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

HealeyDMS-1007830 主要研究者和他的同事们研究了几个非线性弹性的第二梯度问题，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，通常以小的正参数为特征。 本工作的主要目的是：（1）提供合理的模型，特别是在两相脂双层囊泡的情况下，以了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最小值的平衡点（3）得到了这类模型作为弱平衡解的整体能量极小值的存在性，并研究了当小参数趋于零时它们在渐近极限处的结构。目标（2）和（3）与目标（1）有着千丝万缕的联系。 我们采用具有一般本构函数特征的有理连续模型，研究了分叉阈值和全局能量极小化器的结构，并与实验结果进行了比较。 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。 该项目的重点是基本建模和预测数学分析的定量表征的形状和变形模式的某些微米级结构下施加负载，即，脂质双层膜囊泡，形状记忆合金，和薄膜。 每一个都与基础科学和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相关的问题。 例如，脂质双层膜在生物分子系统中无处不在;理解和预测它们的力学行为对于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压力，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要求对其在负载下的多相力学行为有基本的了解。 同样，对于形状记忆合金和薄膜的相变，对其行为的基本理解和数学预测对于表征新型材料的机械性能以及微米尺度的电位传感和驱动都很重要。

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.