Popliteal rippling of layered elastic tubes and scrolls
L. Mahadevan (1), J. Bico (2) and G. McKinley (2)
(1) Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K.
(2) Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Mass. Ave., Cambridge, MA 02139, USA.
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 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.