Initiation of in-plane shear instability under slip-dependent friction

We study the initiation of an unstable homogeneous elastodynamic inplane sh ear process under slip-weakening friction. We assume a linear dependency of the friction at the beginning of the slip, and we make an eigenvalue analy sis in the time domain. We prove that two types of eigenvalue are possible. With the first type, the eigenvalues have a negative square and represent the wave part of the solution. With the second type, they have a positive s quare and lead to the dominant part of the solution. We use a classical met hod based on the normalization of the dominant eigenfunctions in order to g ive the analytical expression of the dominant part of the solution. This an alysis shows that the response of the dominant part will develop on a conti nuous but limited spectral domain. This limit depends on the weakening of t he friction and a coefficient including the ratio of P-wave velocity to S-w ave velocity. We also show that the exponential growth of the dominant part is directly linked to the weakening and the S-wave velocity. Using the exp ression of the dominant part, we give an estimation of the time of initiati on for the crack to reach the steady propagation stage. We perform numerica l tests with a finite-difference method and show very good agreement betwee n the analytical dominant part of the solution and the complete numerical s olution. Finally, in our case, where the initial stress is equal to the sta tic admissible load, we study the crack propagation and observe that the cr ack tips travel asymptotically at P-wave velocity after a short time of app arent P supersonic velocity. The numerical results show that the linearized dynamic description is also valid ahead the crack tips in the propagation regime.