Viscosity solutions for dynamic problems with slip-rate dependent friction
Author:
Ioan R. Ionescu
Journal:
Quart. Appl. Math. 60 (2002), 461-476
MSC:
Primary 35Q72; Secondary 35B25, 35L85, 74H20, 74M10
DOI:
https://doi.org/10.1090/qam/1914436
MathSciNet review:
MR1914436
Full-text PDF Free Access
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Abstract: The dynamic evolution of an elastic medium undergoing frictional slip is considered. The Coulomb law modeling the contact uses a friction coefficient that is a non-monotone function of the slip-rate. This problem is ill-posed, the solution is non-unique and shocks may be created on the contact interface. In the particular case of the one-dimensional shearing of an elastic slab, the (perfect) delay convention can be used to select a unique solution. Different solutions in acceleration and deceleration processes are obtained. To transform the ill-posed problem into a well-posed one and to justify the choice of the perfect delay criterion, a visco-elastic constitutive law with a small viscosity is used here. An existence and uniqueness result is obtained in three dimensions. The assumptions on the functions implied in the contact model are weak enough to include both the normal compliance and the Tresca model. The following conjecture, based on results of numerical simulations, is stated: in the elastic case, the solution chosen by the perfect delay convention is the one obtained from the solutions of the problem with viscosity, when the viscosity tends to zero.
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G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin-New York, 1976
J. H. Dieterich, A constitutive law for rate of earthquake production and its application to earthquake clustering, J. Geophys. Res. 99, No. B2, 2601–2618 (1994)
P. Favreau, I. R. Ionescu, and M. Campillo, On the dynamic sliding with rate and state dependent friction laws, Geophysical Journal 139, 671–678 (1999)
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I. R. Ionescu and J.-C. Paumier, On the contact problem with slip displacement dependent friction in elastostatics, Internat. J. Engrg. Sci. 34, 471–491 (1996)
J. Jarusek and C. Eck, Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions, Math. Models Methods Appl. Sci. 9, 11–34 (1999)
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K. L. Kuttler, Y. Renard, and M. Schillor, Models and simulations of dynamic frictional contact of a beam, Comput. Meth. Appl. Mech. Engrg. 177, 257–272 (1999)
K. L. Kuttler and M. Schillor, Set-valued pseudomonotone maps and degenerate evolution inclusions, Communications in Contemporary Mathematics 1, 87–123 (1999)
J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Analysis 11, 407–428 (1987)
J. T. Oden and J. A. C. Martins, Models and computational methods for dynamic friction phenomena, Comput. Math. Appl. Mech. Engrg. 52, 527–634 (1985)
G. Perrin, J. R. Rice, and G. Zheng, Self-healing slip pulse on a frictional surface, Journal of the Mechanics and Physics of Solids 43, 1461–1495 (1995)
T. Poston and I. Stewart, Catastrophe Theory and its Applications, Pitman, London, 1978
J. R. Rice and A. L. Ruina, Stability of steady frictional slipping, Journal of Applied Mechanics 50, 343–349 (1983)
Y. Renard, Perturbation singulière d’un problème de frottement sec non monotone, Comptes Rendus Acad. Sci. Paris, Sér. I. Math. 326, 131–136 (1998)
Y. Renard, Singular perturbation approach to an elastic dry friction problem with nonmonotone coefficient, Quart. Appl. Math. 58, 303–324 (2000)
A. L. Ruina, Slip instabilities and state variable friction laws, J. Geophysics Res. 88, B12, 10359–10370 (1983)
C. H. Scholtz, The Mechanics of Earthquakes and Faulting, Cambridge Univ. Press, Cambridge, 1990
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© Copyright 2002
American Mathematical Society