CAREER: Geometric and topological mechanics of flexible structures

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

NONTECHNICAL SUMMARY This CAREER award supports theoretical and computation research and educational activities into the design and characterization of flexible structures. The worlds of science and of everyday life are divided into systems in different phases. Despite their vastly different atomic structures, systems in the solid phase, including brick, bone, metal, glass and wood all have in common that they retain a particular shape, unlike fluids which take the shape of their container. Consequently, solids all display universal response when not treated too harshly, such as developing patterns of stress that can support loading forces and transmitting energy through vibrational waves. However, flexible solids like paper, cloth, wires and many organic tissues such as flower petals differ from this seemingly universal behavior by undergoing large deformations even when gently probed. This flexibility leads to vast new phenomena, and especially to achieving function derived from changing shape, such as in unfurling leaves, collapsing tents or the wing of a bird or plane reshaping itself in response to changing wind conditions. This project explores a particular subset of flexible systems. These systems consist of repeating geometrical motifs, such as rigid pieces that rotate against one another at hinges or origami panels that fold along creases. Such structures are referred to as flexible mechanical metamaterials because they display new properties due to these structural elements that are not derived from their chemical composition. This research applies and extends scientific and mathematical principles that permit the design of new structures that can be deformed into sets of shapes. Choosing particular structures leads to new ways for the system to dilate, shear and curve as desired. This research program is complemented by educational and outreach activities. The PI is developing an advanced course on applying topological concepts to research problems. The team is also developing K12 classroom modules that use solid, manipulable systems such as origami and string to realize deep topological and geometrical concepts. Finally, scientific concepts are communicated to the public at large via a number of channels. TECHNICAL SUMMARY This CAREER award supports theoretical and computational research and educational activities into flexible structures whose rational design leads to analytically tractable and universal behavior. A hallmark of soft matter systems as diverse as liquid crystals, granular matter, gels, and various living structures across many scales is the ability to undergo large deformations while still offering some solid-like resistance to strain. These soft structures host a rich and useful set of phenomena, including buckling, memory, multistability, adaptation, frustration and phase transitions. However, their diversity and complexity limits the ability to identify universal principles that can unite the field. This project focuses on rationally designed systems consisting of rigid elements joined via relatively flexible hinges in a geometrically complementary way which allows the structure to undergo a large deformation at low energies. These structures are nonlinear and complex enough to display rich phenomena while being sufficiently rational and controlled to be analytically tractable and governed by universal theories. The team will develop rules for identifying new structures involves formulating compatibility conditions in the language of discrete differential geometry, combinatorics, and tensor calculus. Complementary to this problem, mechanical criticality indicates that the deformation mode can serve as a symmetry of the system, leading to a mechanism field as the mode is activated to different degrees in different parts of the structure. These phenomena are closely informed by the role of curvature, symmetry and topology. The team will identify which nonlinear, non-uniform low-energy deformations are possible in such structures. Finally, these quasistatic geometric properties define a large low-energy space of nonlinear deformations on which high-frequency dynamics can occur. The team will explore how these mechanism fields can be actively driven and dissipate energy in ways that are fundamentally distinct from conventional structures. This research program is complemented by educational and outreach activities. The PI is developing an advanced course on applying topological concepts, including topological insulators, crystal defects and gauge fields to research problems. The team will also develop K12 classroom modules that use solid, manipulable systems to realize deep topological and geometrical concepts such as Euler's formula for polyhedra and Eulerian paths on graphs. Finally, scientific concepts will be communicated to the general public via a number of channels. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.