The Collapse and Expansion of Liquid-Filled Elastic Channels and Cracks

The rate at which fluid drains from a collapsing channel or crack depends on the interaction between the elastic properties of the solid and the fluid flow. The same interaction controls the rate at which a pressurized fluid can flow into a crack. In this paper, we present an analysis for the interaction between the viscous flow and the elastic field associated with an expanding or collapsing fluid-filled channel. We first examine an axisymmetric problem for which a completely analytical solution can be developed. A thick-walled elastic cylinder is opened by external surface tractions, and its core is filled by a fluid. When the applied tractions are relaxed, a hydrostatic pressure gradient drives the fluid to the mouth of the cylinder. The relationship between the change in dimensions, time, and position along the cylinder is given by the diffusion equation, with the diffusion coefficient being dependent on the modulus of the substrate, the viscosity of the fluid, and the ratio of the core radius to the exterior radius of the cylinder. The second part of the paper examines the collapse of elliptical channels with arbitrary aspect ratios, so as to model the behavior of fluid-filled cracks. The channels are opened by a uniaxial tension parallel to their minor axes, filled with a fluid, and then allowed to collapse. The form of the analysis follows that of the axisymmetric calculations, but is complicated by the fact that the aspect ratio of the ellipse changes in response to the local pressure. Approximate analytical solutions in the form of the diffusion equation can be found for small aspect ratios. Numerical solutions are given for more extreme aspect ratios, such as those appropriate for cracks. Of particular note is that, for a given cross-sectional area, the rate of collapse is slower for larger aspect ratios. With minor modifications to the initial conditions and the boundary conditions, the analysis is also valid for cracks being opened by a pressurized fluid.