Asymptotic analysis and scaling of friction parameters

We consider an eigenvalue problem associated to the antiplane shearing on a system of collinear faults under a slip-dependent friction law. Firstly we consider a periodic system of faults in the whole plane. We prove that the first eigenvalues/eigenfunctions of different physical periodicity are all equal and that the other eigenvalues converge to this first common eigenvalue as their physical period becomes indefinitely large. Secondly we consider a large scale fault system composed on a small scale collinear faults periodically disposed. If beta(0)* is the first eigenvalue of the periodic problem in the whole plane, we prove that the first eigenvalue of the microscopic problem behaves as beta(*)(0)/epsilon when epsilon -> 0 regardless the geometry of the domain (here epsilon is the scale quotient). The geophysical implications of this result is that the macroscopic critical slip D-c scales with D-c(epsilon)/epsilon (here D-c(epsilon) is the small scale critical slip).