CAREER: Variational Analysis of Elastic Patterns and Mechanical Metamaterials

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。在复杂力学系统的数学建模中，非凸和奇异摄动的优化方法是普遍存在的，本项目所涉及的问题--约束膜中的应力集中和机械超材料中的形状变化--是非线性力学和变分法的前沿课题。这项工作是跨学科的，成功将来自于将工程和物理的技术与纯粹的数学分析相结合。严格的最优化问题被用来识别最极端的例子，目的是推导出预测填充弹性薄板中褶皱和折叠图案的一般理论，以及用于承重可变形材料设计的一般技术。计划开展面向高中生的外展活动，包括大学生和科学、技术、工程和数学领域的研究人员。该项目旨在培养在变分分析和材料力学交叉领域工作的下一代有效的数学研究人员，支持本科生的研究基础设施，并为研究生和本科生提供支持和指导机会。研究集中在两组来自力学的问题：约束弹性壳和相关一维系统中的应力集中，以及被称为机械超材料的许多物体相互作用的弹性系统的聚集特性。关于应力聚焦：目的是开发皱折状态的变分模型，这种状态最近在受限壳体中观察到，但尚未得到系统的数学处理。在前人对扁壳起皱规律预测的基础上，作者从完全非线性弹性出发，寻求了更一般起皱状态的渐近刻画。关于机械超材料：研究人员的目标是研究这些弹性系统和其他相关的多体弹性系统的有效变形和本征应力-应变规律，目的是预测这些弹性系统在外加载荷下的整体行为。目标是从完全非线性弹性出发，推导出相应的弱极限和无限多物体极限中的储能。学生们在各个层面都参与了该项目，包括通过芝加哥地区的一次高中外展活动，以及由研究人员共同指导的本科生研究数学计算实验室。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.