Popliteal rippling of layered elastic tubes and scrolls

– Motivated by the periodic ripples observed in bent multi-walled nanotubes, we use macroscopic ideas to derive geometric scaling laws for the wavelength and amplitude of ripples in elastic scrolls and multi-walled tubes. Remarkably, our predictions are essentially independent of material properties, and thus have a range of validity varying from the atomic to the macroscopic even for relatively large deformations. We verify this using experimental data that vary over six orders of magnitude in length, ranging from millimeters to nanometers obtained using materials as disparate as rubber and graphite. The mechanical response of a slender strut or a tube is very different from the bulk response of the same material, owing primarily to the geometric separation of scales inherent in the structure. This makes the structure relatively flexible and capable of large elastic deformations, a fact that is at the heart of many new material systems. The scale of these soft modes of deformation, such as bending and twisting, is typically determined by a combination of material and geometric properties and gives rise to all manner of instabilities classified under the general rubric of buckling. Understanding the response of the structures beyond the onset of these instabilities is typically made difficult by the combination of geometric and material nonlinearities. However, for small structures as well for those made of soft materials, a reasonable description of the large-deformation behavior requires a consideration of just the geometric nonlinearities, thus making the problem more tractable. In this letter, we treat a class of such systems motivated by the nonlinear mechanical response of layered tubular structures, and show that our geometric scaling laws are consistent with experimental data gathered from phenomena separated by six orders of magnitude in scale. In fig. 1(a), we see the rippling instability of a bent multi-walled carbon nanotube, where the finite-amplitude periodic ripples have a wavelength that is much larger than the thickness of a single layer, itself a fraction of a nanometer. These deformations are not unique to the nano-world; when a rubber macrotube made by rolling a thin sheet of rubber into a scroll (∗) Current address: Divison of Engineering and Applied Sciences, Harvard University Cambridge, MA 02138, USA. E-mail: [email protected]