Near-tip Behavior of a Shallow Hydraulic Fracture with Small Toughness

This paper presents a study of the coupled problem of steadily moving semi-infinite fractures driven by an incompressible viscous fluid in proximity of a free surface. Since at large times the fluid occupies most of the fracture, except for a small region near the crack tip where the fluid front lags the crack tip, it is assumed that the fluid occupies the entire open fracture space. As a consequence of the geometric asymmetry along the crack line and the negative net fluid pressure in the tip region, a contact sliding zone emerges in front of the crack opening region when the toughness is sufficiently low. The primary objective is to ascertain the structure of the solution in the tip region and the dimension of the sliding zone, with a particular focus on the case of low toughness. A scaling analysis indicates that the solution depends on a single parameter, which can be interpreted as a dimensionless toughness. When the dimensionless parameter is less than a critical value, a sliding zone emerges, and its length increases as the dimensionless toughness decreases. The solution is characterized by a multi-scale structure, comprising a nested boundary layer in the near-tip region. This solution opens the way to construct solutions for hydraulic fractures propagating near a free surface.