Fault finiteness and initiation of dynamic shear instability

We study the initiation of slip instabilities of a finite fault in a homoge neous linear elastic space. We consider the antiplane unstable shearing und er a slip-dependent friction law with a constant weakening rate. We attack the problem by spectral analysis. We concentrate our attention on the case of long initiation, i.e. small positive eigenvalues. A static analysis of s tability is presented for the nondimensional problem. Using an integral equ ation method we determine the first (nondimensional) eigenvalue which depen ds only on the geometry of the problem, In connection with the weakening ra te and the fault length this (universal) constant determines the range of i nstability for the dynamic problem. We give the exact limiting value of the length of an unstable fault for a given friction law, By means of a spectr al expansion we define the 'dominant part' of the unstable dynamic solution , characterized by an exponential time growth. For the long-term evolution of the initiation phase we reduce the dynamic eigenvalue problem to a hyper singular integral equation to compute the unstable eigenfunctions. We use t he expression of the dominant part to deduce an approximate formula for the duration of the initiation phase. Finally, some numerical tests are perfor med. We give the numerical values for the first eigenfunction. The dependen ce of the first eigenvalue and the duration of the initiation on the weaken ing rate are pointed out. The results are compared with those for the full solution computed with a finite-differences scheme. These results suggest t hat a very simple friction law could imply a broad range of duration of ini tiation, They show the fundamental role played by the limited extent of the potentially slipping patch in the triggering of an unstable rupture event. (C) 2000 Published by Elsevier Science B.V, All rights reserved.