Softening of the Euler Buckling Criterion under Discretization of Compliance

Euler solved the problem of the collapse of tall thin columns under unexpectedly small loads in 1744. The analogous problem of the collapse of circular elastic rings or tubes under external pressure was mathematically intractable and has only been fully solved recently. In the context of carbon nanotubes, an additional phenomenon was found experimentally and in atomistic simulations but not explained: the collapse pressure of smaller-diameter tubes deviates below the continuum-mechanics solution [Torres-Dias et al., Carbon 123, 145 (2017)]. Here, this deviation is shown to occur in discretized straight columns and it is fully explained in terms of the phonon dispersion curve. This reveals an unexpected link between the static mechanical properties of discrete systems and their dynamics described through dispersion curves.

Figure 1

(a) Entasis in Greek columns (exaggerated). (b) A schematic of the Pont du Gard. (c) A continuous pillar, buckling (dashed). (d) A discretized pillar, buckling (dashed). (e) The collapse of a continuous elastic ring. (f) A discretized elastic ring (a polygon). (g) A continuous tube. (h) An example of a discretized tube (a nanotube) [3].

Figure 2

A pillar along the $x$ axis is discretized to have compliance only at $N$ angular hinges of stiffness $c$. The lower part of the graph shows a chain of rigid rods numbered $\mu =0$ to $N$ at angles ${\theta }_{\mu }$. Buckling causes $y$-axis displacements of the hinges $y_{\mu }$ ($\mu =1$ to $N$).

Figure 3

The normalized collapse force $F_C^N$ of Eq. (2) is plotted against $N^{-2}$, where $N$ is the number of hinges per half-wavelength of a buckling mode of an infinite chain with $k=\pi$ (solid black line) or, equivalently, the number of hinges in a column of unity length. Results from calculations for integer $N$ are plotted (solid circles), for $N=1$–4, 6, $\mathrm{\infty }$. The green dashed line is the least-squares fit of Eq. (3) to these data, excluding $N=1$. The blue crosses are the numerical data for the collapse of $m$-gons [13], plotted for $m=4$–8, 10, 12, $\mathrm{\infty }$. These data are rescaled by converting pressure to circumferential force and using $m=4N$. The blue dotted line is the least-squares fit to these data. The experimental data for nanotubes [11] are shown by red data points with error bars and fitted by the red dashed line, scaled to these axes as described in the text.