Exotic Continua: Geometry, Topology and Mechanics in Soft Matter

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This grant is being funded by the Condensed-Matter and Materials Theory program in the Division of Materials Research and by the Applied-Mathematics program in the Division of Mathematical Sciences.NONTECHNICALWhile the engineered world is built with straight lines and rigid components, nature is filled with beautiful, soft and undulating shapes. We see these shapes in lichens and in corals, in kale and in sea slugs, in cacti and in flatworms. A natural question is why? Why are frilly, crenelated forms ubiquitous in nature, and what potential evolutionary benefits arise from these shapes? What can we learn from nature to help develop soft and flexible robots for practical applications? These are some of the questions that will be addressed in this research.The short answer is that an undulating, ruffled shape possesses some fascinating mechanical properties that influence the ways in which living organisms grow, move, and otherwise interact with their environment. To understand these mechanical properties and, further, to use them to model natural systems as well as for technological design, we have to turn to ideas and tools from multiple disciplines: mathematics (differential geometry), materials (elasticity), and mechanics (forces and motion). This project will explore connections between these disparate fields and develop new theoretical and practical tools for modeling and designing with soft/flexible materials. This project includes specific applications to growth, plants, marine invertebrates, and soft robotics, and will, therefore, be of interest to researchers in applied mechanics, physics, biology, and engineering. This research is strongly interdisciplinary and offers excellent opportunities for training the next generation of scientists and mathematicians in a variety of technical skills, as well as in their ability to abstract, model, and design complex systems for practical applications.TECHNICALA template, that nature uses repeatedly, is that of a ruffled, undulating shape. We see it in a multitude of living organisms. A natural question is why? More precisely, what is unique about this ruffled, undulating shape that it is so prevalent in nature. The answer seems to lie in some fascinating mechanical properties this shape confers on organisms, properties that arise from interesting topological and geometric considerations. Studying this interplay between topology, geometry and mechanics is the overarching theme of this project.The ruffled shape arises naturally in thin elastic objects that are intrinsically negatively curved. These are examples of exotic continuua, i.e. materials that self-organize into collective states that display local signatures of geometry and topology. The self-organization manifests itself as extreme pliability and strongly nonlinear and spatially inhomogeneous response to external forces. This phenomenon has important biomechanical implications, as well as potential technological applications for biomimetic design. The PI proposes to investigate the mechanical properties of a class of materials that exhibit these features: non-Euclidean elastic thin sheets. This project weaves together three disparate strands of research to create new theoretical and numerical tools for modeling and analyzing the mechanics, growth, and dynamics of soft materials. These ``formerly unrelated" areas are (1) Discrete differential geometry, which develops discrete analogs of concepts in continuous geometry and has been developed in the context of graphics and computer science; (2) The study of Lorentz surfaces with roots in pure mathematics and investigations into the causal structure of 2D spacetimes; and (3) The invariant variational bicomplex, which arose from a study of the interplay between symmetries and calculus of variations, and has roots in analysis, group theory and geometry. The interplay between theoretical questions and practical applications in this project offers excellent opportunities for education and interdisciplinary training to undergraduate and graduate students. The PI will continue ongoing efforts to encourage undergraduates to participate in scientific research, help beginning graduate students transition into starting research, mentor graduate research associates, and actively work with people from groups that are under-represented in the mathematical sciences to help advance their scientific careers. All of these efforts are geared toward developing a diverse, innovative, and broadly trained STEM workforce in our society.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.