Dynamic stability of one-dimensional models of fracture

We examine the linear stability of steady-state propagating fracture in two one-dimensional models. Both of these models include a cohesive force at the crack tip; they differ only in that the dissipative mechanism is a frictional force in the first model and a viscosity in the second. Our strategy is to compute the linear response of this system to a spatially periodic perturbation. As expected, we find no dynamical instabilities in these models. However, we do find some interesting analytic properties of the response coefficient that we expect to be relevant to the analysis of more realistic two-dimensional models