CAREER: Variational Analysis of Elastic Patterns and Mechanical Metamaterials

该奖项全部或部分根据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，本项目中解决的问题--关于受限膜中的应力集中和机械超材料的形状变化--处于非线性力学的前沿和变分法。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融合。严格的优化问题被认为是确定最极端的例子，与推导出一个通用的理论预测图案的褶皱和折叠的包装弹性片材，以及一般技术的承载变形材料的设计的目的。计划开展面向高中生的外联活动，让大学生和科学、技术、工程和数学领域的研究人员参与。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。研究集中在力学的两组问题：应力集中在有限的弹性壳和相关的一维系统，并在许多机构相互作用的弹性系统，称为机械超材料的总属性。关于压力集中：目的是发展一个变分模型的折叠状态，这是最近观察到的限制壳，但尚未收到一个系统的数学处理。基于先前的成功预测的扁壳的折叠模式，调查人员寻求一个渐近表征的更一般的折叠状态从完全非线性弹性。关于机械超材料：受预测kirigami弹性系统响应于所施加载荷的整体行为的问题的激励，研究者旨在表征这些和其它相关的多体弹性系统的有效变形和出现的应力-应变规律。我们的目标是从完全非线性弹性开始，并在无限多个物体的极限中导出相关的弱极限和储能。学生们参与了该项目的各个层面，包括通过在芝加哥地区的高中外展活动，以及在一个数学计算实验室的本科生研究共同指导的调查员。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.