The tip region of a shallow hydraulic fracture

This study develops a traveling-wave solution for the tip region of a shallow hydraulic fracture by analyzing the problem of a semi-infinite fracture propagating at constant velocity parallel to a free surface. The semi-fracture is driven by a far-field bending moment and a fluid pressure governed by the lubrication equation. The geometric and mechanical asymmetry introduced by the free surface gives rise to a mixed-mode fracture, described through a system of coupled elasticity equations for the shear and normal displacement jumps. The analysis assumes that the fluid front coincides with the fracture tip, thereby neglecting the existence of a fluid-lag zone. A scaling analysis reveals that the solution depends on a single parameter, the dimensionless fracture toughness K. The relationship between the far-field bending moment and fracture toughness is established by applying a mixed-mode propagation criterion and accounting for viscous fluid flow. Results indicate the dependence of the fracture-tip response on K: for sufficiently small K, the solution transitions to pure mode II, accompanied by the development of a sliding contact zone whose length increases as toughness decreases. These findings provide new insight into the propagation of shallow hydraulic fractures and highlight the role of substrate elasticity in governing tip processes.