In condensed-matter physics, a Nambu-Goldstone mode is a particular low energy deformation of an ordered phase arising from a symmetry of the ground state. Examples are acoustic phonons and spin waves. In this paper, I use the fact that an isometry is a symmetry of the stretching energy of a shell to derive a set of low energy Goldstone-like deformations. This new approach to understanding the small deformations of shells is then used to revisit the problem of a pinched cylinder and used to study a pinched cone.
C.D. Santangelo, “Nambu-Goldstone modes and diffuse deformations in elastic shells,” to appear in Soft Matter (2013). [ARXIV]
Physicists Narayanan Menon, Benny Davidovitch and Chris Santangelo, together with Tom Russell from PS&E, have been awarded a $1M grant from the WM Keck Foundation to develop the basic science needed to spontaneously deliver ultrathin films to fluid interfaces. The films may then serve as microscopic and functional wall paper or shrink-wrap. The Keck Science and Engineering program funds “endeavors that are distinctive and novel in their approach. It encourages projects that are high-risk with the potential for transformative impact”.
We develop equations that describe the three-dimensional shape of a sheet upon which a series of concentric, curved folds have been inscribed. When there is no stretching outside the fold, the three-dimensional shape of a single fold prescribes the shape of the entire origami structure. To better explore these structures and their mechanical properties, we develop a description of the smooth surface intersecting all the mountain folds. We find that this surface is always saddle-shaped. A series of open, circular folds (as seen on the right figure) with constant fold angle generate a classical minimal surface known as the helicoid.
M.A. Dias and C.D. Santangelo, “The shape and mechanics of curved fold origami structures,” EPL 100, 54005 (2012). [JOURNAL] [ARXIV].