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Research Papers

Self-Excited Oscillations of a Finite-Thickness Elastic Layer Sliding Against a Rigid Surface With a Constant Coefficient of Friction

[+] Author and Article Information
Neda Karami Mohammadi

Department of Mechanical and
Industrial Engineering,
Northeastern University,
Boston, MA 02115
e-mail: nkaramim@yahoo.com

George G. Adams

Professor
Department of Mechanical and
Industrial Engineering,
Northeastern University,
Boston, MA 02115
e-mail: adams@coe.neu.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 27, 2017; final manuscript received November 28, 2017; published online December 12, 2017. Assoc. Editor: Kyung-Suk Kim.

J. Appl. Mech 85(2), 021005 (Dec 12, 2017) (7 pages) Paper No: JAM-17-1603; doi: 10.1115/1.4038640 History: Received October 27, 2017; Revised November 28, 2017

This investigation considers the dynamic stability of the steady-state frictional sliding of a finite-thickness elastic layer pressed against a moving rigid and flat surface of infinite extent. The elastic layer is fixed on its bottom surface; on its entire top surface, the rigid surface slides with constant speed and with a constant friction coefficient. The plane-strain equations of motion for a linear isotropic elastic solid are solved analytically for small dynamic disturbances. The analysis shows that even with a constant (speed-independent) friction coefficient, the steady solution is dynamically unstable for any finite friction coefficient. Eigenvalues with positive real parts lead to self-excited vibrations which occur for any sliding speed and which increase with increasing coefficient of friction. This is in contrast to the behavior of an elastic half-space sliding against a rigid surface in which the instability only occurs if the coefficient of friction is greater than unity. This work and its extensions are expected to be relevant in the theoretical aspects of sliding friction as well as in a variety of areas such as earthquake motion and brake dynamics.

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References

Schallamach, A. , 1971, “ How Does Rubber Slide?,” Wear, 17(4), pp. 301–312. [CrossRef]
Adams, G. G. , 1995, “ Self-Excited Oscillation of Two Elastic Half-Spaces Sliding With a Constant Coefficient of Friction,” ASME J. Appl. Mech., 62(4), pp. 867–872. [CrossRef]
Martins, J. A. C. , Guimarães, J. , and Faria, L. O. , 1995, “ Dynamic Surface Solutions in Linear Elasticity and Viscoelasticity With Frictional Boundary Conditions,” ASME J. Vib. Acoust., 117(4), pp. 445–451. [CrossRef]
Ranjith, K. , and Rice, J. R. , 2001, “ Slip Dynamics at an Interface Between Dissimilar Materials,” J. Mech. Phys. Solids, 49(2), pp. 341–361. [CrossRef]
Prakash, V. , and Clifton, R. J. , 1993, “ Time Resolved Dynamic Friction Measurements in Pressure-Shear,” Exp. Tech. Dyn. Deform. Solids, 165, pp. 33–48.
Prakash, V. , 1998, “ Frictional Response of Sliding Interfaces Subjected to Time Varying Normal Pressure,” ASME J. Tribol., 120(1), pp. 97–102. [CrossRef]
Martins, J. A. C. , Oden, J. T. , and Simões, F. M. F. , 1990, “ A Study of Static and Kinetic Friction,” Int. J. Eng. Sci., 28(1), pp. 29–92. [CrossRef]
Adams, G. G. , 1998, “ Steady Sliding of Two Elastic Half-Spaces With Friction Reduction Due to Interface Stick-Slip,” ASME J. Appl. Mech., 65(2), pp. 470–475. [CrossRef]
Adams, G. G. , 2000, “ Radiation of Body Waves Induced by the Sliding of an Elastic Half-Space Against a Rigid Surface,” ASME J. Appl. Mech., 67(1), pp. 1–5. [CrossRef]
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Brener, E. A. , Weikamp, M. , Spatschek, R. , Bar-Sinai, Y. , and Bouchbinder, E. , 2016, “ Dynamic Instabilities of Frictional Sliding at a Bimaterial Interface,” J. Mech. Phys. Solids, 89, pp. 149–173. [CrossRef]
Behrendt, J. , Weiss, C. , and Hoffmann, N. P. , 2011, “ A Numerical Study on Stick–Slip Motion of a Brake Pad in Steady Sliding,” J. Sound Vib., 330(4), pp. 636–651. [CrossRef]
Achenbach, J. D. , 1973, Wave Propagation in Elastic Solids, Elsevier Science Publishers, Amsterdam, The Netherlands.
Adams, G. G. , 2000, “ Friction Reduction in the Sliding of an Elastic Half-Space Against a Rigid Surface Due to Incident Rectangular Dilatational Waves,” ASME J. Tribol., 122(1), pp. 10–15. [CrossRef]
Ben-Zion, Y. , 2008, “ Collective Behavior of Earthquakes and Faults: Continuum-Discrete Transitions, Progressive Evolutionary Changes, and Different Dynamic Regimes,” Rev. Geophys., 46(4), p. RG4006. [CrossRef]
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Figures

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Fig. 1

A rigid body sliding on an isotropic finite-thickness elastic layer with a constant coefficient of friction

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Fig. 2

Wave propagation in a waveguide showing dilatational (P) and shear (SV) waves

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Fig. 3

The dimensionless instability (ΛR) versus friction coefficient (f) for a thickness-to-wavelength ratio of H = 0.25. The quantity ΛR represents the real part of Λ in the eΛt variation of the wave where t=c2t⌢/ℓ, t̂ is time, c2 is the shear wave speed, and ℓ is the apparent wavelength.

Grahic Jump Location
Fig. 4

The dilatational wave angle (θ1) and shear wave angle (θ2) shown with dashed and solid lines, respectively, for thickness-to-wavelength ratio of H = 0.25

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Fig. 5

Real values of s1 and s3 with dashed and solid lines, respectively, for thickness-to-wavelength ratio of H = 0.25. The quantities sj represent the esjx2 variation of the wave where the dimensionless coordinate x2=x̂2/ℓ and ℓ is the apparent wavelength.

Grahic Jump Location
Fig. 6

The dimensionless instability (ΛR) versus friction coefficient (f) for thickness-to-wavelength ratio of H = 0.5. The quantity ΛR represents the real part of Λ in the eΛt variation of the wave where t=c2t⌢/ℓ, t̂ is time, c2 is the shear wave speed, and ℓ is the apparent wavelength.

Grahic Jump Location
Fig. 7

The dilatational wave angle (θ1) and shear wave angle (θ2) shown with dashed and solid lines, respectively, for thickness-to-wavelength ratio of H = 0.5

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Fig. 8

Real values of s1 and s3 with dashed and solid lines, respectively, for thickness-to-wavelength ratio of H = 0.5. The quantities sj represent the esjx2 variation of the wave where the dimensionless coordinate x2=x̂2/ℓ and ℓ is the apparent wavelength.

Grahic Jump Location
Fig. 9

The dimensionless instability (ΛR) versus friction coefficient (f) for thickness-to-wavelength ratio of H = 1. The quantity ΛR represents the real part of Λ in the eΛt variation of the wave where t=c2t⌢/ℓ, t̂ is time, c2 is the shear wave speed and ℓ is the apparent wavelength.

Grahic Jump Location
Fig. 10

The dilatational wave angle (θ1) and shear wave angle (θ2) shown with dashed and solid lines, respectively, for thickness-to-wavelength ratio of H = 1

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Fig. 11

Real values of s1 and s3 with dashed and solid lines, respectively, for thickness-to-wavelength ratio of H = 1. The quantities sj represent the esjx2 variation of the wave where the dimensionless coordinate x2=x̂2/ℓ and ℓ is the apparent wavelength.

Grahic Jump Location
Fig. 12

The dimensionless instability (ΛR) versus friction coefficient (f) for thickness-to-wavelength ratio of H = 2. The quantity ΛR represents the real part of Λ in the eΛt variation of the wave where t=c2t⌢/ℓ, t̂ is time, c2 is the shear wave speed and ℓ is the apparent wavelength.

Grahic Jump Location
Fig. 13

The dilatational wave angle (θ1) and shear wave angle (θ2) shown with dashed and solid lines, respectively, for thickness-to-wavelength ratio of H = 2

Grahic Jump Location
Fig. 14

Real values of s1 and s3 with dashed and solid lines, respectively, for thickness-to-wavelength ratio of H = 2. The quantities sj represent the esjx2 variation of the wave where the dimensionless coordinate x2=x̂2/ℓ and ℓ is the apparent wavelength.

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