Multiphase Problems of Nonlinear Elasticity

研究人员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常见线索是下面的数学结构：低阶项中的非凸势能，它除了包括载荷之外，还包括小的高阶正则化(或奇异扰动)，其特征是小参数e0。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--用于理解这种结构通常的奇异行为；(2)系统地找到与总势能或亚稳定解的局部最小值相对应的平衡。在讨论(1)时，注意到在真实材料上的力学实验通常是从均匀的试件开始的，该试件在稳定的、准静态的载荷变化下逐渐发展出越来越复杂的结构或微观结构。研究人员使用了以一般本构函数为特征的有理连续介质模型，并寻找了分叉和稳定性交换的阈值，与实验进行了比较。特别是，亚稳态溶液可能是在准静态实验中观察到的。因此，对于(2)，他考虑了一种新的方法，其基础是：(A)任意小的e0的解的严格存在性，通过打破对称的全局分支方法和a-优先级界；(B)有效和可靠的全局数值路径跟踪来寻找亚稳态。该项目的重点是对某些微米级结构在外加载荷下的形状和变形模式的定量表征进行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。其中每一个都与基础科学和技术有直接和重要的联系。脂质双层膜普遍存在于生物分子系统中，了解和预测其力学行为对于理解细胞功能至关重要。该项目的重点是简单的人造薄膜或脂质体。脂质体(封闭膜)作为药物输送载体的未来前景取决于对其力学行为的基本了解。同样，对于形状记忆合金中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和在微米尺度上的电位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.