Initiation of antiplane shear instability under slip dependent friction

We study the initiation of an unstable antiplane elastodynamic shear process under slip-weakening friction. We give an analytical expression of the slip that we intrepret using an eigenvalue analysis. Considering only the part of the solution associated with positive eigenvalues, we define a ''dominant part'' characterized by an exponential growth with time. An explicit formula is given for the dominant part that controls the development of the instability after the application of an initial perturbation on the surface or inside the elastic body. It shows that in response to a small initial perturbation the instability will develop in a limited spectral domain. The limiting wavenumber (or reciprocal critical length) is a function of the parameters of the friction law and the elastic properties. The part of the solution associated with negative eigenvalues (the ''wave part'') becomes rapidly negligible when the instability develops. We found that in the initiation phase the displacement field in the elastic body has a simple exponential dependence on the coordinate perpendicular to the fault. Using the expression of the dominant part, we estimate the duration of the initiation phase. We show the accuracy of the theoretical analysis by comparison with numerical tests computed with an independent technique. Finally, we show how the initiation phase determines the evolution toward the dynamic rupture propagation. We introduce the critical patch length in a natural way. The transition between the initiation and the propagation stages is characterized by an apparent supersonic velocity of the rupture front.