CAREER: Variational Analysis of Elastic Patterns and Mechanical Metamaterials

职业：弹性模式和机械超材料的变分分析

• 批准号: 2350161
• 负责人: Ian Tobasco
• 金额: $ 45万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-10-15 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350161&HistoricalAwards=false
• 关键词: CAREER; Variational; Analysis; Elastic; Patterns

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.

该奖项的全部或部分资金根据《2021 年美国救援计划法案》（公法 117-2）提供。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，该项目解决的问题——约束膜中的应力集中和机械超材料的形状变化——处于非线性力学和变分计算的前沿。这项工作是跨学科的，成功将来自于工程和物理学技术与纯数学分析的结合。考虑严格的优化问题来识别最极端的例子，目的是推导预测包装弹性片中皱纹和折叠图案的通用理论，以及设计承载可变形材料的通用技术。计划针对高中生开展外展活动，其中包括大学生和科学、技术、工程和数学领域的研究人员。该项目的目标是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，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。该研究集中于两组力学问题：有限弹性壳和相关一维系统中的应力集中，以及许多身体相互作用的弹性系统的聚合属性。 机械超材料。关于应力聚焦：目标是开发皱纹折叠状态的变分模型，该模型最近在有限的贝壳中观察到，但尚未接受系统的数学处理。基于先前预测浅贝壳皱纹模式的成功，研究人员寻求从完全非线性弹性开始的更一般的皱纹折叠状态的渐近表征。在机械超材料方面：出于预测剪纸弹性系统响应施加载荷的整体行为的问题，研究人员旨在表征这些和其他相关的许多身体弹性系统的有效变形和出现的应力应变定律。目标是从完全非线性弹性出发，导出无限多个物体极限下的相关弱极限和储存能量。学生们在各个层面上参与该项目，包括通过芝加哥地区的高中推广活动，以及由研究者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价值和更广泛的影响审查标准进行评估，被认为值得支持。