Fault finiteness and initiation of dynamic shear instability

We study the initiation of slip instabilities of a finite fault in a homogeneous linear elastic space. We consider the antiplane unstable shearing under 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 stability is presented for the nondimensional problem. Using an integral equation method we determine the first (nondimensional) eigenvalue which depends only on the geometry of the problem. In connection with the weakening rate and the fault length this (universal) constant determines the range of instability 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 spectral 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 hypersingular integral equation to compute the unstable eigenfunctions. We use the expression of the dominant part to deduce an approximate formula for the duration of the initiation phase. Finally, some numerical tests are performed. We give the numerical values for the first eigenfunction. The dependence of the first eigenvalue and the duration of the initiation on the weakening rate are pointed out. The results are compared with those for the full solution computed with a finite-differences scheme. These results suggest that a very simple friction law could imply a broad range of duration of initiation. They show the fundamental role played by the limited extent of the potentially slipping patch in the triggering of an unstable rupture event.