
-1
-3
-4
-5
-6
comp.2
comp.1
comp.2
-7
-8
-9
-10
-11
-0.025 -0.02 -0.015 -0.01 -0.005
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
0
0.005 0.01 0.015 0.02 0.025
x (mm)
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
normalized contact pressure
normalized contact pressure
Component 2 (%)
comp.2
comp.1
comp.2
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-0.025 -0.02 -0.015 -0.01 -0.005
Contact mechanics I
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
0
0.005 0.01 0.015 0.02 0.025
x (mm)
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Influence of the number of dimples
ESD type
R10
R20
R40
2a (µm)
360
360
360
10% comp 2
lESD (µm)
nb.ESD
11.1
32.4
22.2
16.2
44.4
8.1
70% comp 2
lESD (µm)
nb.ESD
33.3
10.8
66.6
5.4
133.3
2.7
0.5
0.45
0.4
COF
0.35
0.3
0.25
0.2
R10_el
R20_el
R40_el
S20_el
S40_el
0.15
0.1
0
0.2
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
0.4
0.6
Component 2 (%)
0.8
1
Contact mechanics I
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Influence of plastic deformations
0.5
-0.5
-1
0.45
normalized contact pressure
-1.5
0.4
COF
0.35
0.3
0.25
analytic
analytic
R40_el
R40_pl
S40_el
S40_pl
0.2
0.15
0.1
0
0.2
0.4
0.6
Component 2 (%)
0
0.8
-2
-2.5
-3
comp.2
comp.1
-4
-4.5
CComp.1
CComp.2
R40_el: pComp.1, pComp.2
R40_pl: pComp.1, pComp.2
-5
-5.5
1
comp.2
-3.5
-0.04
-0.02
0
x (mm)
0.02
0.04
0.0038
0.011
0.019
0.027
0.035
0.042
0.05
0.0076
0.015
0.023
0.031
0.038
0.046
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Conclusion
Estimation of the upper and lower bound.
The friction coefficient depend on
the CSL geometry
the dissimilarity of the CSL component materials
the compliance of substrate and counter body
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
Contact mechanics I
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Summary
Contact and friction
complicated phenomena;
depend on many material properties;
not yet well elaborated.
Analytical solutions
hertzian contact;
nonlinear material;
friction;
stick-slip instabilities.
Numerical analysis
examples of indendation tests;
analysis of heterogeneous friction.
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Thank you for your attention!
Georges Cailletaud <[email protected]>
Stéphanie Basseville <[email protected]>
Vladislav A. Yastrebov <[email protected]>
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Casal, O. and Forest, S. (2009).
Finite element crystal plasticity analysis of spherical indentation.
Computational Materials Science, 45:774–782.
Dick, T. and Cailletaud, G. (2006).
Analytic and FE based estimations of the coefficient of friction of composite
surfaces.
Wear, 260:1305–1316.
Goryacheva, I. (1998).
Contact mechanics in tribology.
London.
Johnson, K. (1996).
Contact mechanics.
Cambridge.
Laniel, R. (2004).
Simulation des procı̈¿½dı̈¿½s d’indentation et de rayage par ı̈¿½lı̈¿½ments finis et
ı̈¿½lı̈¿½ments disctincts.
Rabinowicz, E.
Friction and Wear.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ
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