Self-Buckling of Pressurized Cylindrical Tubes

Abstract

We investigate the buckling of hollow cylindrical tubes subject to their own weight and internal pressure, inspired by the columnar cells of the palisade mesophyll in dicotyledon leaves which resemble pressurized cylindrical tubes. When the internal pressure in the cylinder is equal to the outside pressure, the problem is usually termed self-buckling, which has been studied extensively for solid rods, hollow cylinders, and thin cylindrical shells. Specifically, we perform FEM simulations and desktop-scale experiments to determine the instability thresholds for different geometrical parameters. We first test our models against self-buckling results without pressure for solid rods and hollow cylindrical tubes, and then proceed to determine the critical buckling pressure for a set of material and geometrical parameters. We find that positive internal pressures can stiffen cylinders that are unstable under their own weight, leading to an effective Young's modulus that we show scales linearly with the applied pressure. On the contrary, cylinders that are stable under self-weight, buckle under a negative pressure, resembling classical results on pressure-induced ring buckling. Our findings offer new insights on the interplay between gravity and pressure for the mechanical instability of hollow cylindrical tubes, which we hope will be useful for the study of both engineering and biological structures under similar loads.

Figures (5)

• Figure 1: (a) The geometry of the cylinder subjected to internal pressure $P$. (b) Experimental setup. The test specimen is clamped on an internal mandrel, which allows for volume control via a syringe pump. (c) Cross-section image of a viburnum leaf showing the internal photosynthetic tissue comprising palisade mesophyll and spongy mesophyll, with the cells pressurized through osmotic water exchange. Scale bar 20 µm. (d) 3D reconstruction of mesophyll tissue in Arabidopsis thaliana mutant cgr2-3 showing palisade cells in light green. Scale bar 50 µm.
• Figure 2: Self-buckling of solid and hollow cylinders. The critical value of $\beta_\textup{cr}$ is shown as a function of $R/L$, while the colorbar indicates different values of $\alpha$. Solid black and dashed blue lines represent the analytical solution for $\alpha=0$ and $\alpha=0.8$, respectively. Solid symbols represent our numerical results for the SVK energy with $\nu=0$ (diamonds) and $\nu=0.49$ (squares), and for the neo-Hookean energy (circles). Triangles are experimental results on cylindrical shells from Calladine1Calladine2, to highlight the different behavior between hollow cylinders and cylindrical shells.
• Figure 3: Critical dimensionless pressure, $P_\textup{cr}/E$, as a function of $\beta/\beta_\textup{cr}$, for different values of $\alpha$ as shown in the colorbar. Numerical results are shown as solid squares, while experiments are shown as circles with corresponding error bars. The two insets show the characteristic buckling modes for $\beta/\beta_\textup{cr}>1$ (bending), and for $\beta/\beta_\textup{cr}<1$ (localized buckling).
• Figure 4: Effective Young's modulus normalized by the Young's modulus of the base material, $E_\textup{e}/E$ as a function of $P/E$ for cylinders subject to positive pressure, for $\alpha\in[0.6,0.8]$. Regression is performed for $\alpha\in[0.7,0.8]$, which yields the solid black line.
• Figure 5: Absolute value of the critical, dimensionless, pressure, $P_\textup{cr}/E$, in the plateau regime versus $h/R_\textup{m}$. Experiments are represented as crosses, while numerical results are represented as circles. Error bars denote the standard deviation of the mean. The solid black line represents the theoretical prediction of the critical pressure for ring buckling.