CAREER: Variational Analysis of Elastic Patterns and Mechanical Metamaterials

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Non-convex and singularly perturbed optimization methods are ubiquitous in the mathematical modeling of complex mechanical systems, and the questions addressed in this project - on stress focusing in confined membranes, and shape change in mechanical metamaterials - are at the cutting edge of nonlinear mechanics and the calculus of variations. The work is interdisciplinary, and success will come from blending techniques from engineering and physics with pure mathematical analysis. Rigorous optimization questions are considered to identify the most extreme examples, with the aim of deriving a general theory for predicting the motifs of wrinkles and folds in packed elastic sheets, as well as general techniques for the design of load-bearing morphable materials. Outreach activities to high school students are planned, involving university students and researchers in science, technology, engineering, and mathematics. With the goal of training the next generation of effective mathematical researchers working at the intersection of variational analysis and the mechanics of materials, this project supports undergraduate research infrastructure, and provide support and mentoring opportunities for graduate and undergraduate students. The research concentrates on two sets of questions from mechanics: on stress focusing in confined elastic shells and related one-dimensional systems, and on the aggregate properties of many body interacting elastic systems known as mechanical metamaterials. On stress focusing: the aim is to develop a variational model of the wrinkle-fold state, which has recently been observed in confined shells but has yet to receive a systematic mathematical treatment. Based on prior successes with predicting the wrinkling patterns of shallow shells, the investigator seeks an asymptotic characterization of the more general wrinkle-fold state starting from fully nonlinear elasticity. On mechanical metamaterials: motivated by the question of predicting the overall behaviors of kirigami elastic systems in response to applied loads, the investigator aims to characterize the effective deformations and emergent stress-strain laws of these and other related many body elastic systems. The goal is to start from fully nonlinear elasticity and derive the relevant weak limits and stored energy in the limit of infinitely many bodies. Students are involved in the project at all levels, including through a high school outreach event in the Chicago area, as well as in a mathematical computing laboratory for undergraduate research co-directed by the investigator. 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. NSF Award ID: 2618358 | Program: 01002526DB NSF RESEARCH & RELATED ACTIVIT,01002425DB NSF RESEARCH & RELATED ACTIVIT,01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Ian Tobasco | Institution: Regents of the University of Michigan - Ann Arbor, ANN ARBOR, MI | Award Amount: $103,434 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2618358 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2618358.html