{'fraction_non_alphanumeric': 0.032648125755743655, 'fraction_numerical': 0.013905683192261185, 'mean_word_length': 4.648464163822526, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Buoyancy-driven crack propagation: the limit of large fracture toughness

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.