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

首席研究员和他的同事们研究了非线性弹性的几个二次梯度问题，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。贯穿这些问题的共同线索是以下数学结构：第一梯度项中的非凸势能和附加的第二梯度正则化，通常以一个小的正参数为特征。这项工作的主要目标是：(1)提供合理模型的分类-特别是在两相脂质双分子层囊泡的情况下-用于理解这种结构在负载下的奇异行为；(2)系统地找到它们对应于总势能局部极小值的平衡点（亚稳定解）；(3)得到了这些模型的整体能量极小值作为弱平衡解的存在性，并研究了它们在小参数趋近于零的渐近极限下的结构。目标(2)和(3)与目标(1)密不可分。我们采用以一般本构函数为特征的连续统模型，研究了分岔阈值和全局能量最小值的结构，并与实验进行了比较。提议的工作是高度跨学科的，需要来自多个领域的工具和观点，例如非线性分析和偏微分方程，分岔理论，变分法，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。该项目侧重于基础建模和预测数学分析，以定量表征某些微米尺度结构在施加载荷下的形状和变形模式，即脂质双层膜囊泡，形状记忆合金和薄膜。这些学科都与基础科学技术有着直接而重要的联系，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无处不在；理解和预测它们的机械行为对理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、温度或成分变化下的行为。脂质体囊泡（封闭膜）作为药物递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，对其行为的基本理解和数学预测对于表征新材料的机械性能以及在微米尺度上的潜在传感和驱动是重要的。

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.