Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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$ \Gamma$-convergence for a fault model with slip-weakening friction and periodic barriers


Authors: Ioan R. Ionescu, Daniel Onofrei and Bogdan Vernescu
Journal: Quart. Appl. Math. 63 (2005), 747-778
MSC (2000): Primary 35J25, 74Q05; Secondary 35P99, 74B10
DOI: https://doi.org/10.1090/S0033-569X-05-00981-7
Published electronically: October 6, 2005
MathSciNet review: 2187930
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a three-dimensional elastic body with a plane fault under a slip-weakening friction. The fault has $ \epsilon$-periodically distributed holes, called (small-scale) barriers. This problem arises in the modeling of the earthquake nucleation on a large-scale fault.

In each $ \epsilon$-square of the $ \epsilon$-lattice on the fault plane, the friction contact is considered outside an open set $ T_\epsilon$ (small-scale barrier) of size $ r_\epsilon<\epsilon$, compactly inclosed in the $ \epsilon$-square. The solution of each $ \epsilon$-problem is found as local minima for an energy with both bulk and surface terms. The first eigenvalue of a symmetric and compact operator $ K^\epsilon$ provides information about the stability of the solution.

Using $ \Gamma$-convergence techniques, we study the asymptotic behavior as $ \epsilon$ tends to 0 for the friction contact problem. Depending on the values of $ c=:\lim_{\epsilon\rightarrow 0} r_\epsilon /\epsilon^2$ we obtain different limit problems.

The asymptotic analysis for the associated spectral problem is performed using $ G$-convergence for the sequence of operators $ K^\epsilon$. The limits of the eigenvalue sequences and the associated eigenvectors are eigenvalues and respectively eigenvectors of a limit operator.

From the physical point of view our result can be interpreted as follows:

i) if the barriers are too large (i.e. $ c = \infty$), then the fault is locked (no slip),

ii) if $ c >0$, then the fault behaves as a fault under a slip-dependent friction. The slip weakening rate of the equivalent fault is smaller than the undisturbed fault. Since the limit slip-weakening rate may be negative, a slip-hardening effect can also be expected.

iii) if the barriers are too small (i.e. $ c=0$), then the presence of the barriers does not affect the friction law on the limit fault.


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Additional Information

Ioan R. Ionescu
Affiliation: Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France
Email: ionescu@univ-savoie.fr

Daniel Onofrei
Affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts 01609
Email: onofrei@wpi.edu

Bogdan Vernescu
Affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts 01609
Email: vernescu@wpi.edu

DOI: https://doi.org/10.1090/S0033-569X-05-00981-7
Keywords: $\Gamma$-convergence, slip-weakening friction, Steklov problem
Received by editor(s): March 11, 2005
Published electronically: October 6, 2005
Article copyright: © Copyright 2005 Brown University

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