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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/
2NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
3NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
4NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
5NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
6NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
7NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
8NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
0 2 4 6Displacement / x (µm)
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
9NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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)
10NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
11NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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 =
12NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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 =
13NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
14NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
15NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
16NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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]
17NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
18NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
19NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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”
20NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
21NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
<
=
22NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
=
=
23NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
- Subshear crack like rupture μ s=0.16, μ d=0.12, D=0.0002
Subshear crack like rupture
Ω2
Ω1
||Velocity|| iso values
Nucleation zoneτ =0
Frictional interfaceBlue = sticking
Red=slidingn
n
t m s
t m s
<
=
24NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
=
=
25NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
- Supershear pulse like rupture μ s=0.3, μ d=0.05, D=0.00001
Supershear pulse like rupture
Ω2
Ω1
||Velocity|| iso values
Nucleation zoneτ =0
Frictional interfaceBlue = sticking
Red=slidingn
n
t m s
t m s
<
=
26NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
Supershear pulse like rupture
Red=sliding
Blue = sticking
P wavesRupture-tips
S waves
τ0
σn0
τ0
σn0
Mach cone
Ω2
Ω1
Vrupture≈2200 m/sp
s
V 2613 m / s
V 1255 m / s
=
=
27NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
- Subshear pulse like rupture μ s=0.3, μ d=0.05, D=0.0001
Subshear pulse like rupture
Ω2
Ω1
||Velocity|| iso values
Nucleation zoneτ =0
Frictional interfaceBlue = sticking
Red=slidingn
n
t m s
t m s
<
=
28NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
Subshear pulse like rupture
Blue = sticking
P wavesRupture-tips
S waves
τ0
σn0
τ0
σn0
Red=sliding
Ω2
Ω1
Vrupture≈1110 m/sp
s
V 2613 m / s
V 1255 m / s
=
=
29NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
30NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
Subshear Crack to supershear Pulse
Crack
Pulses
Supershearspeed
Subshearspeed
(200000 finite elements)
Video
31NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
Coulomb friction law μ =0.16 No propagation (for this simulation !)
||Velocity|| iso values
P waves
S waves
32NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
Thank you for your attention
Aup du seuil (Grenoble – Chartreuse)
33NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
34NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
36NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
37NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture….
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
0 2 4 6Displacement / x (µm)
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