Multiphase Problems of Nonlinear Elasticity

希利0707715 研究人员研究了几个非线性弹性的多相问题，并将其应用于脂质双层膜、形状记忆合金和薄膜。 一个共同的线程贯穿这些问题是以下的prosticalstructure：一个非凸的势能在低阶项，其中包括加载，除了一个小的，高阶正则化（或奇异摄动），其特征在于一个小参数e 0。 本工作的主要目标是：（1）提供合理的模型类-特别是在多相脂质双层膜的情况下-为理解这种结构的经常奇异的行为;（2）系统地找到平衡对应于局部最小的总势能或亚稳定的解决方案。 在讨论（1）时，注意到对真实的材料进行的力学实验通常开始于一个在稳定的、准静态的载荷变化下逐渐发展出越来越复杂的图案或微观结构的复杂试样。 研究者采用具有一般本构函数特征的理性连续体模型，寻找分叉和稳定性交换的阈值，并与实验进行比较。 特别是，亚稳态的解决方案是那些可能观察到准静态实验。因此，对于（2），他认为一个新的方法的基础上：（a）严格存在的解决方案，为任意小的e 0通过突破性的全球分歧的方法和a-优先界;（B）有效和可靠的全球numericalpath-following找到亚稳态。 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。脂质双层膜在生物分子系统中普遍存在，理解和预测其力学行为对于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欠载力学行为的基本理解。 同样，对于形状记忆合金中的相变和薄膜的起泡/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电位传感和致动都是重要的。

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.