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

• Published in: Journal of fluid mechanics Vol. 580; pp. 359 - 380
• Main Authors: ROPER, S. M., LISTER, J. R.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.06.2007
• Subjects: Buoyancy; Earth sciences; Earth, ocean, space; Exact sciences and technology; Fluid dynamics; Fluid mechanics; Fracture toughness; Hydrostatic pressure; Mechanics; Tectonics. Structural geology. Plate tectonics; Viscous flow; volume; bulk modulus; viscosity; fracture propagation; mathematical models; fluid phase; flow mechanism; magnitude; hydrostatic pressure; buoyancy; dimensions
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S0022112007005472

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=Kc /(3μQm 3/2)1/4 describes the relative magnitudes of resistance to fracture and resistance to viscous flow, where Kc 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.