  The dynamics of rapid brittle cracks is commonly studied in the framework of
linear elastic fracture mechanics where nonlinearities are neglected. However,
recent experimental and theoretical work demonstrated explicitly the importance
of elastic nonlinearities in fracture dynamics. We study two simple
one-dimensional models of fracture in order to gain insights about the role of
elastic nonlinearities and the implications of their exclusion in the common
linear elastic approximation. In one model we consider the decohesion of a
nonlinear elastic membrane from a substrate. In a second model we follow the
philosophy of linear elastic fracture mechanics and study a linearized version
of the nonlinear model. By analyzing the steady state solutions, the
velocity-load relations and the response to perturbations of the two models we
show that the linear approximation fails at finite crack tip velocities. We
highlight certain features of the breakdown of the linear theory and discuss
possible implications of our results to higher dimensional systems.
