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 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 E tends to 0 for the friction contact problem. Depending on the values of c =: lim(epsilon -> 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 = infinity), 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.