Dynamics of a fluid‐driven crack in three dimensions by the finite difference method

The finite difference method is applied to the study of the dynamics of a three‐dimensional fluid‐filled crack excited into resonance by the sudden failure of a small barrier of area Δ S on the crack surface. The impulse response of the crack is examined for various ratios of crack width to crack length and for several values of the crack stiffness C = ( b /μ)( L / d ), where b is the bulk modulus of the fluid, μ is the rigidity of the solid, and L and d are the crack length and crack thickness, respectively. The motion of the crack is characterized by distinct time scales representing the duration of brittle failure and the periods of acoustic resonance in the lateral and longitudinal dimensions of the source. The rupture has a duration proportional to the area of crack expansion and is the trigger responsible for the excitation of the crack into resonance; the resonant periods are proportional to the crack stiffness and to the width and length of the crack. The crack wave sustaining the resonance is analogous to the tube wave propagating in a fluid‐filled borehole. It is dispersive, showing a phase velocity that decreases with increasing wavelength. Its wave speed is always lower than the acoustic velocity of the fluid and shows a strong dependence on the crack stiffness, decreasing as the stiffness increases. The initial motion of the crack surface is an opening, and the radiated far‐field compressional wave starts with a strong but brief compression which has a duration proportional to the crack stiffness and size of the rupture area; the amplitude of this pulse increases with the area of rupture but decreases with increasing stiffness. Flow into the newly created cavity triggers a pressure drop in the fluid, which produces a partial collapse of the wall propagated over the crack surface at the speed of the crack wave. The collapse of the crack surface generates a weak long‐period component of dilatation following the compressional first motion in the far‐field P wave train; the dilatational component is clearer in the signal from stiffer cracks when seen in the direction of the rupture. The energy loss by radiation is stronger for high frequencies, resulting in a progressive enrichment of the crack response in lower frequencies over the duration of resonance. These source characteristics translate into a far‐field signature that is marked by a high‐frequency content near its onset and dominated by a longer‐period component in its coda. The source duration shows a strong dependence on the fluid viscosity and associated viscous damping at the crack wall.