Nonlinear Problems of Second-Gradient Elasticity for Multi-Phase Structures and Solids

多相结构和固体的二阶梯度弹性非线性问题

• 批准号: 1007830
• 负责人: Timothy Healey
• 金额: $ 24.56万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007830&HistoricalAwards=false
• 关键词: Nonlinear; Problems; Second; Gradient; Elasticity

HealeyDMS-1007830 The principal investigator and his colleagues study severalsecond-gradient problems of nonlinear elasticity, withapplications to multi-phase structures and solids -- especiallylipid bilayer vesicles, shape-memory alloys and thin structures. A common thread running through these problems is the followingmathematical structure: a non-convex potential energy in thefirst-gradient term and an additive second-gradientregularization, often characterized by a small positiveparameter. The main goals of this work are: (1) to provideclasses of rational models -- particularly in the case oftwo-phase lipid bilayer vesicles -- for understanding the oftenexotic behavior of such structures under loadings; (2) tosystematically find their equilibria corresponding to localminima of the total potential energy (meta-stable solutions); (3)to obtain the existence of global energy minima as weakequilibrium solutions of those models and study their structurein the asymptotic limit as the small parameter approaches zero. Goals (2) and (3) are inextricably linked to (1). We employrational continuum models, characterized by general constitutivefunctions, and study thresholds of bifurcation and the structureof global energy minimizers to compare with experiment. Theproposed work is highly interdisciplinary, requiring tools andperspectives from several fields, e.g., nonlinear analysis andpartial differential equations, bifurcation theory, calculus ofvariations, nonlinear continuum mechanics, computational methods,symmetry ideas, biophysics, and materials science. The project focuses on fundamental modeling and predictivemathematical analysis for the quantitative characterization ofshape and deformation patterns of certain micron-scale structuresunder applied loading, namely, lipid-bilayer membrane vesicles,shape-memory alloys, and thin films. Each of these has directand important connections to basic science and technology -- inparticular, to problems associated with bio-molecular structuresand advanced materials. For example, lipid-bilayer membranes areubiquitous in bio-molecular systems; understanding and predictingtheir mechanical behavior is crucial for understanding cellfunction. The project focuses on understanding the behavior ofpure, man-made membranes or liposomes under changes in osmoticpressure, temperature, or composition. The future promise ofliposome vesicles (closed membranes) as vehicles for drugdelivery demands a fundamental understanding of their multi-phasemechanical behavior under loading. Likewise for phasetransitions in shape-memory alloys and wrinkling of thin films --a fundamental understanding and mathematical prediction of theirbehavior are important for characterizing the mechanicalproperties of novel materials and for potential sensing andactuation at the micron scale.

Healey DMS-1007830首席研究员和他的同事们研究了几个非线性弹性的二次梯度问题，并将其应用于多相结构和固体--特别是脂质双层囊泡、形状记忆合金和薄结构。贯穿这些问题的一个共同线索是下面的数学结构：第一个梯度项中的非凸势能和一个附加的第二级正则化，通常以一个小的正参数为特征。这项工作的主要目标是：(1)提供有理模型的解密--特别是在两相脂质双层囊泡的情况下--以了解这类结构在负载下的常见行为；(2)系统地找到与总势能(亚稳定解)的局部极小值相对应的平衡；(3)获得这些模型的全局能量极小值的存在性，并在小参数趋于零时研究其结构的渐近极限。目标(2)和(3)与(1)有着千丝万缕的联系。我们使用了以一般本构函数为特征的有理连续模型，研究了分叉阈值和全局能量最小化的结构，并与实验进行了比较。拟议的工作是高度跨学科的，需要从几个领域的工具和观点，例如，非线性分析和偏微分方程，分叉理论，变分，非线性连续介质力学，计算方法，对称思想，生物物理学和材料科学。该项目专注于定量描述某些微米级结构在外加载荷下的形状和变形模式的基本建模和预测数学分析，即脂质双层膜泡、形状记忆合金和薄膜。其中每一项都与基础科学和技术有直接而重要的联系，特别是与生物分子结构和先进材料有关的问题。例如，脂质双层膜在生物分子系统中普遍存在；了解和预测它们的机械行为对于了解细胞功能至关重要。该项目的重点是了解纯人造膜或脂质体在渗透压、温度或组成变化时的行为。脂质体囊泡(封闭膜)作为药物输送载体的未来前景需要对其在负载下的多相力学行为有一个基本的了解。同样，对于形状记忆合金中的相变和薄膜的起皱，对它们行为的基本了解和数学预测对于表征新材料的机械性能以及微米级的电势传感和驱动都是重要的。