instabilities and dynamic rupture in a frictional · pdf filensf 2008 workshop on friction -...

1NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….

Instabilities and Dynamic Rupturein a Frictional Interface

Laurent BAILLET

LGIT (Laboratoire de Géophysique Interne et Tectonophysique)Grenoble – [email protected]://www-lgit.obs.ujf-grenoble.fr/users/lbaillet/


Outline

• Variational formulation of the Signorini problem with friction

• 2D Model descriptionP

V=Velocity of the rigid surface- Local dynamics : stable/unstable state

Limit cycleinfluence on velocity & stress

• 3D Simulation of braking

Aim : describe tools for numerical simulation which enable the understanding of the appearance of the vibration of structure generated by the frictional contact between two bodies

- Study of friction coefficient

• Dynamic rupture in a frictional interface


Unilateral contact

Coulomb friction law

imp

0 00 0

( u ) D ( u )

div ( u ) f u

( u ).n Pu u

u( t ) u ; u( t ) u

s e

s r

s

=

+ =

==

= =

&&

& &

The problem of unilateral contact with Coulomb friction law consists in finding thedisplacement and the second order stress tensorsatisfying the equation of the mechanics

( u )su

impuPr

f

applied pressurePur

[ ] [ ]n nu.n 0, 0, . u.n 0 ons s G£ £ =

n t

n t

sticking u 0

sliding 0 s.t. u

t m s

t m s x xt

é ù< Þ =ê úë ûé ù= Þ $ ³ = -ê úë û

r

r

&

&

impuPr

G t ns

Variational formulation of the Signorini problem with friction

tr

nur


PLASTD (freeware developed by L.BAILLET) is based on a dynamic explicit method and includes large deformations and non linear material behavior

⇒ analysis in TIMEThe formulation is spatially discretized using the finite element method and is temporally discretized by the central difference method (explicit scheme).

The equations of motions are developed via the principle of virtual work at time t

where M and C are respectively the mass and damping matricesis the nodal vectors of external forces and the velocity and acceleration vectors.

Remark : assuming a diagonal form of the mass and the damping matrices, displacements and velocities can be updated without equation solving.

with2D : quadrilateral

extt t t t

Finite element formulation

M u C u K u F+ + =&& &t t t t t

t 2

t t t tn

u 2u uu

tu u

u2 t

D D

D D

D

D

+ -

+ -

- +ìïï =ïïí -ïï =ïïî

&&

&

tu&&tu&ext

tF


with λt the contact forces vector acting on the nodes of theslave surface,

Gt+Δt the global matrix of the constraint (normal and tangential contact conditions)

Xt+Δt=Xt+ut+Δt-ut the coordinate vector at time t+Δt.

Lagrange multiplier method equations set is built up using equation of motion at time t and the displacement constraints acting on the contacting surfaces at time t+Δt (implicit contact treatment)

Lagrange multiplier method

Ω2

Ω1int T extt t t t t t t

t t t t

M u C u F G F

G X 0D

D D

l+

+ +

ì + + + =ïïïíï £ïïî

&& &

Forces conditions :No bonding

Coulomb’s friction law with μ constant

i

i i

( n )

( t ) ( n )

0l

l ml

ìï £ïïíï £ïïî

ur

r ur

i i

( n ) ( t )i slave nodes( , )l l =

rur


2D Model description

400 (40*10) elements de 0.002x0.0025 m

No thermal effect

No physico-chemical effect

Perfectly smooth surfaces (no roughness)

E = 10 000 MPaν = 0.3ρ = 2000 kg/m3

μ : Coulomb coefficientP = 1 MPaV = 2 m/s

Constant interface Coulomb friction coefficient μ

⇒ Contact with friction of a elastic body on a rigid surface

P pressure

V=ConstantVelocity of the rigid surface

L=0.1 mh

= 0.

02 m

x

yRigid

The friction law used is the classical Coulomb friction model without regularization of the tangential force versus the tangential velocity component.

Pad : deformable body E, ν


V=Velocity of the rigid surface

Local dynamics : stable/unstable state

Stable state=No instabilityLocal sliding

Unstable state=periodic steady state.

Local sliding

Local sticking

Local separation

µ = 0.05 P = 1 MPa V = 2 m/s

V=Velocity of the rigid surface

µ = 0.6 P = 1 MPa V = 2 m/s

⇒ Instabilities characterized by the appearance of sliding-sticking-separation waves

i i

( t ) ( n )l m l=r ur

i i

( t ) ( n )l m l=r ur

i i

( t ) ( n )l m l<r ur

i i

( t ) ( n ) 0l l= =r ur


Local dynamics : Limit cycleD

ispl

acem

ent /

y (µ

m)

0

1

2

3

-2 0 2 4 6Displacement / x (µm)

2

1

0

1

2

3

0 2 4 6Displacement / x (µm)

Dis

plac

emen

t / y

(µm

)

2

1

P

V

µ = 0.4

P = 1 MPa

V = 2 m/sx

y

0

1

2

3


Dis

plac

emen

t / y

(µm

)

1

2

⇒ at whatever node of the surface of the pad there is a limit cycle decomposes into a :

1=displacement with contact (sliding or sticking)

2=displacement corresponding to the elastic return (no contact with the rigid surface)

Pad : deformable body E, ν


Influence of local dynamics

Global interface movement

“normal impact” : high normal velocity (Vn ≥ 1.5 m/s)“sliding” : higher sliding velocity (Vt = 3 m/s > 2 m/s)“high pressure” : σn max ≈ 10 MPa >> P=1 MPa“repetitive instabilities” : high frequency (F = 68 kHz)

Local interface movementinstabilities (stick-slip-separation)

Understand generation of instabilities (noise, earthquake…)

Understand particle detachment (wear)


Study of friction coefficient

Impose a constant local friction coefficient µinterface at the contact nodes

Calculate a global friction coefficient µapparent = µ* (≅ experimental coefficient)

µ* = |Σ Ti / Σ Ni|

Evolution of µ* with respect of V et P ?

P

VNi

→ Ti

→

i

( t )lr

i

i

i

( t )( n )

int erface( n )if 0, cons tan tl

l ml

¹ £ =

rur

ur

i

( n )lur


Study of friction coefficient : influence of V

A : Stick-slip waves + separation appearing: V ⇒ µ* B : Stick-slip-separation waves: V ⇒ µ* C : Slip-separation waves: V ⇒ µ*

µ* ≤ µinterface

µinterface = 0.4

P = 1 MPaB

A

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6Velocity V (m/s)

Glo

bal f

rictio

n co

ef.

µ*

0.4

C P

V

NT

µ*

int erface 0.4m =


Study of friction coefficient : influence of P

A : Stick-slip-separation waves: P ⇒ µ* B : Stick-slip waves + separation disappearing: P ⇒ µ* C : Slip + stick disappearing: P ⇒ µ*

µ* ≤ µinterface

V = 2 m/s

µinterface = 0.4

BA

0

0.1

0.2

0.3

0.4

0 10 20 30 40Distributed pressure (MPa)

Glo

bal f

rictio

n co

ef.

µ*

4

CP

V

NT

µ*

int erface 0.4m =


ωF

x

yz

Friction coefficient of Coulomb type 0.5

3D Simulation of braking

Normal force imposed on the pad F [N] 25000 Equivalent pressure [MPa] 7.8 Disc speed ω [rad/s] 47.6 Time step of the simulation Δt [s] 0.1*10-6

Numerical damping β2 0.6 Viscous damping βv 0.25*10-6

Simulation parameters

Disc (steel)

Brake Pad

3D Contact with friction between two deformable bodies

The boundary conditions of the model :-the basic force F is applied to all the nodesof the upper area of the brake pad

-the upper area nodes of the brake padare constrained in x and y direction.-the nodes belonging to the inner disc radius are constrained in the z-axisand have an imposed rotative speed ω.

⇒ Next slide : video of the change from a stable state to an unstable stateCharacterized by a contact where the nodes do not stick and

stay in a sliding contact on the moving disc during the simulationCharacterized by the appearance of contact zones (sliding

and sticking) and of separated zones with the disc area


ω

Sliding

Stick-slip-Separation

Sliding-separation

Sliding-separation

Contact zones of the brake pad and their status

Fourier transform of the acceleration atone surface point of the pad and disc

15000 Hz

41000 150000

Movie

IterationsNor

mal

and

tang

entia

l co

ntac

t for

ces

Isovalues of the speed on zTotal time of the simulation = 5ms

Friction generates instabilities characterized byappearance of stick-slip-separation waves

⇒ gives periodic spectrum with the main peak placedat 15 kHz and whole number multiples.

ω


Modal analysisNatural mode (1,4) of the disc(f= 15195 Hz)

Normal speed on z ⇒ enables the disc vibration to be visualised⇒ the disc vibration frequency is 15 000Hz

Vibrations of the disc

z

The disc vibrates with a 15kHz mode which is the same frequency as the contact area phenomenon of the brake pad and the disc

z


Dynamic rupture in a frictional interface

Dynamic rupture converts elastic strain stored in the media to kinetic energy (wave), dissipative energy near the failure surface (heat), dissipation in the bulk(creation of new surface area and inelastic strain) [Shi et al., JMPS 2008]

Rupture on a frictional interface in homogeneous solid can occur in either- crack-like mode- pulse-like mode

[Lykotrafitis et al., Science 2006]

Crack-like mode : slipping region expands continuously until the rupture terminates

Pulse-like mode : small portion of the interface slips at any given time

Speed of rupture propagation = interest topic

subshear or supershear ?[Xia et al., Science 2004]


Model configuration

Nucleation zone

Uniform compressive normal stress σn0Uniform shear stress τ0Nucleation process that initiates the dynamic rupture events

Two identical elastic media

p( 1 )EV 2613 m / s

( 1 )( 1 2 )n

r n n-= =

+ -

sE GV 1255 m / s

2 ( 1 )r n r= = =

+

RC 1174 m / s=

Plane strain problem, speed of P waves

speed of S waves

speed of Rayleigh waves

Frictional interfaces

τ0

σn0

τ0

σn0

Ω2

Ω1


Simulation stages

Stickingfrictional interfaces

τ0

σn0

τ0

σn0

Ω2

Ω1

Ω2

Ω1

!! Amplification of the deformed shape !!

τ0

σn0

τ0

σn0

Ω2

Ω1

Nucleation zoneτ =0 at t=0

Dynamic rupturepropagation

t>0


Friction laws

Coulomb friction law

δ

μ

Bi-linear slip-weakening friction(μ s, μ d) are the static and dynamic friction coefficients,D is the critical slip

μsμd

D

μ

μs =μd

δ

δ tangential relative displacementμ friction coefficient

[Ohnaka, Science 2004] “Earthquake rupture is a mixture of frictional slip failure and the fracture of initially intact rock”


Simulation results

By varying parameters (μ s, μ d, D) we obtain four different rupture modes

- Supershear crack like rupture- Subshear crack like rupture

- Supershear pulse like rupture- Subshear pulse like rupture

Same uniform compressive normal stress σn0 , same uniform shear stress τ0, same nucleation length Lnucleation

δ

μ

μ0=0.15

0.3

0.0001 0.0002

Bi-linear slip-weakening friction


Nucleation zoneτ =0

Ω2

Ω1

||Velocity|| iso values

- Supershear crack like rupture μ s=0.16, μ d=0.11, D=0.0001

Supershear crack like rupture

Frictional interfaceBlue = sticking

Red=slidingn

n

t m s

t m s

<

=


Supershear crack like rupture

Red=sliding

Blue = sticking

P waves

Rupture-tips

τ0

σn0

τ0

σn0

Mach cone

Ω2

Ω1

Vrupture≈2135 m/s

p

s

V 2613 m / s

V 1255 m / s

=

=


- Subshear crack like rupture μ s=0.16, μ d=0.12, D=0.0002

Subshear crack like rupture

Ω2

Ω1




Red=slidingn

n

t m s

t m s

<

=


Subshear crack like rupture

Red=sliding

Blue = sticking

P wavesRupture-tips

S waves

τ0

σn0

τ0

σn0

Ω2

Ω1

Vrupture≈1250 m/sp

s

V 2613 m / s

V 1255 m / s

=

=


- Supershear pulse like rupture μ s=0.3, μ d=0.05, D=0.00001

Supershear pulse like rupture

Ω2

Ω1




Red=slidingn

n

t m s

t m s

<

=


Supershear pulse like rupture

Red=sliding

Blue = sticking

P wavesRupture-tips

S waves

τ0

σn0

τ0

σn0

Mach cone

Ω2

Ω1


s

V 2613 m / s

V 1255 m / s

=

=


- Subshear pulse like rupture μ s=0.3, μ d=0.05, D=0.0001

Subshear pulse like rupture

Ω2

Ω1




Red=slidingn

n

t m s

t m s

<

=


Subshear pulse like rupture

Blue = sticking

P wavesRupture-tips

S waves

τ0

σn0

τ0

σn0

Red=sliding

Ω2

Ω1


s

V 2613 m / s

V 1255 m / s

=

=


Discussion

Crack-like = slip rate everywhere behind the propagating rupture fronts

Pulse-like = slip rate is non zero only in narrow regions behind the rupture

Ruptures

RupturesTime

Tan

gent

ial r

elat

ive

disp

lace

men

tNucleation zone

TIME

Tang

entia

l rel

ativ

e di

spla

cem

ent

Nucleation zone


Subshear Crack to supershear Pulse

Crack

Pulses

Supershearspeed

Subshearspeed

(200000 finite elements)

Video


Coulomb friction law μ =0.16 No propagation (for this simulation !)


P waves

S waves


Thank you for your attention

Aup du seuil (Grenoble – Chartreuse)


Conclusion

Modelisation developed describes:

⇒ the instabilities generated by contact, the appearance of body vibrations responsible for squealing

⇒ the local cinematic of the contact surface and therefore :• the distribution of the contact pressures, stress, deformation• the tribological state of the instantaneous contact zones :

sticking, sliding, separation• how the instantaneous zones ensure continuous macroscopicsliding

⇒ at the same time what occurs in the contact (tribology) and exchanges withthe outside (acoustic)

⇒ a model which can be parameterized to separate the role of the mechanism(boundary conditions), from the role of the first bodies (Young Modulus, Poisson coefficient) and from that of the third body (rheology)

35NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….Percentage error of the natural mode frequencies

for the three mesh, type M1, M2, M3

Convergence study for the mechanicaland vibration aspect


ωF

x

yz

3D Simulation of braking

Friction coefficient of Coulomb type 0.5 Normal force imposed on the pad F [N] 25000 Equivalent pressure [MPa] 7.8 Disc speed ω [rad/s] 47.6 Time step of the simulation Δt [s] 0.1*10-6

Numerical damping β2 0.6 Viscous damping βv 0.25*10-6

Simulation parameters

Disc :Young’s modulus E [MPa] 210000Poisson’s coefficientυ 0.3Volumic mass ρ [Kg/m3 ] 7800Interior radius ri [mm] 60Exterior radius re [mm] 150Thickness e [mm] 20

Pad:

Young’s modulus E [MPa] 10000Poisson’s coefficientυ 0.3Volumic mass ρ [Kg/m3 ] 2500Width (x), length (y), height (z) [mm] 40 x 80 x 20

Characteristics of the brake pad and the disc

Contact with friction between two deformable bodies

The boundary conditions of the model :-the basic force F is applied to all the nodesof the upper area of the brake pad

-the upper area nodes of the brake padare constrained in x and y direction.-the nodes belonging to the inner disc radius are constrained in the z-axisand have an imposed rotative speed ω.

⇒ Next slide : video of the change from a stable state to an unstable stateCharacterized by a contact where the nodes do not stick and

stay in a sliding contact on the moving disc during the simulationCharacterized by the appearance of contact zones (sliding

and sticking) and of separated zones with the disc area


Local dynamics : influence on velocity & stress

“normal impact” : high normal velocity (Vn ≥ 1.5 m/s)“sliding” : higher sliding velocity (Vt = 3 m/s > 2 m/s)“repetitive instabilities” : high frequency (F = 68 kHz)“high pressure” : Pmax≈ 10 MPa >> P=1 MPa

0

1

2

3


Dis

plac

emen

t / y

(µm

)

SlipStickSeparation

“impact”µ = 0.4P = 1 MPaV = 2 m/s

P

V

µ

Increase of the contact stress due - as much to the reduction of the contact area- as to the kinematics (impact) of the surfaces

instabilities and dynamic rupture in a frictional · pdf filensf 2008 workshop on friction -...

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