Interaction of faults under slip-dependent friction. Non-linear eigenvalue analysis

• Published in: Mathematical methods in the applied sciences Vol. 28; no. 1; pp. 77 - 100
• Main Authors: Ionescu, Ioan R., Wolf, Sylvie
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 10-01-2005; Wiley; Teubner
• Subjects: domains with cracks; earthquake initiation; Exact sciences and technology; Mathematical analysis; Mathematics; mixed finite element method; non-linear eigenvalue problem; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Rayleigh quotient; Sciences and techniques of general use; slip-dependent friction; unilateral conditions; wave equation; Wave equation; Rayleigh quotient; Slip dependent friction; Existence of solution; Eigenfunction; Linear time; Mathematical method; Discretization method; Unbounded solution; Non linear equation; Applied mathematics; Earthquakes; Unilateral; Domain with crack; Eigenvalue problem; Mixed finite element; Numerical convergence
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.550

We analyse the evolution of a system of finite faults by considering the non‐linear eigenvalue problems associated to static and dynamic solutions on unbounded domains. We restrict our investigation to the first eigenvalue (Rayleigh quotient). We point out its physical significance through a stability analysis and we give an efficient numerical algorithm able to compute it together with the corresponding eigenfunction.
We consider the anti‐plane shearing on a system of finite faults under a slip‐dependent friction in a linear elastic domain, not necessarily bounded. The static problem is formulated in terms of local minima of the energy functional. We introduce the non‐linear (static) eigenvalue problem and we prove the existence of a first eigenvalue/eigenfunction characterizing the isolated local minima. For the dynamic problem, we discuss the existence of solutions with an exponential growth, to deduce a (dynamic) non‐linear eigenvalue problem. We prove the existence of a first dynamic eigenvalue and we analyse its behaviour with respect to the friction parameter. We deduce a mixed finite element discretization of the non‐linear spectral problem and we give a numerical algorithm to approach the first eigenvalue/eigenfunction. Finally we give some numerical results which include convergence tests, on a single fault and a two‐faults system, and a comparison between the non‐linear spectral results and the time evolution results. Copyright © 2004 John Wiley & Sons, Ltd.