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Research Papers

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

[+] Author and Article Information
Fanbo Meng, Jiexi Huang

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109

M. D. Thouless

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109;
Department of Materials Science and Engineering,
University of Michigan,
Ann Arbor, MI 48109

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 29, 2015; final manuscript received July 10, 2015; published online July 22, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(10), 101009 (Jul 22, 2015) Paper No: JAM-15-1275; doi: 10.1115/1.4031048 History: Received May 29, 2015

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.

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Figures

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Fig. 1

Schematic figures of the axisymmetric model. The origin is located at the midpoint of the cylinder. (a) A tube has an initial inner radius of R1o, an initial outer radius of R2o and an initial half length of Lo. (b) Uniform tensile tractions, σo, are applied to the outer surface of the tube, resulting in increases in the inner and outer radius. Then, the tube is filled with fluid at the ambient pressure. (c) The applied tractions are fully relaxed, resulting in a uniform fluid pressure, P = Po, being established throughout the core. (d) The fluid flows out of the tube driven by the difference between the internal fluid pressure, P, and the ambient pressure. The fluid pressure, P, decreases during this process, and its reduction is accompanied by changes in the dimensions of the tube. The tube returns to its original shape as P approaches the ambient pressure.

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Fig. 4

(a) The elliptical channel has initial radii of ao and bo. (b) A remote tension of σo is applied to open the channel, increasing its size. While the channel is held open, it is filled with fluid at the ambient pressure. (c) The applied tension is relaxed to zero, resulting in a fluid pressure of Po. The dimensions of the channel are related to those in (b) by the conservation of volume for an incompressible fluid. This is the initial state from which flow occurs. (d) The pressure relaxes back to the ambient pressure as the fluid flows out of the channel to restore the original shape (the dimensions in the schematics are exaggerated for ease of visualization, and are not to scale).

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Fig. 2

Profile of the inner surface of a cylindrical channel as a function of distance from the outlet at different times. Plots are shown for an infinitely long channel, and for a channel with Lo/R1o=1000.

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Fig. 3

A cartoon based on the analysis showing how a liquid-filled cylindrical tube collapses by Poiseuille from the center of the tube to the exit, when the remote tractions are removed (the shape change and dimensions are exaggerated, and do not reflect real data)

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Fig. 6

A collapse front defined by Δψ/Δψmax=50% travels from the exit of the channel toward the center. Numerical solutions for the position of the collapse front as a function of time are given in this figure. The error bars are comparable with the thickness of the lines. When the original aspect ratio is less than about 30, the motion of the collapse front is described accurately by the analytical solution given in the text. As the aspect ratio increases, the front moves increasingly slowly. However, the numerical solutions suggest there is an asymptotic solution appropriate for cracks (φo→∞).

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Fig. 7

The change in aspect ratio along an elliptical channel with an initial aspect ratio of 100, at different times

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Fig. 8

(a) The relative expansion of the major axis of a crack as a function of position, at different times after relaxation of an applied strain. (b) The relative expansion of the minor axis of a crack as a function of position, at different times after relaxation of an applied strain.

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Fig. 9

An expansion front, defined by the point where b is equal to 50% of its maximum value, bmax, travels from the entrance of a channel under a pressure Po. Numerical solutions for the position of this expansion front as a function of time are given in this figure for very long channels with different aspect ratios.

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Fig. 10

(a) The relative expansion of a cylindrical channel in an infinite substrate as a function of position, at different times after being opened by a fluid under a pressure Po. (b) The relative expansion of the minor axis of an elliptical channel in an infinite substrate with an aspect ratio of 100, as a function of position, at different times after being opened by a fluid under a pressure Po. (c) The relative opening of a crack in an infinite substrate as a function of position, at different times after being opened by a fluid under a pressure Po.

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Fig. 5

A schematic showing the model used in the numerical calculations for deformation of an elliptical channel. A long channel is segmented into thin slices. Each slice has a length of δLo, and the jth element has a major axis of aj, a minor axis of bj, and a pressure of Pj. The values of aj and bj are related by Eq. (44), while Eqs. (46) and (47) are used to calculate the evolution of the channel profile.

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Fig. 11

The width at the center of a long crack, as a function of time and different initial strains. There is an incubation period while the collapse front travels along the channel. This incubation period decreases with the initial strain and has the approximate form of τc≈1.8×106(σo/E)-3. Once the crack has started to collapse, its width follows an approximate form of b/ao≈320τ-3.

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