At first sight the subject of elasticity is dull and prosaic. However, a closer look at one’s wastepaper basket, or a peek outside the window shows that there are actually many experimental phenomena challenging investigation – the structure of a crumpled sheet of paper, the rippling of a leaf, the coils of a pea tendril, the cracking of mud, and more –  all beckon the curious scientist willing to veer away from the beaten path, hoping for the occasional sweep of generalization arising from geometry and topology.

One early question we addressed was the approximate shape of a Mobius band, linking topology, geometry and elasticity.  Later studies involved elastic patterns and defects in thin filaments, plates and shells. We provided a  theoretical and experimental description of the fundamental singularity in thin elastic sheets, that is ubiquitous in crumpled paper, origami, drapery, etc. This has helped resolve how approximately isometric deformations of a two-dimensional surface help pack it in three-dimensions.  We also derived a general nonlinear theory of how fine scales such as wrinkling arise in slender structures combining geometry and physics, with predictions for the wavelength and amplitude that were verified by our experiments and data that range over 16 orders of magnitude in scale, from nanotube wrinkling to tectonic subduction.

An exploration of the implications of these instabilities in different physical situations followed, e.g. how nested self-similar wrinkles arise in polymer skins, a theory of self -organized origami to create foldable structures such as the Miura-ori and related fine-scale instabilities in supported and unsupported thin films. Our work on how singularities  and fine scales   form has led to substantial interest in these problems from the engineering community who now use the design principles we uncovered to build surfaces with unusual frictional, optical and other properties, and in the mathematical community interested in a rigorous analysis of the scaling laws.

Another theme is the study of soft material interfaces via the use of simple geometric and scaling arguments combined with approximate analysis and experiments, complementing the large-scale computational approaches that are commonly used. We uncovered a fundamentally new instability in the simplest of these systems, a half-space that is compressed uniformly (in-plane strain) can fold into a scale-free sulcus akin to the structures in the brain – unexplained for more than a century – and yet as ubiquitous and general as buckling and cracking. Our work used experiment, theory, and computations to show that it is akin to a first-order transition, but without a barrier, with a number of physical and mathematical implications that are only just beginning to be explored. In studying soft interfaces and thin solid films, we showed how a confined elastic meniscus can lose stability to the formation of fingers (digits)  in a manner that is similar to the classical Saffman-Taylor instability in hydrodynamics, but via a sub-critical instability that opens the way for patterning thin films digitally.

Related Publications

Geometry and physics of wrinkling,  Cerda, E. and L. Mahadevan,  Physical Review Letters , 90 (7) 074302, 2003 (Physical Review Focus Article). [View PDF] [Download PDF]
Conical dislocations in crumpling Cerda, E., S. Chaieb, F. Melo and L. Mahadevan,  Nature , 401, 46-49, 1999. [View PDF] [Download PDF]
Wrinkling of a stretched elastic sheet, Cerda, E., K. Ravi-Chandar and L. Mahadevan,  Nature , 419, 579, 2002. [View PDF] [Download PDF]
Topology, geometry and mechanics of strongly stretched and twisted filaments: solenoids, plectonemes, and artificial muscle fibers. N. Charles, M. Gazzola, L. Mahadevan Phys. Rev. Lett. , 123, 208003, 2020. [DOI] [View PDF] [Download PDF]
Geometric localization in supported elastic struts TCT Michaels, R. Kusters, AJ Dear, C. Storm, JC Weaver, L. MahadevanProceedings of the Royal Society A 475, 20190370. 2019. [DOI] [View PDF] [Download PDF]
Periodic folding of thin sheets Mahadevan, L., and J.B. Keller,  SIAM Journal on Applied Mathematics , 55(6) , 1609-1624, 1995. [View PDF] [Download PDF]
The shape of a Möbius band Mahadevan, L., and J.B. Keller,  Proceedings of the Royal Society of London, Series A , 1440, 149-162, 1993. [View PDF] [Download PDF]
Rings, rackets and kinks in filamentous assemblies, Cohen, A. and L. Mahadevan,  Proceedings of the National Academy of Sciences (USA) , 100, 12141-46, 2003. [View PDF] [Download PDF]
Popliteal instability of bent multi-walled elastic tubes, Mahadevan, L., J. Bico and G. McKinley,  Europhysics Letters , 65 (3), 323-29, 2004. [DOI] [View PDF] [Download PDF]
Elements of Draping Cerda, E., L. Mahadevan and J. Passini,  Proceedings of the National Academy of Sciences (USA) , 101 (7), 1806-10, 2004. [View PDF] [Download PDF]
Solenoids and Plectonemes in stretched and twisted elastomeric filaments,  A. Ghatak and L. Mahadevan,  Physical Review Letters  , 95, 057801, 2005. [View PDF] [Download PDF]
Self-organized origami, L. Mahadevan and S. Rica,  Science , 307, 1740, 2005. [View PDF] [Download PDF]
Crack-front instability in a confined elastic film M. Adda Bedia and L. Mahadevan,  Proceedings of the Royal Society of London, series A , 462, 3233, 2006. [View PDF] [Download PDF]
Mechanical basis for fibrillar bundle morphology. T. Michaels, E. Memet, and L. Mahadevan, Soft Matter , 16, 9306-18, 2020. [ONLINE ARTICLE] [DOI] [View PDF] [Download PDF]
Confined elastic developable surfaces: cylinders, cones and the elastica E. Cerda and L. Mahadevan,  Proceedings of the Royal Society of London (A),  461, 671-700, 2005. [View PDF] [Download PDF]
Geometry, mechanics and electronics of singular structures in graphene V. Pereira, H-Y Liang, A. Castro-Neto and L. Mahadevan,  Physical Review Letters , 105, 156603, 2010. [View PDF] [Download PDF]
The shape and motion of a ruck in a rug J. Kolinski, P. Aussillous and L. Mahadevan,  Physical Review Letters , 103, 174302, 2009. [View PDF] [Download PDF]
Self-similar nested wrinkling patterns in skins K. Efimenko, M. Rackaitis, E. Manias, A. Vaziri, L. Mahadevan and J. Genzer,  Nature — Materials , 4, 293-97, 2005. [View PDF] [Download PDF]
Elastic-instability enabled locomotion. A. Nagarkar, W-K. Lee, D. Preston, M. Nemitz, N. Deng, G. Whitesides and L. Mahadevan, Proc. Natl. Acad. Sci., e2013801118, 2021. [ONLINE ARTICLE] [DOI] [View PDF] [Download PDF]
 Static adhesion hysteresis in elastic structures. E. Memet, F. Hilitski, Z. Dogic and L. Mahadevan, Soft Matter, DOI: 10.1039/d0sm02192j, 2021. [DOI] [View PDF] [Download PDF]
Compact reconfigurable kirigami.
G.P. T. Choi , L.H. Dudte, and L. Mahadevan. Phys. Rev. Res.3, 043030, 2021.
[View PDF] [Download PDF]
Geometric control of topological dynamics in a singing saw. Suraj Shankar, Petur Bryde and L. Mahadevan.  Proceedings of  the National Academy of  Sciences, Vol. 119 | No. 17 [ONLINE ARTICLE] [DOI] [View PDF] [Download PDF]