Buoyancy-driven crack propagation: the limit of large fracture toughness

Abstract

We study steady vertical propagation of a crack filled with buoyant viscous fluid through an elastic solid with large effective fracture toughness. For a crack fed by a constant flux Q , a non-dimensional fracture toughness K = K c /(3μ Qm 3 /2) 1/4 describes the relative magnitudes of resistance to fracture and resistance to viscous flow, where K c is the dimensional fracture toughness, μ the fluid viscosity and m the elastic modulus. Even in the limit K ≫ 1, the rate of propagation is determined by viscous effects. In this limit the large fracture toughness requires the fluid behind the crack tip to form a large teardrop-shaped head of length O ( K 2/3 ) and width O ( K 4/3 ), which is fed by a much narrower tail. In the head, buoyancy is balanced by a hydrostatic pressure gradient with the viscous pressure gradient negligible except at the tip; in the tail, buoyancy is balanced by viscosity with elasticity also playing a role in a region within O ( K 2/3 ) of the head. A narrow matching region of length O ( K −2/5 ) and width O ( K −4/15 ), termed the neck, connects the head and the tail. Scalings and asymptotic solutions for the three regions are derived and compared with full numerical solutions for K ≤ 3600 by analysing the integro-differential equation that couples lubrication flow in the crack to the elastic pressure gradient. Time-dependent numerical solutions for buoyancy-driven propagation of a constant-volume crack show a quasi-steady head and neck structure with a propagation rate that decreases like t −2/3 due to the dynamics of viscous flow in the draining tail.