We present the analytical solution for a fundamental fracture mode in the form of a self-similar, self-healing pulse. The existence of such a fracture mode was strongly suggested by recent numerical simulations of seismic ruptures but, to our knowledge, no formal proof of their origin has been proposed yet. We present a two-dimensional, anti-plane solution for fixed rupture and healing speeds that satisfies both the wave equation and crack boundary conditions for a simple Coulomb friction law in the absence of any rate or state dependence. This solution is an alternative to the classic self-similar crack solution by Kostrov. In practice, the self-healing impulsive mode rather than the expanding crack mode is selected depending on details of fracture initiation and is thereafter self-maintained. We discuss stress concentration, fracture energy, and rupture velocity and compare them to the case of a crack. The analytical study is complemented by various numerical examples and comparisons. On more general grounds, we argue that an infinity of marginally stable fracture modes may exist in addition to the crack solution or the impulsive fracture described here.

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