Nonlinear Problems for Highly Deformable Elastic Solids and Structures

本研究项目涉及机械和生物力学系统变形的建模、分析和数值探索，特别是高度可变形的薄表面结构和固体。这些在生物分子系统中自然发生，也在人造薄膜和弹性体中发生。脂质双分子膜在生物分子系统中无处不在，准确建模和预测其在外界刺激下的机械反应对于理解细胞功能行为以及脂质体的行为至关重要，后者可以用作营养和药物输送的载体。同样，对高度可变形结构和材料的非线性响应的基本理解也很重要，例如，在传感器的设计以及许多其他工程应用中。该研究旨在提供新的连续力学模型和数学分析的新方法，从而对此类系统的行为进行定量理解。这个项目集中在建模，计算和分析高度可变形，薄弹性结构和固体。特别是不可压缩固体、薄弹性表面、两相脂质双层囊泡和广义棒模型的问题。这项工作的主要目标是：(i)提供基于连续力学的新类型模型，（ii）系统地发现负载和其他参数变化时的全局平衡并评估其稳定性（局部能量最小值），以及（iii）识别新现象。目标（ii）需要严格的存在性结果和系统的数值计算。特别是，前者需要解决偏微分方程和变分学中的新问题。目标(i)和（ii）告知并丰富对方；目标（iii）由目标(i)和（ii）实现。这项工作是高度跨学科的，需要来自多个领域的工具和观点，包括非线性连续介质力学、生物物理学、材料科学、非线性椭圆偏微分方程、分岔理论、变分法、数值方法和对称思想。

This research project concerns the modeling, analysis, and numerical exploration of deformations of mechanical and biomechanical systems, in particular highly deformable thin-surface structures and solids. These occur naturally in bio-molecular systems and also in man-made thin films and elastomers. Lipid-bilayer membranes are ubiquitous in bio-molecular systems, and the accurate modeling and prediction of their mechanical response under external stimuli is crucial for understanding the behavior of cell function and also that of liposomes, the latter of which can be used as vehicles for nutrient and drug delivery. Likewise, a fundamental understanding of the nonlinear response of highly deformable structures and materials is important, for example, in the design of sensors as well as for many other engineering applications. The research aims to provide new classes of continuum-mechanical models and novel approaches to their mathematical analysis, leading to a quantitative understanding of the behavior of such systems.This project centers on the modeling, computation, and analysis of highly deformable, thin elastic structures and solids. In particular, classes of problems for incompressible solids, thin elastic surfaces, two-phase lipid-bilayer vesicles, and generalized rod models will be addressed. The main goals of the work are: (i) to provide new classes of continuum-mechanics-based models, (ii) to systematically find global equilibria as loading and other parameters vary and assess their stability (local energy minima), and (iii) to identify new phenomena. Goal (ii) entails rigorous existence results as well as systematic numerical computation. In particular, the former will entail addressing new questions in both partial differential equations and the calculus of variations. Goals (i) and (ii) inform and enrich the other; goal (iii) is enabled by goals (i) and (ii). The proposed work is highly interdisciplinary, requiring tools and perspectives from several fields, including nonlinear continuum mechanics, biophysics, materials science, nonlinear elliptic partial differential equations, bifurcation theory, calculus of variations, numerical methods, and symmetry ideas.