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
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Contact mechanics I: basics
Georges Cailletaud1 Stephanie Basseville1,2
Vladislav A. Yastrebov1
1Centre des Materiaux, MINES ParisTech, CNRS UMR 76332Laboratoire d’Ingenierie des Systemes de 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 manyother trobological topics;
Issak Newton [1687]Newton’s third law for bodiesinteraction;
chemical reactivity;absorbtioncapabilities;surface energy;compatibility ofsurfaces;
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
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 yieldstress interdependence
[Rabinowicz, ]
Young’s modulus and melting temperatureinterdependence [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ 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 expansionand 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 thenormal load; coefficient of proportionality isinverse of the material hardness;
sliding distance:contact area might be 3(!) times as great asthe 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:real area of contact is proportional to thenormal load; coefficient of proportionality isinverse of the material hardness;
sliding distance:contact area might be 3(!) times as great asthe 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]
Ar ∼ F
Ar - real contact area, F -applied load
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:real area of contact is proportional to thenormal load; coefficient of proportionality isinverse of the material hardness;
sliding distance:contact area might be 3(!) times as great asthe 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]
Ar =F
p
Ar - real contact area, F -applied load; p - hardness.
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:real area of contact is proportional to thenormal load; coefficient of proportionality isinverse of the material hardness;
sliding distance:contact area might be 3(!) times as great asthe 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]
Ar =F
p
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:real area of contact is proportional to thenormal load; coefficient of proportionality isinverse of the material hardness;
sliding distance:contact area might be 3(!) times as great asthe 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]
Ar =F
p
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:real area of contact is proportional to thenormal load; coefficient of proportionality isinverse of the material hardness;
sliding distance:contact area might be 3(!) times as great asthe 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]
Ar =F
p
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 doesnot depend on normalload.
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 = cFα, α ∈ˆ
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 doesnot depend on normalload.
Fig. 1. For very small sliding, theforce of friction is not proportionalto 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 = cFα, α ∈ˆ
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 doesnot depend on normalload.
Fig. 1. For very small sliding, theforce of friction is not proportionalto 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 = cFα, α ∈ˆ
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 doesnot depend on normalload.
Fig. 1. For very small sliding, theforce of friction is not proportionalto 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 = cFα, α ∈ˆ
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 doesnot depend on normalload.
Fig. 1. For very small sliding, theforce of friction is not proportionalto 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 = cFα, 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 isdetermined 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 isopposite 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 surfaceroughnesses.
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 isopposite 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 surfaceroughnesses.
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 isopposite 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 surfaceroughnesses.
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 dependon the apparent area of contact.
Exceptions:very smooth and very clean surfaces.
First approximation:
Friction coefficient does not dependon sliding surface roughness.
Exceptions:very smooth or very rough surfaces(fig. 1).
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 :: apparent area and roughness
First approximation:
Friction coefficient does not dependon the apparent area of contact.
Exceptions:very smooth and very clean surfaces.
First approximation:
Friction coefficient does not dependon sliding surface roughness.
Exceptions:very smooth or very rough surfaces(fig. 1).
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 :: apparent area and roughness
First approximation:
Friction coefficient does not dependon the apparent area of contact.
Exceptions:very smooth and very clean surfaces.
First approximation:
Friction coefficient does not dependon 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 dependon time.
Exceptions:creeping materials.
First approximation:
Friction coefficient does not dependon sliding velocity.
Exceptions:if material behaves differently atdifferent loading rate, then thefriction depends on the slidingvelocity;
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 dependon time.
Exceptions:creeping materials.
First approximation:
Friction coefficient does not dependon sliding velocity.
Exceptions:if material behaves differently atdifferent loading rate, then thefriction depends on the slidingvelocity;
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 dependon time.
Exceptions:creeping materials.
First approximation:
Friction coefficient does not dependon sliding velocity.
Exceptions:if material behaves differently atdifferent loading rate, then thefriction depends on the slidingvelocity;
Static coefficient of friction evolution with time Kinetic friction decreases with increasing slidingvelosity
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 :: velocity
First approximation:
Friction coefficient does not dependon sliding velocity.
Exceptions:if material behaves differently atdifferent loading rate, then thefriction depends on the slidingvelocity;
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
Friction coefficient slightly decreses withincreasing velocity of sliding, titanium on
titanium [Rabinowicz, ]
First approximation:
Friction coefficient does not dependon sliding velocity.
Exceptions:if material behaves differently atdifferent loading rate, then thefriction depends on the slidingvelocity;
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
Friction coefficient slightly decreses withincreasing velocity of sliding, titanium on
titanium [Rabinowicz, ]
First approximation:
Friction coefficient does not dependon sliding velocity.
Exceptions:if material behaves differently atdifferent loading rate, then thefriction depends on the slidingvelocity;
Friction coefficient dependence on velocity ofsliding 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
Friction coefficient increases and decreases withincreasing velocity of sliding, hard on soft (steel
on lead, steel on indium) [Rabinowicz, ]
First approximation:
Friction coefficient does not dependon sliding velocity.
Exceptions:if material behaves differently atdifferent loading rate, then thefriction depends on the slidingvelocity;
Friction coefficient dependence on velocity ofsliding 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
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 31/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Transition toward the slip
Definition of slidingRelative peripheral velocity of the surfaces at their point of contactSliding of non-conforming elastic bodies
S
Fixed slider
V
P
Q
Qx
z
S2
1
2a
Sliding contact
QuestionThe tangential traction due to the friction at thecontact surface influences the size and shape of thecontact area or the distribution of normal pressure ?
Evaluation of the elastic stresses and displacementsBasic premise of the Hertz theory
Relationship between the tangential traction and thenormal pressure
|q(x, y)|p(x, y)
=|Q|P
= µ Coulomb’s law
ApplicationCylinder 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
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 33/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Partial slip
Relation between slip zone (c) and contact zone (a)
−a +a
q1(x) = µ p0
(
1−x
2
a2
)1/2
+ =
−a +a−c +c
q2(x) = µc
ap0
(
1−x
2
c2
)1/2
q(x) = q1(x) − q2(x)
−a +a−c +c
glissement
zone collee
c
a=
s1−
Q
µP
If x < c : stick condition. The local contact shear stress is
τxz = µp0
r1− (
x
a)2 −
c
aµp0
r1− (
x
c)2
If c < x < a: slip condition. The local contact shear stress is
τxz = µp0
r1− (
x
a)2
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 34/68
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.
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
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 35/68
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 systemstiffness, damping, inertia
Friction dynamic at the interfaceDifference between static (µs ) and dynamic (µd ) friction coefficientµs and µd dependence on the sliding velocity and time
A simple stick-slip model
���������������������������
���������������������������
m
kF
X
v
Friction law
F
v
F
F
s
d
Fig : Plot of frictional force vs. time.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 36/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Friction instability : “stick-slip” 3/4
During sliding, the problem is:
mx − Fd = −kx x(0) =Fs
kx(0) = v
and the solution is
x(t) =1
k{(Fs − Fd )cos(ωt) + Fd} +
v
ωsin(ωt), ω =
rk
m
or the velocity v is negligible compared to dx/dt:
x(t) ≈1
k{(Fs − Fd )cos(ωt) + Fd}
Characteristic time of sliding
Tinertia = 2π
rm
k
The force F oscillates between Fs and 2Fd − Fs .
F
t
F
d2F − Fs
Tinertia
s
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 37/68
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 regionsof stable and unstable motion.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 38/68
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 39/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Examples
(a) Vickers indenta-tion test, palladiumglasses
(b) Contact zone underVickers indenter, zirconiumglasses
(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 anelastic-plastic indentation cone
daPlastic
Elastic
dh
dcc
da
a
da
aβ
rcore
du(r)
AssumptionsWithin 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 ofstress (in the external zone)
The radial displacement on r=a during an increment dhmust accommodate the volume of material.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 41/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Characteristic
In the plastic zone: a ≤ r ≤ cσrY = −2ln(c/r)− 2/3
σθY = −2ln(c/r) + 1/3
In the elastic zone: r ≥ cσrY = − 2
3
`cr
´3σθY = 1
3
`cr
´3where Y denotes the value of the yield strees of material in simple shear and simple compression.
Core pressure
pY = −[ σr
Y ]r=a = 23 + 2ln(c/a)
Radial displacement
du(r)dc = T
E {3(1− ν)(c2/r2)− 2(1− 2ν)(r/c)}
Conservation volume
2πa2du(a) = πa2dh = πa2tan(β)da
9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;
Pressure in the core,for an incompressible material
p
Y=
2
3
1 + ln
„Etan(β)
3Y
«ff
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Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Unloading indentation: elastic strain energy
Example: spherical indenter
R R’
aa
R ρ
a aBefore loading Under loading After unloading
Residual depth δ − δ′: Estimation of the energy dissipated ∆W in one cycle of the load∆W = α
RPdδ where α is the hysteresis-loss factor. (α = 0, 4% for hard bearing steel)
W = 25
„9E∗
2P5
16R
«4a3“
1R −
1ρ
”= 3P
E∗
a =“
3P4E′
”1/3
δ = a2
R =“
9P2
16RE′3
”1/3
9>>>>>>>>>>>=>>>>>>>>>>>;
δ′ = 9πPpm
16E′2with pm = 0.38Y in fully plastic state
PPY
= 0, 38(δ′/δY )2
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 43/68
Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites
Sharp indentation
Schematic illustration of a typical P-hreponse of an elasto-plastic material
Characterisation of P-h reponseDuring the loading,