Multiphase Problems of Nonlinear Elasticity

研究者研究了非线性弹性的几个多相问题，并将其应用于脂质双层膜、形状记忆合金和薄膜。贯穿这些问题的一个共同的线程是以下的数学结构：一个低阶项的非凸势能，它包括负载，除了一个小的，高阶正则化（或奇异扰动），以一个小参数e0为特征。这项工作的主要目标是：(1)提供一类合理的模型-特别是在多相脂质双层膜的情况下-用于理解这种结构的奇异行为；(2)系统地寻找与总势能局部极小值相对应的平衡点或元稳定解。在解决(1)中，观察到真实材料的力学实验通常从一个均匀的样品开始，在稳定的准静态加载变化下逐渐发展出越来越复杂的图案或微观结构。研究者采用以一般本构函数为特征的理性连续体模型，并寻找分岔阈值和稳定性交换阈值与实验进行比较。特别是，亚稳定解是在准静态实验中可能观察到的解。因此，对于(2)，他考虑了一种新的方法，该方法基于：(a)通过对称破缺的全局分岔方法和a-优先界得到任意小e 0解的严格存在性；(b)高效可靠的全局数值路径跟踪方法来寻找元稳定状态。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。这些都与基础科学技术有着直接而重要的联系。脂质双分子膜在生物分子系统中普遍存在；了解和预测它们的力学行为对理解细胞功能至关重要。该项目侧重于简单的人造膜或脂质体。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机械行为的基本理解。同样，对于形状记忆合金的相变和薄膜的起皱/起泡，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。

Healey0707715 The investigator studies several multi-phase problems ofnonlinear elasticity, with applications to lipid-bilayermembranes, shape-memory alloys and thin films. A common threadrunning through these problems is the following mathematicalstructure: a non-convex potential energy in the lower-orderterms, which includes the loading, in addition to a small,higher-order regularization (or singular perturbation),characterized by a small parameter e 0. The main goals of thiswork are: (1) to provide classes of rational models --particularly in the case of multi-phase lipid-bilayer membranes-- for understanding the often exotic behavior of suchstructures; (2) to systematically find equilibria correspondingto local minima of the total potential energy or meta-stablesolutions. In addressing (1), observe that mechanicalexperiments on real materials often begin with a homogeneousspecimen that progressively develops increasingly complexpatterns or micro-structure under steady, quasi-static changes inloading. The investigator employs rational continuum models,characterized by general constitutive functions, and looks forthresholds of bifurcation and exchange of stability to comparewith experiment. In particular, meta-stable solutions are theones presumably observed in quasi-static experiments. Accordingly for (2) he considers a new methodology based upon:(a) rigorous existence of solutions for arbitrarily small e 0via symmetry-breaking global bifurcation methods and a-prioribounds; (b) efficient and reliable global numericalpath-following to find meta-stable states. 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 direct andimportant connections to basic science and technology. Lipid-bilayer membranes are ubiquitous in bio-molecular systems;understanding and predicting their mechanical behavior is crucialfor understanding cell function. The project focuses on simple,man-made membranes or liposomes. The future promise of liposomevesicles (closed membranes) as vehicles for drug delivery demandsa fundamental understanding of their mechanical behavior underloading. Likewise for phase transitions in shape-memory alloysand wrinkling/blistering of thin films -- a fundamentalunderstanding and the mathematical prediction of their behaviorare important for characterizing the mechanical properties ofnovel materials and for potential sensing and actuation at themicron scale.