Γ-CONVERGENCE FOR A FAULT MODEL WITH SLIP-WEAKENING FRICTION AND PERIODIC BARRIERS

• Published in: Quarterly of applied mathematics 2005-12, Vol.63 (4), p.747-778
• Main Authors: IONESCU, IOAN R., ONOFREI, DANIEL, VERNESCU, BOGDAN
• Format: Article
• Language: English
• Subjects: Applied sciences; Buildings. Public works; Decreasing sequences; Earthquakes; Eigenvalues; Eigenvectors; Exact sciences and technology; Fault planes; Friction; Fundamental areas of phenomenology (including applications); Geotechnics; Homogenization; Local minimum; Mathematical methods in physics; Mechanical contact (friction...); Nucleation; Numerical approximation and analysis; Ordinary and partial differential equations, boundary value problems; Physics; Soil mechanics. Rocks mechanics; Solid mechanics; Structural and continuum mechanics; Symmetry; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
• ISSN: 0033-569X; 1552-4485
• DOI: 10.1090/S0033-569X-05-00981-7

We consider a three-dimensional elastic body with a plane fault under a slip-weakening friction. The fault has ∊-periodically distributed holes, called (smallscale) barriers. This problem arises in the modeling of the earthquake nucleation on a large-scale fault. In each ∊-square of the ∊-lattice on the fault plane, the friction contact is considered outside an open set T∊ (small-scale barrier) of size r∊ < ∊, compactly inclosed in the ∊-square. The solution of each ∊-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∊ provides information about the stability of the solution. Using Γ-convergence techniques, we study the asymptotic behavior as ∊ tends to 0 for the friction contact problem. Depending on the values of c =: lim∊→0T→/∊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→. 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 = ∞), 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.