$\Gamma$-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 $\epsilon$ 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.