Geometry & elasticity

Filaments, sheets, and shells, acquire their characteristic mechanical softness due to their slender geometry. As a result, thin elastic structures display a stunning array of complex morphologies, instabilities, and dynamics across scales, from crumpled paper and ruffled leaves to ripples in atomically thin graphene. How can we understand and control such phenomena? What are the morphological, statistical, mechanical, and dynamical consequences? Can we design new (meta)materials with prescribed shapes and programmable responses simply by manipulating material geometry and architecture? While the basic equations of elasticity theory are well-established, they remain largely intractable to analytical solutions due to strong geometric nonlinearities. One approach has been to view the large-scale patterns and dynamics of thin elastic sheets as unusual “collective phenomena”, only now emerging from an interplay of geometry and elasticity rather than microscopic composition and interactions. Our work has shown that a geometric focus allows thin elastic structures to be fruitfully viewed as condensed matter. This offers a new perspective on classic problems and brings forth a powerful set of tools, from the renormalization group to topological band theory to impress upon diverse problems in mechanics, whose seemingly common-place nature hides their deep complexity.

Mechanical phase transitions

Mechanical instabilities, e.g., buckling, wrinkling, etc., lead to large changes in shape and response for small changes in external forces and stresses, very similar to phase transitions. In the presence of thermal fluctuations, as relevant for graphene and other 2D materials, these instabilities become true critical phenomena with unconventional scaling and size dependent mechanics. Our work has shown that subtle thermodynamic features, including boundary conditions, system size, and ensemble choice can have a drastic impact on the force response and buckling of thermalized sheets, with profound consequences for nanomechanical metrology.

Geometric incompatibility & mechanics
Thin sheets perforated by holes and slits are easy to moph; a feature exploited by the Japanese art of kirigami (kiri: cut, kami: paper). How can we understand this soft response? Via an analogy to electrostatics, we have shown that a hole in an externally loaded sheet is a source of geometric frustration and its buckling response is a mechanism of stress relief. Our quantitative description of nonlinear kirigami mechanics provides simple design rules for locally relaxing stresses in patterned elastic sheets.

Acoustics & dynamics

Acoustic devices and resonators, including musical instruments, often use specialized designs to sustain sound or vibrations. In contrast, a carpenter’s handsaw becomes an instrument (the singing saw) and manages to hold a note simply when the blade is bent in an S-shape. How does geometry accomplish this feat? We have shown that spatially varying curvature in a thin shell can localize and trap acoustic modes along inflection lines, insulating them from dissipative decay. These modes are topologically protected, akin to edge states in topological insulators, and allow the saw to function as a geometrically tuned high-quality oscillator. Our work has uncovered a simple mechanism for designing robust, yet reconfigurable, high-quality resonators across scales, from macroscopic instruments to nanoscale devices, just by using geometry.

Relevant publications