Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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
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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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DOI: https://doi.org/10.1090/qam/1914436
Article copyright: © Copyright 2002 American Mathematical Society


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