Contact mechanics I: basics

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

- 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

- 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 ∈ ˆ

3

; 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 ∈ ˆ

3

; 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 ∈ ˆ

3

; 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 ∈ ˆ

3

; 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 ∈ ˆ

3

; 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 ₊

[Song, Yovanovich, 1987]

−F n = C 1 e ^c

^u

Hertz 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 2R

x

+ 1

2R

y

z

= −

„ 1

2R

x

+ 1 2R

y

«

where R

and R

are the principal radii of curvature of the surface i.

Separation between the two surfaces

h = z

− z

= Ax

+ By

Displacement

u

+ u

+ h = δ

+ δ

[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

, a << l The surfaces are frictionless, q − x = q

= 0

Applications

1 Solids of revolution

2 Two-dimensional contact of cylindrical bodies Note

1

E

= 1 − ν

E

+ 1 − ν

E

Hertz theory : Solids of revolution

Simple case of solids of revolution Principal radii of curvature

R

= R

= R

, i = 1,2 Boundary conditions for the displacement

u

+ u

= δ − (1/2R)r

Pressure distribution

p = p

{1 − (r/a)

}

Consequences

Pressure

p

= 6PE

P

R

!

Radius of the contact circle

a =

„ 3PR

4E

«

Displacement

δ = 9P

16RE

!

Hertz theory : Solids of revolution

Distributions of stresses

σr

p0

(x = 0, z) =

0

(x = 0, z)

= −(1 + ν){1 − (z/a)tan

(a/z)}

+

(1 + z

/a

)

σz

p0

(x = 0,z) = −(1 + z

/a

)

Maximum shear stress τ

=

|σ

− σ

|

(τ

)

= 0.31p

at 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

+

2 1

R

=

1

+

2

p(x) = p

(1 − (x/a)

)

a =

q

p

= q

πR

2D contact of cylindrical bodies

Example : cylinder/plate

Distributions of normal pressure (Hertz) and tangential stress

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σ

(x = 0, z) = −

{(a

+ 2z

)(a

+ z

)

− 2z}

σ

(x = 0,z) = −p

a(a

+ z

)

τ

(x = 0, z) = p

a{z − z

(a

− z

)

}

σ

= σ

= σ

= 0

Macroscopic friction 1/2

Tresca

8

> >

> >

> <

> >

> >

> :

|F

| ≤ g

If |F

| < g, then V

= 0

If |F

| = g, ∃λ > 0 such V

= −λF

Coulomb

8

> >

> >

> <

> >

> >

> :

|F

| ≤ µ|F

|

If |F

| < µ|F

|, then V

= 0 stick

If |F

| = µ|F

|, ∃λ > 0 such V

= −λF

slip

Macroscopic friction 2/2

Regularized Coulomb [Oden, Pires, 1983], [Raous, 1999]

|F t | = −µφ ⁱ (V slide ) |F n | φ ¹ = √ ^V

V

+ε

φ ² = tanh ^V

_ε

Variable friction

8

> >

> >

> <

> >

> >

> :

|F

| ≤ C

(u

)

If |F

| < C

(u

)

, then V

= 0

If |F

| = C

(u

)

, ∃λ > 0 such V

= −λR

Transition 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 ) =

q

1 − (

)

q(x ) = ±µp(x ) = ±µ

q 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

= µp

r 1 − ( x

a )

− c a µp

r 1 − ( x

c )

If c < x < a: slip condition. The local contact shear stress is

τ

= µp

r

1 − ( x

a )

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

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

) and dynamic (µ

) friction coefficient µ

and µ

dependence 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

= −kx x(0) = F

k x(0) = ˙ v and the solution is

x(t) = 1

k {(F

− F

)cos(ωt) + F

} + v

ω sin(ωt), ω =

r k

m

or the velocity v is negligible compared to dx/dt:

x(t) ≈ 1

k {(F

− F

)cos(ωt) + F

}

Characteristic time of sliding

T

= 2π

r m

k

The force F oscillates between F

and 2F

− F

.

F

t F

2F − Fd _s

T_inertia

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

= −

`

r

´

σθ

Y

=

`

r

´

where Y denotes the value of the yield strees of material in simple shear and simple compression.

Core pressure

p

Y

= −[

]

=

+ 2ln(c/a) Radial displacement

du(r)

dc

=

{3(1 − ν)(c

/r

) − 2(1 − 2ν)(r/c)}

Conservation volume

2πa

du(a) = πa

dh = πa

tan(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

: 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 =

„

9E∗2 P5 16R

«

4a

“

1 R

−

”

=

a = “

3P 4E0

”

δ =

= “

9P2 16RE03

”

9

> >

> >

> >

> >

> >

> =

> >

> >

> >

> >

> >

> ;

δ

=

16E02

with p

= 0.38Y in fully plastic state

P

PY

= 0,38(δ

/δ

)

Sharp indentation

Schematic illustration of a typical P-h reponse of an elasto-plastic material

Characterisation of P-h reponse During the loading, P = Ch

Kick’s law During the unloading,

dPu dh

˛

˛

˛

initial slope

h

Residual 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

Rε

for σ ≥ σ

with

E Young’s modolus R a strength coefficient n the strain hardening exponent σ

the initial yield stress Assumption :

The theory of plasticity with the von Mises effect stress.

Parameters for an elasto-plastic behavior

E, ν, σ

, 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

, n) or (E, ν, σ

,n) or (E, ν, σ

, σ

)

Load P

P = P(h, E

, σ

, n) or P = P(h, E

, σ

,n) or P = P(E, ν, σ

, σ

) with

E

= 1 − ν

E + 1 − ν

E

!

Unload

P

= P

(h,h

, E, ν, E

, ν

, σ

,n) or P

= P

(h,h

, E

, σ

, 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

,n) C =

h2

= σ

Π

“

, n ”

P = P(h,E

, σ

, n) C =

h2

= σ

Π

“

E∗ σy

,

”

P = P(E, ν, σ

, σ

) C =

h2

= σ

Π

“

E∗ σr

,

r

”

with Π are dimensionless functions.

And then h

!

Determination of h _r

Application of the dimensional analysis during the unload

dP

dh = dP

dh (h, h

,E

, σ

, n) thus

dP

dh = E

hΠ

„ h

h , σ

E

, n

«

Consequently,

dP

dh

˛

˛

˛

˛

= E

h

Π

„ 1, σ

E

, n

«

= E

h

Π

„ σ

E

, n

«

Or

P

= P

(h, h

, E

, σ

, n) = E

Π

„ h

h , σ

E

, n

«

Finaly,

P

= 0 implies 0 = Π

„ h

h

, σ

E

,n

«

whether h

h

= Π

„ σ

E

, n

«

and then h

!

Π

,i =1,2,3 can be used to relate the indentaion reponse to mechanical properties.

FE Vickers indentation test

Maximum penetration h

3.11 µm

Parameters

Elastic E=81600MPa, ν = 0,36

Elasto-plastic model E=81600MPa, ν = 0,36, σ

= 1610MPa

Drucker-Prager model E=81600MPa, ν = 0,36, σ

= 1600MPa, σ

= 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

: 3.5 µ m

[Casal and Forest, 2009]

Contact reponse

(d)

(e)

von Mises stress fields

Figure: (a) f.c.c and (b) h.c.p crystals. Penetration depth: h

= 1.25µm

Plastic zone morphology

(a)

(b)

Figure: Penetration depth: h

= 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

(1−ν

) (1−ν

)

(1−2ν

)

[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 C_Comp.2 P1-10: p_Comp.1, p_Comp.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: p_Comp.1, p_Comp.2 R20_el: p_Comp.1, p_Comp.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: p_Comp.1, p_Comp.2 R20_el: p_Comp.1, p_Comp.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

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

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.

Thank you for your attention!

Georges Cailletaud <georges.cailletaud@mines-paristech.fr>

St´ ephanie Basseville <stephanie.basseville@mines-paristech.fr>

Vladislav A. Yastrebov <vladislav.yastrebov@mines-paristech.fr>

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 proc¨ı¿ ½ d¨ı¿ ½ s d’indentation et de rayage par ¨ı¿ ½ l¨ı¿ ½ ments finis et

¨ı¿ ½ l¨ı¿ ½ ments disctincts.

Rabinowicz, E.

Friction and Wear.