Fracture is ubiquitous in a crystalline material. Inspired by the observed phenomenological similarities between the spatial profile of a fractured surface and velocities in hydrodynamic turbulence, we set up a hydrodynamic description for the dynamics of fracture surface propagation mode I or opening fracture front. We consider several related continuum hydrodynamic models and use them to extract the similarities between the profile of a fractured surface and velocities in hydrodynamic turbulence. We conclude that a fractured surface should be generically self-similar with an underlying multifractal behaviour. This article is part of the theme issue ‘Statistical physics of fracture and earthquakes’.