Singular perturbation approach to an elastic dry friction problem with non-monotone coefficient
Author:
Yves Renard
Journal:
Quart. Appl. Math. 58 (2000), 303-324
MSC:
Primary 74M10; Secondary 34A60, 34E15
DOI:
https://doi.org/10.1090/qam/1753401
MathSciNet review:
MR1753401
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Abstract: In this paper we study a one-dimensional dynamic model of dry friction with slip velocity dependent coefficient. In many cases, this model has more than one solution. We introduce a perturbed friction condition which allows us to regain the uniqueness of the solution. We show that the perturbed problem’s solutions pointwise converge to a particular solution of the initial problem when the perturbation parameter tends to zero. The singular perturbation approach provides the analysis of a criterion used to select a solution of the problem, and suggests a method to study more elaborated dry friction problems.
W. R. Brace and J. D. Byerlee, Stick-slip as a mechanism for earthquakes, Science 153, 990–992 (1966)
M. Campillo, I. R. Ionescu, J.-C. Paumier, and Y. Renard, On the dynamic sliding with friction of a rigid block and of an infinite elastic slab, Physics of the Earth and Planetary Interiors 96, 15–23 (1996)
- Klaus Deimling, Multivalued differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1992. MR 1189795
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
C. Gao and D. Kuhlmann-Wilsdorf, On stick-slip and velocity dependence of friction at low speeds, ASME Journal of Tribology 112, 355–360 (1990)
- I. R. Ionescu and J.-C. Paumier, On the contact problem with slip rate dependent friction in elastodynamics, European J. Mech. A Solids 13 (1994), no. 4, 555–568. MR 1298058
A. I. Leonov and A. Srinivasan, Self-oscillations of an elastic plate sliding over a smooth surface, Internat. J. Engrg. Sci. 31, 453–473 (1993)
E. Rabinowicz, Friction and Wear, Study of the Stick—slip Process, Davies, Elsevier, New York, 1949, pp. 149–164
E. Rabinowicz, The intrinsic variables affecting the stick-slip process, Proceedings of the Royal Physical Society 71, 668–675 (1958)
Y. Renard, Modélisation des instabilités liées au frottement sec des solides élastiques, aspects théoriques et numériques, thèse de doctorat, LMC-IMAG Grenoble (1998)
- Walter Rudin, Analyse réelle et complexe, Masson, Paris, 1980 (French). Translated from the first English edition by N. Dhombres and F. Hoffman; Third printing. MR 662565
D. Tabor, Friction : The present state of our understanding, J. of Lubrication Technology 103, 169–179 (1981)
A. N. Tikhonov, A. B. Vasiléva, and A. G. Sveshnikov, Differential Equations, Chapter VII, Asymptotics of solutions of differential equations with respect to a small parameter, Springer-Verlag, New York, 1980, pp. 81–213
W. R. Brace and J. D. Byerlee, Stick-slip as a mechanism for earthquakes, Science 153, 990–992 (1966)
M. Campillo, I. R. Ionescu, J.-C. Paumier, and Y. Renard, On the dynamic sliding with friction of a rigid block and of an infinite elastic slab, Physics of the Earth and Planetary Interiors 96, 15–23 (1996)
K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992
L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, 1992
C. Gao and D. Kuhlmann-Wilsdorf, On stick-slip and velocity dependence of friction at low speeds, ASME Journal of Tribology 112, 355–360 (1990)
I. R. Ionescu and J.-C. Paumier, On the contact problem with slip rate dependent friction in elastodynamics, European J. Mech. A Solids 13, 555–568 (1994)
A. I. Leonov and A. Srinivasan, Self-oscillations of an elastic plate sliding over a smooth surface, Internat. J. Engrg. Sci. 31, 453–473 (1993)
E. Rabinowicz, Friction and Wear, Study of the Stick—slip Process, Davies, Elsevier, New York, 1949, pp. 149–164
E. Rabinowicz, The intrinsic variables affecting the stick-slip process, Proceedings of the Royal Physical Society 71, 668–675 (1958)
Y. Renard, Modélisation des instabilités liées au frottement sec des solides élastiques, aspects théoriques et numériques, thèse de doctorat, LMC-IMAG Grenoble (1998)
W. Rudin, Analyse réelle et complexe, Masson, Paris, 1992
D. Tabor, Friction : The present state of our understanding, J. of Lubrication Technology 103, 169–179 (1981)
A. N. Tikhonov, A. B. Vasiléva, and A. G. Sveshnikov, Differential Equations, Chapter VII, Asymptotics of solutions of differential equations with respect to a small parameter, Springer-Verlag, New York, 1980, pp. 81–213
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© Copyright 2000
American Mathematical Society
