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

Roper, S.M. and Lister, J.R. (2007) Buoyancy-driven crack propagation: the limit of large fracture toughness. Journal of Fluid Mechanics, 580, pp. 359-380. (doi: 10.1017/S0022112007005472)

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=Kc/(3μQm³/2)^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 xs226B 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Roper, Dr Steven
• Authors: Roper, S.M., and Lister, J.R.
• Subjects: T Technology > TA Engineering (General). Civil engineering (General)
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Fluid Mechanics
• Publisher: Cambridge University Press
• ISSN: 0022-1120
• ISSN (Online): 1469-7645
• Published Online: 21 May 2007
• Copyright Holders: Copyright © 2007 Cambridge University Press
• First Published: First published in Journal of Fluid Mechanics 580:359-380
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher.