Γ-convergence for a Fault Model with Slip-weakening Friction and Periodic Barriers

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 →0 r / 2 we obtain different limit problems. The asymptotic analysis for the associated spectral problem is performed using Gconvergence 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), Received March 11, 2005. 2000 Mathematics Subject Classification. Primary 35J25, 74Q05; Secondary 35P99, 74B10.