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

Laboratoire de Géophysique Interne et Tectonophysique Observatoire de Grenoble Université Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France
Laboratoire de Mathématiques Appliquées Université de Savoie Campus Scientifique, 73376 Le Bourget-du-lac, Cedex, France

Abstract

We study the initiation of an unstable homogeneous elastodynamic inplane shear process under slip-weakening friction. We assume a linear dependency of the friction at the beginning of the slip, and we make an eigenvalue analysis 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 square and lead to the dominant part of the solution. We use a classical method based on the normalization of the dominant eigenfunctions in order to give the analytical expression of the dominant part of the solution. This analysis shows that the response of the dominant part will develop on a continuous but limited spectral domain. This limit depends on the weakening of the friction and a coefficient including the ratio of P-wave velocity to S-wave 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 expression of the dominant part, we give an estimation of the time of initiation for the crack to reach the steady propagation stage. We perform numerical tests with a finite-difference method and show very good agreement between the analytical dominant part of the solution and the complete numerical solution. Finally, in our case, where the initial stress is equal to the static admissible load, we study the crack propagation and observe that the crack tips travel asymptotically at P-wave velocity after a short time of apparent P supersonic velocity. The numerical results show that the linearized dynamic description is also valid ahead the crack tips in the propagation regime.