Instabilities and Dynamic Rupture in a Frictional Interface
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1 Instabilities and Dynamic Rupture in a Frictional Interface Laurent BAILLET LGIT (Laboratoire de Géophysique Interne et Tectonophysique) Grenoble France laurent.baillet@ujf-grenoble.fr NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 1
2 Outline Variational formulation of the Signorini problem with friction P 2D Model description V=Velocity of the rigid surface - Local dynamics : stable/unstable state Limit cycle influence on velocity & stress - Study of friction coefficient 3D Simulation of braking Dynamic rupture in a frictional interface 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 NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 2
3 Variational formulation of the Signorini problem with friction The problem of unilateral contact with Coulomb friction law consists in finding the displacement andu the second order stress tensor s(u) satisfying the equation of the mechanics Unilateral contact Coulomb friction law s(u) = D e(u) div s( u) + f = ru& s(u).n = P u = u imp u(t ) = u ; u(t & ) = u& [ u.n] 0, s 0, s.[ u.n] = 0 on G n n u imp NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. P r P r P ur f applied pressure u imp t < ms n Þ sticking éu ù ê& r = 0 s n ë t úû G t r t t = ms n Þ sliding $ x ³ 0 s.t. éu r ù ê& = - xt n ur ë t úû 3
4 Finite element formulation PLASTD (freeware developed by L.BAILLET) is based on a dynamic explicit method and includes large deformations and non linear material behavior analysis in TIME The 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 M u& + C u& + K u = F ext t t t t with ìï u& = ï í ï u& n = ïî u - 2u + u t+ Dt t t- Dt t 2 u - Dt u t+ Dt t- Dt 2Dt 2D : quadrilateral where M and C are respectively the mass and damping matrices ext F t is the nodal vectors of external forces and u& the velocity and acceleration vectors. u& t t Remark : assuming a diagonal form of the mass and the damping matrices, displacements and velocities can be updated without equation solving. NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 4
5 Lagrange multiplier method 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) ì M u& + C u& + F + G l = F ï í ï Gt+ DtX t+ Dt 0 ïî int T ext t t t t+ Dt t t Ω 1 ur r (n) (t) ( l, l ) i = slave nodes i i Ω 2 with λ t the contact forces vector acting on the nodes of the slave surface, G t+δt the global matrix of the constraint (normal and tangential contact conditions) X t+δt =X t +u t+δt -u t the coordinate vector at time t+δt. Forces conditions : ur (n) ìï l ï í ï ïî i r ï (t) l i 0 ml ur (n) i No bonding Coulomb s friction law with μ constant NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 5
6 2D Model description y x h = 0.02 m Constant interface Coulomb friction coefficient μ P pressure Rigid Pad : deformable body E, ν V=ConstantVelocity of the rigid surface L=0.1 m 400 (40*10) elements de 0.002x m No thermal effect No physico-chemical effect Perfectly smooth surfaces (no roughness) E = MPa ν = 0.3 ρ = 2000 kg/m 3 μ : Coulomb coefficient P = 1 MPa V = 2 m/s Contact with friction of a elastic body on a rigid surface The friction law used is the classical Coulomb friction model without regularization of the tangential force versus the tangential velocity component. NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 6
7 Local dynamics : stable/unstable state µ = 0.05 P = 1 MPa V = 2 m/s Stable state=no instability Local sliding l r (t) i = ml ur (n) i V=Velocity of the rigid surface µ = 0.6 P = 1 MPa V = 2 m/s Unstable state=periodic steady state. Local sliding l r (t) i = ml Local sticking l r (t) i < ml ur (n) i ur (n) i V=Velocity of the rigid surface Local separation Instabilities characterized by the appearance of sliding-sticking-separation waves NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. l r (t) i ur (n) = l = i 0 7
8 Local dynamics : Limit cycle y x P Pad : deformable body E, ν µ = 0.4 P = 1 MPa V = 2 m/s Displacement / y (µm) Displacement / x (µm) Displacement / y (µm) Displacement / x (µm) 6 Displacement / y (µm) V Displacement / x (µm) 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) NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 8
9 Influence of local dynamics Global interface movement Local interface movement instabilities (stick-slip-separation) normal impact : high normal velocity (V n 1.5 m/s) sliding : higher sliding velocity (V t = 3 m/s > 2 m/s) high pressure : σ nmax 10 MPa >> P=1 MPa repetitive instabilities : high frequency (F = 68 khz) Understand generation of instabilities (noise, earthquake ) Understand particle detachment (wear) NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 9
10 Study of friction coefficient P N i Impose a constant local friction coefficient µ interface at the contact nodes ur l l r (t) Calculate a global friction coefficient µ apparent = µ* ( experimental coefficient) µ* = Σ T i / Σ N i Evolution of µ* with respect of V et P? NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 10 T i l r (t) (n) i if l ¹ 0, ur m i (n) int erface = constan t l ur (n) i i i V
11 0.4 Study of friction coefficient : influence of V µ interface = 0.4 Global friction coef. µ* B C A Velocity V (m/s) P = 1 MPa T µ* P N V m int erface = 0.4 µ* µ interface A : Stick-slip waves + separation appearing: B : Stick-slip-separation waves: C : Slip-separation waves: V µ* V µ* V µ* NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 11
12 0.4 Study of friction coefficient : influence of P µ interface = 0.4 Global friction coef. µ* 0.3 B 0.2 A C Distributed pressure (MPa) V = 2 m/s T µ* P N V m int erface = 0.4 µ* µ interface A : Stick-slip-separation waves: B : Stick-slip waves + separation disappearing: C : Slip + stick disappearing: P µ* P µ* P µ* NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 12
13 3D Simulation of braking z y ω F Disc (steel) Brake Pad x 3D Contact with friction between two deformable bodies The boundary conditions of the model : -the basic force F is applied to all the nodes of the upper area of the brake pad -the upper area nodes of the brake pad are constrained in x and y direction. -the nodes belonging to the inner disc radius are constrained in the z-axis and have an imposed rotative speed ω. Friction coefficient of Coulomb type 0.5 Normal force imposed on the pad F [N] 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 β Viscous damping β v 0.25*10-6 Simulation parameters Next slide : video of the change from a stable state to an unstable state Characterized by a contact where the nodes do not stick and stay in a sliding contact on the moving disc during the simulation Characterized by the appearance of contact zones (sliding and sticking) and of separated zones with the disc area NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 13
14 Total time of the simulation = 5ms Contact zones of the brake pad and their status Sliding ω ω Stick-slip- Separation Normal and tangential contact forces Sliding-separation Sliding-separation Movie Iterations Isovalues of the speed on z Fourier transform of the acceleration at one surface point of the pad and disc Hz Friction generates instabilities characterized by gives periodic spectrum with the main peak placed NSF appearance 2008 Workshop of stick-slip-separation friction - Laurent waves BAILLET - LGIT at 15 khz - Instabilities and whole and number Dynamic multiples. Rupture. 14
15 Vibrations of the disc z z Normal speed on z enables the disc vibration to be visualised the disc vibration frequency is Hz Modal analysis Natural mode (1,4) of the disc (f= Hz) The disc vibrates with a 15kHz mode which is the same frequency as the contact area phenomenon of the brake pad and the disc NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 15
16 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] NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 16
17 Model configuration Ω 1 σ n0 τ 0 Nucleation zone Two identical elastic media Plane strain problem, speed of P waves V p (1- n)e = = 2613 m / s r(1 + n)(1-2 n) speed of S waves Frictional interfaces Ω 2 V s = E G 2 r(1 + n) = r = 1255 m / s τ 0 σ n0 speed of Rayleigh waves C R = 1174 m / s Uniform compressive normal stress σ n0 Uniform shear stress τ 0 Nucleation process that initiates the dynamic rupture events NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 17
18 Simulation stages Ω 1 Sticking frictional interfaces Ω 1 σ n0 τ 0 Ω 2 Ω 2 τ 0 σ n0!! Amplification of the deformed shape!! σ n0 τ 0 Ω 1 Ω 2 τ 0 Nucleation zone τ =0 at t=0 Dynamic rupture propagation t>0 σ n0 NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 18
19 Friction laws μ μ s =μ d μ μ s μ d δ D δ δ tangential relative displacement μ friction coefficient Coulomb friction law Bi-linear slip-weakening friction (μ s, μ d ) are the static and dynamic friction coefficients, D is the critical slip [Ohnaka, Science 2004] Earthquake rupture is a mixture of frictional slip failure and the fracture of initially intact rock NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 19
20 Simulation results Same uniform compressive normal stress σ n0, same uniform shear stress τ 0, same nucleation length L nucleation By varying parameters (μ s, μ d, D) we obtain four different rupture modes 0.3 μ - Supershear crack like rupture - Subshear crack like rupture μ 0 =0.15 δ - Supershear pulse like rupture - Subshear pulse like rupture Bi-linear slip-weakening friction NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 20
21 Supershear crack like rupture - Supershear crack like rupture μ s =0.16, μ d =0.11, D= Ω 1 Nucleation zone τ =0 Frictional interface Blue = sticking t Red=sliding t < = ms ms n n Ω 2 Velocity iso values NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 21
22 Supershear crack like rupture σ n0 τ 0 Ω 1 Red=sliding Mach cone Blue = sticking P waves Rupture-tips Ω 2 V rupture 2135 m/s τ 0 σ n0 V V p s = = 2613 m / s 1255 m / s NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 22
23 Subshear crack like rupture - Subshear crack like rupture μ s =0.16, μ d =0.12, D= Ω 1 Nucleation zone τ =0 Frictional interface Blue = sticking t Red=sliding t < = ms ms n n Ω 2 Velocity iso values NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 23
24 Subshear crack like rupture σ n0 τ 0 Ω 1 Red=sliding Blue = sticking S waves Rupture-tips P waves τ 0 Vp = 2613 m / s σ n0 Vs = 1255 m / s NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 24 Ω 2 V rupture 1250 m/s
25 Supershear pulse like rupture - Supershear pulse like rupture μ s =0.3, μ d =0.05, D= Ω 1 Nucleation zone τ =0 Frictional interface Blue = sticking t Red=sliding t < = ms ms n n Ω 2 Velocity iso values NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 25
26 Supershear pulse like rupture σ n0 τ 0 Ω 1 S waves Mach cone Red=sliding Blue = sticking Rupture-tips Ω 2 P waves V rupture 2200 m/s τ 0 Vp = 2613 m / s σ n0 Vs = 1255 m / s NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 26
27 Subshear pulse like rupture - Subshear pulse like rupture μ s =0.3, μ d =0.05, D= Nucleation zone τ =0 Ω 1 Frictional interface Blue = sticking t Red=sliding t < = ms ms n n Ω 2 Velocity iso values NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 27
28 Subshear pulse like rupture σ n0 τ 0 Ω 1 S waves Red=sliding Blue = sticking Rupture-tips Ω 2 P waves V rupture 1110 m/s τ 0 Vp = 2613 m / s σ n0 Vs = 1255 m / s NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 28
29 Discussion Crack-like = slip rate everywhere behind the propagating rupture fronts Nucleation zone Ruptures Tangential relative displacement Pulse-like = slip rate is non zero only in narrow regions behind the rupture TIME Ruptures Tangential relative displacement Nucleation zone NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 29 Time
30 Subshear Crack to supershear Pulse Video Subshear speed Supershear speed Crack Pulses ( finite elements) NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 30
31 Coulomb friction law μ =0.16 No propagation (for this simulation!) P waves S waves Velocity iso values NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 31
32 Thank you for your attention Aup du seuil (Grenoble Chartreuse) NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 32
33 NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 33
34 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 macroscopic sliding at the same time what occurs in the contact (tribology) and exchanges with the 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) NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 34
35 Convergence study for the mechanical and vibration aspect Percentage error of the natural mode frequencies NSF 2008 Workshop on friction for the - Laurent three mesh, BAILLET type M1, - LGIT M2, -M3 Instabilities and Dynamic Rupture. 35
36 3D Simulation of braking ω z F Contact with friction between two deformable bodies The boundary conditions of the model : -the basic force F is applied to all the nodes of the upper area of the brake pad -the upper area nodes of the brake pad are constrained in x and y direction. -the nodes belonging to the inner disc radius are constrained in the z-axis and have an imposed rotative speed ω. y x Disc : Young s modulus E [MPa] Poisson s coefficient υ 0.3 Volumic mass ρ [Kg/m 3 ] 7800 Interior radius r i [mm] 60 Exterior radius r e [mm] 150 Thickness e [mm] 20 Pad : Young s modulus E [MPa] Poisson s coefficient υ 0.3 Volumic mass ρ [Kg/m 3 ] 2500 Width (x), length (y), height (z) [mm] 40 x 80 x 20 Characteristics of the brake pad and the disc Friction coefficient of Coulomb type 0.5 Normal force imposed on the pad F [N] 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 β Viscous damping β v 0.25*10-6 Simulation parameters Next slide : video of the change from a stable state to an unstable state Characterized by a contact where the nodes do not stick and stay in a sliding contact on the moving disc during the simulation Characterized by the appearance of contact zones (sliding and sticking) and of separated zones with the disc area NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 36
37 Local dynamics : influence on velocity & stress P µ V µ = 0.4 P = 1 MPa V = 2 m/s Displacement / y (µm) impact Slip Stick Separation Displacement / x (µm) normal impact : high normal velocity (V n 1.5 m/s) sliding : higher sliding velocity (V t = 3 m/s > 2 m/s) repetitive instabilities : high frequency (F = 68 khz) high pressure : P max 10 MPa >> P=1 MPa Increase of the contact stress due - as much to the reduction of the contact area - as to the kinematics (impact) of the surfaces NSF 2008 Workshop on friction - Laurent BAILLET - LGIT - Instabilities and Dynamic Rupture. 37
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