CAREER: Variational Analysis of Elastic Patterns and Mechanical Metamaterials

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。非凸和奇摄动优化方法在复杂机械系统的数学建模中无处不在，本项目所解决的问题-在封闭膜中的应力集中，以及机械超材料的形状变化-处于非线性力学和变分学的前沿。这项工作是跨学科的，成功将来自工程和物理技术与纯数学分析的结合。严格的优化问题被认为是识别最极端的例子，目的是推导出一个一般的理论来预测褶皱和褶皱的图案，以及设计承重变形材料的一般技术。计划向高中生开展推广活动，包括大学生和科学、技术、工程和数学方面的研究人员。为了培养下一代在变分分析和材料力学交叉领域工作的高效数学研究人员，该项目支持本科生研究基础设施，并为研究生和本科生提供支持和指导机会。本研究主要集中在两组力学问题上：局限弹性壳和相关一维系统的应力聚焦，以及被称为机械超材料的许多体相互作用弹性系统的聚集特性。关于应力聚焦：其目的是建立一种褶皱状态的变分模型，这种变分模型最近在有限的壳中被观察到，但尚未得到系统的数学处理。基于先前成功预测浅壳的起皱模式，研究者寻求从完全非线性弹性开始的更一般的起皱-折叠状态的渐近表征。在机械超材料方面：由于预测基里伽米弹性系统在外加载荷下的整体行为的问题，研究者的目标是表征这些和其他相关的许多体弹性系统的有效变形和紧急应力-应变规律。目标是从完全非线性弹性出发，在无限多物体的极限下推导出相关的弱极限和存储能量。学生们在各个层面都参与了这个项目，包括在芝加哥地区的一个高中外展活动，以及在一个由研究者共同指导的本科生研究的数学计算实验室。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.