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Contact mechanics and elements of tribology

Lecture 7.

Two words about friction

Vladislav A. Yastrebov

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

@ Centre des Mat´eriaux February 11, 2016

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Coulomb’s friction

One-dof frictional problem

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Coulomb’s friction

One-dof frictional problem

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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

•µsstatic andµkkinetic coefficients of friction.

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Theory and simulations

?Theory

1 M. Renardy, Ill-posedness at the boundary for elastic solids sliding under Coulomb friction, J. Elast. 27:281-287 (1992).

2 G. G. Adams, Self-excited oscilations of two elastic half-spaces sliding with a constant coefficient of friction, J. Appl. Mech. 62:867-872 (1995).

3 J. A. C. Martins, F. M. F. Sim ˜oes, On some sources of instability/ill-posedness in elasticity problems with Coulomb’s friction, in Contact Mechanics, eds. M.

Raous, M. Jean, J.-J. Moreau, 95-106 (1995).

?Simulation

1 D. J. Andrews, Y. Ben-Zion, Wrinkle-like slip pulse on a fault between different materials, J. Geophys. Res., 102:553-571 (1997).

2 R. A. Harris, S. M. Day, Effect of a low-velocity zone on a dynamic rupture, Bull.

Seismol. Soc. Am., 87:1267-1280 (1997).

3 A. Cochard, J. R. Rice, Fault rupture between dissimilar materials: ill-posedness, regularization, and slip-pulse response, J. Geophys. Res. 105:25891 (2000).

?Theory

1 K. Ranjith, J. R. Rice, Slip dynamics at an interface between dissimilar materials.

J. Mech. Phys. Solids 49:341-361 (2001).

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Rate and state friction and regularization

Rate and state friction law Ratevt=|vt|– relative slip velocity

Stateθ–≈internal time Dieterich–Ruina–Perrin (1979, 83, 95)

Frictional resistance

σct=|σn|fs+bθ+aln(vt/v0) Evolution of the state variable θ˙=−vt

L

hθ+lnvt

v0

i

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

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One story

Spliting of slip pulses,

[1] D. J. Andrews, Y. Ben-Zion, Wrinkle-like slip pulse on a fault between different materials, J.

Geophys. Res., 102:553-571 (1997)

Slip pulse splitting[1]

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One story

Spliting of slip pulses,

[1] D. J. Andrews, Y. Ben-Zion, Wrinkle-like slip pulse on a fault between different materials, J.

Geophys. Res., 102:553-571 (1997)

Problems with mesh convergence

Mesh convergence problems[2]

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One story

Spliting of slip pulses,

[1] D. J. Andrews, Y. Ben-Zion, Wrinkle-like slip pulse on a fault between different materials, J.

Geophys. Res., 102:553-571 (1997)

Problems with mesh convergence Regularized Prakash-Clifton law σ˙t=−vt

Lt+µσn)

[2] A. Cochard, J. R. Rice, Fault rupture between dissimilar materials:

ill-posedness, regularization, and slip-pulse response, J. Geophys. Res.

105:25891 (2000)

Theory

[3] K. Ranjith, J. R. Rice, Slip dynamics at an interface between dissimilar materials. J. Mech. Phys.

Solids 49:341-361 (2001)

Using Prakash-Clifton friction law regularizes the problem[2]

Table for two combinations of materials[2]

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Introduction: Regularized friction law

Elastodynamic frictional contact problem may be ill-posed/unstable For two identical materials: no problems

Two different materials:

If generalized Rayleigh speed is defined: unstable

If generalized Rayleigh speed is not-defined: stable forf <fc

Critical friction exists.

V.A. Yastrebov 10/109

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Introduction: Regularized friction law

Elastodynamic frictional contact problem may be ill-posed/unstable For two identical materials: no problems

Two different materials:

If generalized Rayleigh speed is defined: unstable

If generalized Rayleigh speed is not-defined: stable forf <fc

Critical friction exists.

Possible solutions

Abandon the proportionality of frictional strength to contact pressure

|Ft|,µ|Fn| Viscous analogy

Experiment based Prakash-Clifton friction law [1]

t=−|V|+V0

L

Ft−µmax(0,|Fn|) Non-local friction [2]

|Ft(x)|=µ2L1 RL

L

|Fn(x−s)|w(x−s)ds

[1] Prakash, Clifton. Time resolved dynamic friction measurements in pressure-shear. In Experimental Techniques in the Dynamics of Deformable Solids, edited by Ramesh, vol. 165:33-48, ASME, New York (1993).

[2] J. A. C. Martins, F. M. F. Sim ˜oes, On some sources of instability/ill-posedness in elasticity problems with Coulomb’s friction, in Contact Mechanics, eds. M. Raous, M. Jean, J.-J. Moreau, 95-106 (1995).

V.A. Yastrebov 11/109

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Introduction: Regularized friction law

V.A. Yastrebov 12/109

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Introduction: lab experiments

1 K. Xia,A. J. Rosakis, and H. Kanamori, Science 303, 1859 (2004).

2 S. M. Rubinstein, G. Cohen, andJ. Fineberg, Nature 430, 1005 (2004).

3 S. M. Rubinstein, G. Cohen, andJ. Fineberg, Phys. Rev. Lett. 98, 226103 (2007).

4 O. Ben-David, G. Cohen, andJ. Fineberg, Science 330, 211 (2010).

5 S. Latour, T. Gallot, S. Catheline, C. Voisin, F. Renard, E. Larose, and M. Campillo, Europhys. Lett. 96, 59003 (2011).

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Conclusion

It was an overview

It is up to you to dig and understand the details

“The devil is in the detail” or “God is in the detail”

Good luck to you & Bon courage !

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