Buckling and collapse of open and closed cylindrical shells

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The buckling of a finite section of a cylindrical shell resembling a two-dimensional contact lens, and the collapse of a tubular shell of infinite extent are considered. The deformation is due, respectively, to the application of an edge force or to a negative transmural pressure. In both cases, the shell develops elastic bending moments due to the deformation from a specified resting shape according to a linear constitutive equation, accompanied by in-plane and transverse shear tensions. In the case of a section of a shell with a flat resting shape, classical results due to Euler and Love show that, as the applied edge force is increased beyond a sequence of thresholds, an infinite family of deformed shapes becomes possible corresponding to buckled states that bifurcate from the zero-curvature resting configuration. It is shown here that a corresponding infinite family of shapes is also possible for a finite shell whose resting shape is a section of circle. These shapes, however, no longer arise from bifurcations, but rather constitute disconnected solution branches of a nonlinear boundary-value problem. A closed cylindrical shell whose cross-section has a circular resting shape exhibits similar bifurcations when the difference between the exterior and interior pressure exceeds a sequence of thresholds, but a shell with a non-circular resting shape deforms into a multitude of shapes described by isolated solution branches. The computed two-dimensional buckled shapes are used to reconstruct the three-dimensional shape of a slowly collapsing fluid-conveying vessel. The reconstruction procedure involves stacking together cross-sections at axial positions that are found by integrating the differential equation determining the axial pressure distribution in unidirectional pressure-driven flow, subject to a constant flow rate. The dimensionless coefficient relating the local pressure gradient to the flow rate is computed by solving the Poisson equation governing unidirectional viscous flow using a boundary-element method, and expressing the flow rate as a boundary integral involving the shear stress which is available from the solution of the boundary-integral equation. In an appendix, the energy of the bending state is discussed with reference to specific choices made by previous authors in various branches of science and engineering.