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

By S.M. Roper and J.R. Lister

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/(3μQm<sup>3</sup>/2)<sup>1/4</sup> 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<sup>2/3</sup>) and width O(K<sup>4/3</sup>), 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<sup>2/3</sup>) of the head. A narrow matching region of length O(K<sup>-2/5</sup>) and width O(K<sup>−4/15</sup>), 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<sup>−2/3</sup> due to the dynamics of viscous flow in the draining tail

Topics: TA
Publisher: Cambridge University Press
Year: 2007
OAI identifier: oai:eprints.gla.ac.uk:25320
Provided by: Enlighten

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