Vladislav A. Yastrebov

Contact mechanics and elements of tribology

Lecture 5.

Computational Contact Mechanics

Vladislav A. Yastrebov

MINES ParisTech, PSL Research University, Centre des Mat´eriaux, CNRS UMR 7633, Evry, France

@ Centre des Mat´eriaux February 8, 2016

Outline

Introduction

Governing equations

Optimization methods

Resolution algorithm

Examples

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Aircraft’s engine GP 7200 www.safran-group.com

[1] M. W. R. Savage J. Eng. Gas Turb. Power, 134:012501 (2012)

V.A. Yastrebov 3/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

High speed train TGVwww.sncf.com

Wilde/ANSYSwildeanalysis.co.uk

V.A. Yastrebov 4/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Bearings www.skf.com

[1] F. Massi, J. Rocchi, A. Culla, Y. Berthier Mech. Syst. Signal Pr., 24:1068-1080 (2010)

V.A. Yastrebov 5/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Helical gearwww.tpg.com.tw

www.mscsoftware.com

V.A. Yastrebov 6/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Assembled breaking system www.brembo.com

www.mechanicalengineeringblog.com

V.A. Yastrebov 7/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Tire-road contactwww.michelin.com

[1] M. Brinkmeier, U. Nackenhorst, S. Petersen, O. von Estorff, J. Sound Vib., 309:20-39 (2008)

V.A. Yastrebov 8/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Deep drawingwww.thomasnet.com

[1] G. Rousselier, F. Barlat, J. W. Yoon Int. J. Plasticity, 25:2383-2409 (2009)

V.A. Yastrebov 9/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Crash-testwww.porsche.com

[1] O. Klyavin, A. Michailov, A. Borovkov www.fea.ru

V.A. Yastrebov 10/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Human articulations www.sportssupplements.net

J. A. Weiss, University of Utah Musculoskeletal Research Laboratories

V.A. Yastrebov 11/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Sand duneswww.en.wikipedia.org

E. Azema et al, LMGC90

V.A. Yastrebov 12/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Damage at electric contact zone www.taicaan.com

Simulation of electric current www.comsol.com

V.A. Yastrebov 13/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

San-Andreas fault, by M. Rightmire www.sciencedude.ocregister.com

[1] J.D. Garaud, L. Fleitout, G. Cailletaud Colloque CSMA (2009)

V.A. Yastrebov 14/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Drill Bit toolRobitRocktools; extraction of geothermal energy (SINTEF,NTNU)

[1] T. Saksala, Int. J. Numer. Anal. Meth.

Geomech. (2012)

V.A. Yastrebov 15/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Impact crater, Arizona www.MrEclipse.cometmaps.google.com

Simulation of formation of Copernicus crater Yue Z., Johnson B. C., et al. Projectile remnants in central peaks of lunar impact

craters. Nature Geo 6 (2013)

V.A. Yastrebov 16/75

Industrial and natural contact problems

1

Assembled parts, e.g. engines

2

Railroad contacts

3

Gears and bearings

4

Breaking systems

5

Tire-road contact

6

Metal forming

7

Crash tests

8

Biomechanics

9

Granular materials

10

Electric contacts

11

Tectonic motions

12

Deep drilling

13

Impact and fragmentation

14

etc.

Impact crater, Arizona www.MrEclipse.cometmaps.google.com

Simulation of formation of Copernicus crater Yue Z., Johnson B. C., et al. Projectile remnants in central peaks of lunar impact

craters. Nature Geo 6 (2013)

V.A. Yastrebov 17/75

Equilibrium and contact conditions

Balance of momentum



 

 

 

 

 

 

 

 

∇ · = + σ f

= 0 in Ω

σ = · n = t

on Γ

u = u

on Γ

? on Γ

Frictionless contact conditions (intuitive)

1

No penetration

2

No adhesion

3

No shear transfer

u f u

f 1

2

c1 c2

Equilibrium and contact conditions

Balance of momentum



 

 

 

 

 

 

 

 

∇ · = + σ f

= 0 in Ω

σ = · n = t

on Γ

u = u

on Γ

? on Γ

Frictionless contact conditions (intuitive)

1

No penetration

2

No adhesion

3

No shear transfer

n v

u f u

f 1

2 c1

c2

Equilibrium and contact conditions

Balance of momentum



 

 

 

 

 

 

 

 

∇ · = + σ f

= 0 in Ω

σ = · n = t

on Γ

u = u

on Γ

? on Γ

Frictionless contact conditions (intuitive)

1

No penetration

2

No adhesion

3

No shear transfer

u f u

f 1

2 c1

c2

Equilibrium and contact conditions

Balance of momentum



 

 

 

 

 

 

 

 

∇ · = + σ f

= 0 in Ω

σ = · n = t

on Γ

u = u

on Γ

? on Γ

Frictionless contact conditions (intuitive)

1

No penetration

2

No adhesion

3

No shear transfer

u f u

f 1

2 c1

c2

Equilibrium and contact conditions

Balance of momentum



 

 

 

 

 

 

 

 

∇ · = + σ f

= 0 in Ω

σ = · n = t

on Γ

u = u

on Γ

? on Γ

Frictionless contact conditions (intuitive)

1

No penetration

2

No adhesion

3

No shear transfer

u f u

f 1

2 c1

c2

Gap function

Gap function g

gap = – penetration asymmetric function defined for

• separation g > 0

• contact g = 0

• penetration g < 0 governs normal contact

Master and slave split

Gap function is determined for all slave points with respect to the master surface

g=0 g<0

g>0

penetration non-contact

contact

n n n

Gap between a slave point and a master surface

V.A. Yastrebov 23/75

Gap function

Gap function g

gap = – penetration asymmetric function defined for

• separation g > 0

• contact g = 0

• penetration g < 0 governs normal contact

Master and slave split

Gap function is determined for all slave points with respect to the master surface

Normal gap

g

= n · h

r

− ρ(ξ

) i , n is a unit normal vector, r

slave point, ρ(ξ

) projection point at master surface

g=0 g<0

g>0

penetration non-contact

contact

n n n

Gap between a slave point and a master surface

n

( (

r

Definition of the normal gap

Consider existence and uniqueness

V.A. Yastrebov 24/75

Frictionless or normal contact conditions

No penetration

Always non-negative gap g ≥ 0

No adhesion

Always non-positive contact pressure σ

≤ 0

Complementary condition

Either zero gap and non-zero pressure, or non-zero gap and zero pressure

g σ

= 0

No shear transfer (automatically) σ

= 0

σ^∗n=(σ=·n)·n=σ=: (n⊗n) σ^∗∗_t =σ=·n−σnn=n·σ=·

=I−n⊗n

g

n 0

Scheme explaining normal

contact conditions

Frictionless or normal contact conditions

No penetration

Always non-negative gap g ≥ 0

No adhesion

Always non-positive contact pressure σ

≤ 0

Complementary condition

Either zero gap and non-zero pressure, or non-zero gap and zero pressure

g σ

= 0

No shear transfer (automatically) σ

= 0

σ^∗n=(σ=·n)·n=σ=: (n⊗n) σ^∗∗_t =σ=·n−σnn=n·σ=·

=I−n⊗n

g = 0,

< 0 co nt ac t

non-contact

restricted regions

g

n

g > 0,

= 0 0

Improved scheme explaining

normal contact conditions

Frictionless or normal contact conditions

In mechanics:

Normal contact conditions

≡

Frictionless contact conditions

≡

Hertz

-Signorini

conditions

≡

Hertz

-Signorini

-Moreau

conditions also known in optimization theory as Karush

-Kuhn

-Tucker

conditions

g = 0,

< 0 co nt ac t

non-contact

restricted regions

g

n

g > 0,

= 0 0

Improved scheme explaining normal contact conditions

g ≥ 0, σ

≤ 0, gσ

= 0

1Heinrich Rudolf Hertz(1857–1894) a German physicist who first formulated and solved the frictionless contact problem between elastic ellipsoidal bodies.

2Antonio Signorini(1888–1963) an Italian mathematical physicist who gave a general and rigorous mathematical formulation of contact constraints.

3Jean Jacques Moreau(1923) a French mathematician who formulated a non-convex optimization problem based on these conditions and introduced pseudo-potentials in contact mechanics.

4William Karush(1917–1997),⁵Harold William Kuhn(1925) American mathematicians, 6Albert William Tucker(1905–1995) a Canadian mathematician.

Relative sliding

Recall:

• Convective coordinate in parent space ξ

∈ ( − 1; 1)

• Mapping to real space

ρ(ξ

, ξ

, t) =

8

X

i=1

N

(ξ

, ξ

)ρ

(t)

6 3 7 4

8

2

3

4 1

6 5

7 8

( )

e

e

e

*1 * 2}

} ,

*1 * ,2 1

2

1 -1

-1 1

Relative sliding

Recall:

• Convective coordinate in parent space ξ

∈ ( − 1; 1)

• Mapping to real space

ρ(ξ

, ξ

, t) =

8

X

i=1

N

(ξ

, ξ

)ρ

(t)

6 3 7 4

8

2

3

4 1

6 5

7 8

( )

e

e

e

*1 * 2}

} ,

*1 * ,2 1

2

1 -1

-1 1

Tangential slip velocity v

must take into account:

• only tangential component

• relative rigid body motion

• master’s deformation

v

= ∂ ^ρ

∂ξ

ξ ˙

+ ∂ ^ρ

∂ξ

ξ ˙

where ∂ρ/∂ξ

are the tangent vectors of the local basis and ξ

are the convec- tive coordinates.

Relative slip between a slave point and a

deformable master surface

Relative sliding: example

Consider a one-dimensional example:

P is a projection of A on segment BC.

x

= ξx

+ (1 − ξ)x

(1)

Velocity of the projection point

˙

x

= ξ x ˙

+ (1 − ξ) ˙ x

| {z }

∂xP

∂t

+ (x

− x

) ˙ ξ

| {z }

∂xP

∂ξξ˙

Substract the velocity of point x

for fixed ξ

v

= x ˙

−

∂t

= (x

− x

) ˙ ξ =

ξ ˙

Compute tangential slip increment

∆g

=

(ξ

− ξ

)

A

B C

x P

Example of a one-dimensional

relative slip

Relative sliding: example

Consider a one-dimensional example:

P is a projection of A on segment BC.

x

= ξx

+ (1 − ξ)x

(1)

Velocity of the projection point

˙

x

= ξ x ˙

+ (1 − ξ) ˙ x

| {z }

∂xP

∂t

+ (x

− x

) ˙ ξ

| {z }

∂xP

∂ξξ˙

Substract the velocity of point x

for fixed ξ

v

= x ˙

−

∂t

= (x

− x

) ˙ ξ =

ξ ˙

Compute tangential slip increment

∆g

=

(ξ

− ξ

)

A

B C

x P

Example of a one-dimensional relative slip

Ship-river analogy

Relative sliding: example

Consider a one-dimensional example:

P is a projection of A on segment BC.

x

= ξx

+ (1 − ξ)x

(1)

Velocity of the projection point

˙

x

= ξ x ˙

+ (1 − ξ) ˙ x

| {z }

∂xP

∂t

+ (x

− x

) ˙ ξ

| {z }

∂xP

∂ξξ˙

Substract the velocity of point x

for fixed ξ

v

= x ˙

−

∂t

= (x

− x

) ˙ ξ =

ξ ˙

Compute tangential slip increment

∆g

=

(ξ

− ξ

)

A

B C

x P

Example of a one-dimensional relative slip

Ship-river analogy

Relative sliding: example

Consider a one-dimensional example:

P is a projection of A on segment BC.

x

= ξx

+ (1 − ξ)x

(1)

Velocity of the projection point

˙

x

= ξ x ˙

+ (1 − ξ) ˙ x

| {z }

∂xP

∂t

+ (x

− x

) ˙ ξ

| {z }

∂xP

∂ξξ˙

Substract the velocity of point x

for fixed ξ

v

= x ˙

−

∂t

= (x

− x

) ˙ ξ =

ξ ˙

Compute tangential slip increment

∆g

=

(ξ

− ξ

)

A

B C

x P

Example of a one-dimensional relative slip

Ship-river analogy Li derivative: the change of a vector field along the change of

another vector field

Amontons-Coulomb’s friction

No contact g > 0, σ

= 0 Stick | v

| = 0

Inside slip surface / Coulomb’s cone

f = | σ

| − µ | σ

| < 0 Slip | v

| > 0

On slip surface / Coulomb’s cone

f = | σ

| − µ | σ

| = 0 Complementary condition

Either zero velocity and negative slip criterion, or non-zero velocity and zero slip criterion

| v

|

| σ

| − µ | σ

|

= 0

n t

v

n 1

0 0

Scheme explaining frictional contact

conditions

Amontons-Coulomb’s friction

No contact g > 0, σ

= 0 Stick | v

| = 0

Inside slip surface / Coulomb’s cone

f = | σ

| − µ | σ

| < 0 Slip | v

| > 0

On slip surface / Coulomb’s cone

f = | σ

| − µ | σ

| = 0 Complementary condition

Either zero velocity and negative slip criterion, or non-zero velocity and zero slip criterion

| v

|

| σ

| − µ | σ

|

= 0

st ic k

n

restricted regions slip

t

v

stick

t

n 1

restricted

region

0 0

slip

Improved scheme explaining

frictional contact conditions

Amontons-Coulomb’s friction

No contact g > 0, σ

= 0 Stick | v

| = 0

Inside slip surface / Coulomb’s cone

f = | σ

| − µ | σ

| < 0 Slip | v

| > 0

On slip surface / Coulomb’s cone

f = | σ

| − µ | σ

| = 0 Complementary condition

Either zero velocity and negative slip criterion, or non-zero velocity and zero slip criterion

| v

|

| σ

| − µ | σ

|

= 0

Scheme of 2D frictional contact

t₂

t1 n

v

n

t2

t1

stick

slip slip

stick

Scheme of 3D frictional contact

| v

| ≥ 0, | σ

| − µ | σ

| ≤ 0, | v

|

| σ

| − µ | σ

|

= 0

More friction laws

• Static criteria

stick

t

0 n

stick

t

0 n

slip slip

stick

t

0 n

slip max ^t

stick

t

0 n

max t

slip

(a) (b) (c) (d)

(a) Tresca (b) Amontons-Coulomb (c) Coulomb-Orowan (d) Shaw

• Kinetic criteria

stick

n

slip

t

vt

0

s

k

stick

n

slip

t

vt

0

s

k

(a) (b)

stick

n slip

t

vt

0

s

(c)

stick

n

slip

t

0

s

k

(d)

log( +v )0 gt

(a,b) velocity weakening (c) velocity weakening-strengthening (d) Linear slip weakening

• µ

static and µ

kinetic coefficients of friction.

Rate and state friction and regularization

• Rate and state friction law

Rate v

= | v

| – relative slip velocity State θ – ≈ internal time

Dieterich–Ruina–Perrin (1979, 83, 95) Frictional resistance

σ

= | σ

| µ

+ bθ + a ln(v

/v

) Evolution of the state variable

θ ˙ = −

L

h θ + ln

t v0

i

• Prakash-Clifton friction law (1992,2000)

Viscous type evolution of frictional resistance σ

σ ˙

= −

L

(σ

+ µσ

)

0 0.1 0.2 0.3 0.4 0.5

Slip velocity

Slip velocity

0.2 0.3 0.4 0.5 0.6 0.7 0.8

500 600 700 800 900 1000 1100 1200

Frictional resistance

Time Resistance

Rate and state friction law

0 100 200 300 400 500

Contact pressure

Contact pressure

0 100 200 300 400 500

500 600 700 800 900 1000 1100 1200

Frictional resistance

Time Resistance

Prakash-Clifton regularization

Rate and state friction and regularization

• Rate and state friction law

0 0.1 0.2 0.3 0.4 0.5

Slip velocity

Slip velocity

0.2 0.3 0.4 0.5 0.6 0.7 0.8

500 600 700 800 900 1000 1100 1200

Frictional resistance

Time Resistance

Rate and state friction and regularization

• Rate and state friction law

Rate v

= | v

| – relative slip velocity State θ – ≈ internal time

Dieterich–Ruina–Perrin (1979, 83, 95) Frictional resistance

σ

= | σ

| µ

+ bθ + a ln(v

/v

) Evolution of the state variable

θ ˙ = −

L

h θ + ln

t v0

i

• Prakash-Clifton friction law (1992,2000)

Viscous type evolution of frictional resistance σ

σ ˙

= −

L

(σ

+ µσ

)

0 0.1 0.2 0.3 0.4 0.5

Slip velocity

Slip velocity

0.2 0.3 0.4 0.5 0.6 0.7 0.8

500 600 700 800 900 1000 1100 1200

Frictional resistance

Time Resistance

Rate and state friction law

0 100 200 300 400 500

Contact pressure

Contact pressure

0 100 200 300 400 500

500 600 700 800 900 1000 1100 1200

Frictional resistance

Time Resistance

Prakash-Clifton regularization

Rate and state friction and regularization

• Prakash-Clifton friction law (1992,2000)

0 100 200 300 400 500

Contact pressure

Contact pressure

0 100 200 300 400 500

500 600 700 800 900 1000 1100 1200

Frictional resistance

Time Resistance

Rate and state friction and regularization

• Prakash-Clifton friction law (1992,2000)

0 100 200 300 400 500

Contact pressure

Time Contact pressure

0 0.5 1 1.5 2

500 600 700 800 900 1000 1100 1200

Normalized frictional resistance

Time Normalized resistance

From strong to weak form

• Balance of momentum and boundary conditions

∇ · = + σ f

= 0 in Ω = Ω

∪ Ω

+ B.C.

u f u

f 1

2 c1 c2

Two solids in contact

From strong to weak form

• Balance of momentum and boundary conditions

∇ · = + σ f

= 0 in Ω = Ω

∪ Ω

+ B.C.

• Balance of virtual works Z

∂Ω

n · σ = · δu dΓ + Z

Ω

h f

· δu − σ = ·· δ ∇ u i dΩ = 0

u f u

f 1

2 c1 c2

Two solids in contact

From strong to weak form

• Balance of momentum and boundary conditions

∇ · = + σ f

= 0 in Ω = Ω

∪ Ω

+ B.C.

• Balance of virtual works Z

∂Ω

n · σ = · δu dΓ = Z

Ω

h f

· δu − = σ ·· δ ∇ u i dΩ = 0

Z

Γc1

n · = σ · δρ dΓ

+ Z

Γc2

ν · σ = · δr dΓ

+ Z

Γf

σ

· δu dΓ

n v

u f u

f 1

2 c1

c2

Two solids in contact

From strong to weak form

• Balance of momentum and boundary conditions

∇ · = + σ f

= 0 in Ω = Ω

∪ Ω

+ B.C.

• Balance of virtual works Z

∂Ω

n · = σ · δu dΓ ⇒ Z

Ω

h f

· δu − σ = ·· δ ∇ u i dΩ = 0

Z

Γc1

n · = σ · δρ dΓ

+ Z

Γc2

ν · = σ · δr dΓ

= Z

Γf

σ

· δu dΓ

= Z

Γc1

n · = σ · δ(ρ − r) dΓ

= Z

Γc1

σ

δg

+ ∼ σ

δ ∼ ξ dΓ

n v

u f u

f 1

2 c1

c2

Two solids in contact

From strong to weak form

• Balance of momentum and boundary conditions

∇ · = + σ f

= 0 in Ω = Ω

∪ Ω

+ B.C.

• Balance of virtual works Z

∂Ω

n · = σ · δu dΓ ⇒ Z

Ω

h f

· δu − σ = ·· δ ∇ u i dΩ = 0

Z

Γc1

n · = σ · δρ dΓ

+ Z

Γc2

ν · = σ · δr dΓ

= Z

Γf

σ

· δu dΓ

= Z

Γc1

n · = σ · δ(ρ − r) dΓ

= Z

Γc1

σ

δg

+ ∼ σ

δ ∼ ξ dΓ

n v

u f u

f 1

2 c1

c2

Two solids in contact

Z

Ω

σ = · · δ ∇ u dΩ + Z

Γc1

σ

δg

+ ∼ σ

δ ∼ ξ dΓ

| {z } Contact term

= Z

Γf

σ

· δu dΓ + Z

Ω

f

· δu dΩ

From strong to weak form

• Balance of momentum and boundary conditions

∇ · = + σ f

= 0 in Ω = Ω

∪ Ω

+ B.C.

• Balance of virtual works Z

Ω

σ = · · δ ∇ u dΩ

| {z }

Change of the internal energy

+ Z

Γc1

σ

δg

+ ∼ σ

δ ∼ ξ dΓ

| {z } Contact term

=

Z

Γf

σ

· δu dΓ

| {z }

Virtual work of external forces

+ Z

Ω

f

· δu dΩ

| {z }

Virtual work of volume forces

n v

u f u

f 1

2 c1

c2

Two solids in contact

• Functional space

u ∈ H

(Ω) Hilbert space of the first order

and u satisfy boundary and contact conditions.

Towards variational inequality

Contact term Z

Γc1

σ

δg

+ ∼ σ

δ ξ ∼ dΓ

Z

Γc1

σ

δg

dΓ

≤ 0

Contact configuration

Z

Ω

= σ · · δ ∇ u dΩ + Z

Γc1

∼ σ

δ ∼ ξ ^dΓ

1

≥

Z

Γf

σ

· δu dΓ + Z

Ω

f

· δu dΩ,

K = n

δu ∈ H

(Ω)

δu = 0 on Γ

, g

(u + δu) ≥ 0 on Γ

o

Towards variational inequality

Contact term Z

Γc1

σ

δg

+ ∼ σ

δ ξ ∼ dΓ

Z

Γc1

σ

δg

dΓ

≤ 0

Virtual change of the configuration Z

Ω

= σ · · δ ∇ u dΩ + Z

Γc1

∼ σ

δ ∼ ξ ^dΓ

1

≥

Z

Γf

σ

· δu dΓ + Z

Ω

f

· δu dΩ,

K = n

δu ∈ H

(Ω)

δu = 0 on Γ

, g

(u + δu) ≥ 0 on Γ

o

Towards variational inequality

Contact term Z

Γc1

σ

δg

+ ∼ σ

δ ξ ∼ dΓ

Z

Γc1

σ

δg

dΓ

≤ 0

Normal term in separation

Z

Ω

= σ · · δ ∇ u dΩ + Z

Γc1

∼ σ

δ ∼ ξ ^dΓ

1

≥

Z

Γf

σ

· δu dΓ + Z

Ω

f

· δu dΩ,

K = n

δu ∈ H

(Ω)

δu = 0 on Γ

, g

(u + δu) ≥ 0 on Γ

o

Towards variational inequality

Contact term Z

Γc1

σ

δg

+ ∼ σ

δ ξ ∼ dΓ

Z

Γc1

σ

δg

dΓ

≤ 0

Normal term in sliding

Z

Ω

= σ · · δ ∇ u dΩ + Z

Γc1

∼ σ

δ ∼ ξ ^dΓ

1

≥

Z

Γf

σ

· δu dΓ + Z

Ω

f

· δu dΩ,

K = n

δu ∈ H

(Ω)

δu = 0 on Γ

, g

(u + δu) ≥ 0 on Γ

o

Back to variational equality (unconstrained)

• Constrained minimization problem Z

Ω

= σ · · δ ∇ u dΩ + Z

Γc1

∼ σ

δ ∼ ξ ^dΓ

1

≥

Z

Γf

σ

· δu dΓ + Z

Ω

f

· δu dΩ, K = n

δu ∈ H

(Ω)

δu = 0 on Γ

, g

(u + δu) ≥ 0 on Γ

o

• Use optimization theory to convert to Z

Ω

σ = · · δ ∇ u dΩ + Z

Γ¹c

C(σ

, σ

, g

, ∼ ξ ^{, δu)}

| {z } Contact term

dΓ

= Z

Γf

σ

· δu dΓ + Z

Ω

f

· δu dΩ,

Unconstrained functional space K = n

δu ∈ H

(Ω)

δu = 0 on Γ

o

Contact term

is defined on the potential contact zone Γ

.

Optimization methods: recall

Functional to be minimized F(x) under constraint g(x) ≥ 0 Penalty method

Lagrange multipliers method

Augmented Lagrangian method

Optimization methods: recall

Functional to be minimized F(x) under constraint g(x) ≥ 0 Penalty method

• New functional

F

(x) = F(x) + − g(x)

= F(x) +



 



 



0, if g(x) ≥ 0 non-contact g

(x), if g(x) < 0 contact where is the penalty parameter.

• Stationary point must satisfy

∇ F

(x) = ∇ F(x) + 2 − g(x) ∇ g(x) = 0

• Solution tends to the precise solution as → ∞

Lagrange multipliers method Augmented Lagrangian method

Macaulay bracketshxi=







x, ifx≥0 0, otherwise

Optimization methods: recall

Functional to be minimized F(x) under constraint g(x) ≥ 0

Penalty method F

(x) = F(x) + − g(x)

Lagrange multipliers method

• New functional called Lagrangian L(x, λ) = F(x) + λg(x)

• Saddle point problem min

max

λ

{ L(x, λ) } −→ x

←− min

g(x)≥0

{ F(x) }

• Stationary point

∇

L =

"

∇

F(x) + λ ∇

g(x) g(x)

#

= 0 need to verify λ ≤ 0

Augmented Lagrangian method

Macaulay bracketshxi=







x, ifx≥0 0, otherwise

Optimization methods: recall

Functional to be minimized F(x) under constraint g(x) ≥ 0

Penalty method F

(x) = F(x) + − g(x)

Lagrange multipliers method L(x, λ) = F(x) + λg(x)

Augmented Lagrangian method

[Hestnes 1969], [Powell 1969], [Glowinski & Le Tallec 1989], [Alart & Curnier 1991], [Simo & Laursen 1992]

• New functional, augmented Lagrangian

L

(x, λ) = F(x) +



 



 



λg(x) + g

(x) , if λ + 2g(x) ≥ 0, contact

−

4

λ

, if λ + 2g(x) < 0, non-contact

• Stationary point

∇

L

=



 

 

 



 

 

 





 

 



∇

F(x) + λ ∇

g(x) + 2g(x) ∇ g(x) g(x)



 

 

 = 0, if contact



 

 



∇

F(x)

−



 

 

 = 0, if non-contact

Macaulay bracketshxi=







x, ifx≥0 0, otherwise

Optimization methods: example

Functional : f (x) = x

+ 2x + 1 Constrain : g(x) = x ≥ 0

Solution : x

= 0

Optimization methods: example

Functional : f (x) = x

+ 2x + 1 Constrain : g(x) = x ≥ 0

Solution : x

= 0

Penalty method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Penalty method

F

(x) = F(x) + − g(x)

Penalty method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Penalty method

F

(x) = F(x) + − g(x)

= 0

Penalty method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Penalty method

F

(x) = F(x) + − g(x)

= 1

Penalty method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Penalty method

F

(x) = F(x) + − g(x)

= 10

Penalty method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Penalty method

F

(x) = F(x) + − g(x)

= 50

Penalty method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Penalty method

F

(x) = F(x) + − g(x)

Advantages ,

simple physical interpretation simple implementation

no additional degrees of freedom

“mathematically” smooth functional

Drawbacks /

practically non-smooth functional

solution is not exact:

too small penalty →

large penetration

too large penalty →

ill-conditioning of the

tangent matrix

user has to choose penalty

properly or automatically and/or

adapt during convergence

Lagrange multipliers method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Lagrange multipliers method

L(x, λ) = F(x) + λg(x) → Saddle point → min

x

max

λ

L(x, λ) Need to check that λ ≤ 0

-2 -1 0 1 2

-3 -2 -1 0 1

X λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

Lagrange multipliers method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Lagrange multipliers method

L(x, λ) = F(x) + λg(x) → Saddle point → min

x

max

λ

L(x, λ) Need to check that λ ≤ 0

Advantages ,

exact solution

no adjustable parameters

Drawbacks /

Lagrangian is not smooth

additional degrees of freedom

not fully unconstrained: λ ≤ 0

Augmented Lagrangian method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Augmented Lagrangian method

L

(x, λ) = F(x) +



 



 



λg(x) + g

(x) , if λ + 2g(x) ≥ 0, contact

−

4

λ

, if λ + 2g(x) < 0, non-contact

-2 -1 0 1 2

-3 -2 -1 0 1

X λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

Yellow line separates contact and non-contact regions

Augmented Lagrangian method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Augmented Lagrangian method

L

(x, λ) = F(x) +



 



 



λg(x) + g

(x) , if λ + 2g(x) ≥ 0, contact

−

4

λ

, if λ + 2g(x) < 0, non-contact

-2 -1 0 1 2

-3 -2 -1 0 1

X λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

Yellow line separates contact and non-contact regions

Augmented Lagrangian method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Augmented Lagrangian method

L

(x, λ) = F(x) +



 



 



λg(x) + g

(x) , if λ + 2g(x) ≥ 0, contact

−

4

λ

, if λ + 2g(x) < 0, non-contact

-2 -1 0 1 2

-3 -2 -1 0 1

X λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

-2 -1 0 1 2

-3 -2 -1 0 1

X

λ

Yellow line separates contact and non-contact regions

Augmented Lagrangian method: example

F(x) = x

+ 2x + 1, g(x) = x ≥ 0, x

= 0 Augmented Lagrangian method

L

(x, λ) = F(x) +



 



 



λg(x) + g

(x) , if λ + 2g(x) ≥ 0, contact

−

4

λ

, if λ + 2g(x) < 0, non-contact

Advantages ,

exact solution smooth functional (!) fully unconstrained

Drawbacks /

additional degrees of freedom

quite sensitive to parameter

need to adjust during

convergence

Application to contact problems: weak form

Z

Ω

= σ · · δ ∇ u dΩ + Z

Γ¹c

C

|{z}

Contact term dΓ

=

Z

Γf

σ

· δu dΓ + Z

Ω

f

· δu dΩ,

K = n

δu ∈ H

(Ω)

δu = 0 on Γ

o

Penalty method

Pressure: σ

= g

, Shear: σ

=



 



 



g

, if stick | σ

| < µ | σ

| µg

δg

t

/ | δg

t

| , if slip | σ

| = µ | σ

|

Contact term C = C(g

, g

t

, δg

, δg

t

) = σ

δg

+ σ

· δg

t

Application to contact problems: weak form

Z

Ω

= σ · · δ ∇ u dΩ + Z

Γ¹c

C

|{z}

Contact term dΓ

=

Z

Γf

σ

· δu dΓ + Z

Ω

f

· δu dΩ,

K = n

δu ∈ H

(Ω)

δu = 0 on Γ

o

Augmented Lagrangian method

Contact term C = C(g

, g

t

, λ

, λ

, δg

, δg

t

, δλ

, δλ

)

C=































−¹

λnδλn−λ_t·δλ_t

, if non-contactλn+gn≥0

λˆnδgn+gnδλn+λˆt·δg

t+g

t

·δλˆt, if stick|ˆλ_t| ≤µ|σˆn|

λˆnδgn+gnδλn+µσˆn−µσˆn

λˆt

|λˆt|

·δg

t

−¹







λt+µσˆn

λˆt

|λˆt|









·δλ_t, if slip|ˆλ_t| ≥µ|σˆn|

where λ ˆ

= λ

+ g

and λ ˆ

= λ

+ g

t

.

Friction . . . .

Friction . . . .

The scream

Friction: Return mapping algorithm

Return mapping algorithm in 2D

t i+1

trial

f

trial ti+1

in

1

g

g

g

ti+1

s

i+1n

i+1

f 0 <

f 0 <

s

s

g

g

s

g

g

g

g

g

i+1n

As in plasticity^[1]

[1] Simo J.C. and Hughes T.J.. Computational inelasticity. Springer (2006)

Friction: Return mapping algorithm

Return mapping algorithm in 2D

n

1

trial ti+1 ti+1 ti

in i+1n ni

i+1n

trial ti+1 ti

t t

1

in i

n i+1n i+1n

n

St

Analogy with non-associated plastic flow^[2]

[2] Curnier A. A theory of friction. International Journal of Solids and Structures 20 (1984)

Friction: Return mapping algorithm

Return mapping algorithm in 3D

(

trial

sⁱ_t

i+1

(

trial

y y

i+1

i+1

i

i+1 i

e

e

e

e

i

i

i i i

i+1

* i

i

(a) (b)

(d) (c)

i

x x

x x

e

e

e

e

triali+1

s_tⁱf

g

g g

g g

g

g g g

g g g

g

g g

g g g g

f

sⁱ

sⁱ sⁱ

sⁱ sⁱ

t t t

t

t t

t t

t t

t t

t

t

t t t

t t t t

t

t t t

t t

t t

trial

i+1 i

i

i

i i

i

i

s

i+1 i

i+1 i+1 i+1

i+1

i+1

i+1

i+1 i

( (

n n

n n

n n

n

*

*

*

* i i

y y

A sligthly more messy thing

Application to contact problems: linearization

• Non-linear equation

R(u, f ) = 0

• Contains δg

, δg

• Use Newton-Raphson method

• Initial state at step i

R(u

, f

) = 0

• Should be also satisfied at step i + 1

R(u

, f

) = R(u

+ δu, f

) = 0

• Linearize

R(u

+ δu, f

) = R(u

, f

) + ∂R(u)

∂u δu = 0

• Finally

δu = −

" ∂R(u)

∂u

#

| {z } contains ∆δg

, ∆δg

t

R(u

)