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Morphological instabilities of a stretched twisted ribbon
Finding strategies toward developing functional shapes at smaller and smaller scales has been a focus of intense research in elastic materials. Our investigations show that a wide variety of shapes and instabilities can be obtained by simply varying the applied twist and tension [1]. Due to its geometry which is intermediate between rod and plate, a ribbon can coil and form loops but wrinkles and stress localization can also been seen yielding a surprisingly rich variety of shapes. Using the twist angle, the tension, the thickness and the length as control parameters, the various configurations including longitudinal and transverse wrinkling, and creasing can be rationalized in a 4D phase diagram. We show that this experimental phase diagram can be quantitatively explained using a theoretical framework based on (i) a covariant form of the Föppl-von Kàrmàn equations which is necessary to obtain the correct stress field ; and (ii) a far from threshold analysis which describes a longitudinally-wrinkled state with negligible compression [2]. Our study has both impact on fundamental theory of thin elastic materials [3] and development of smart, reliable, and efficient strategy to build complex structures starting with graphene sheets, and flat semiconductor nanoribbons and biomaterials.
Figure 1 Typical morphologies of ribbons subjected to twist and stretching : (a) helicoid, (b,c) longitudinally wrinkled helicoid, (d) creased helicoid, (e) formation of loops and selfcontact zones, (f) cylindrical wrapping, (g) transverse buckling and (h) twisted towel shows transverse buckling/wrinkling (from Ref.[2]).
[1] J. Chopin and A. Kudrolli, Phys. Rev. Lett. 111, 174302 (2013),
URL : http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.107.208304
[2] J. Chopin, V. Démery, and B. Davidovitch, ArXiv e-prints, 1403.0267 (2014),
URL : http://arxiv.org/abs/1403.0267
[3] C. Santangelo, Journal Club for Condensed Matter Physics (2014),
URL : http://www.condmatjournalclub.org/?p=2330