Nonlinear Problems of Elasticity for Multiphase Solids and Shells

建议：DMS-0406161PI：Timothy J Healey Institution：Cornell University标题：多相固体和贝壳的非线性弹性问题摘要。希利计划研究非线性弹性的各种问题，并将其应用于多相、形状记忆固体和非线性弹性壳--特别是生物膜或生物壳。对于第一类问题，来自以全局能量最小化技术为主的领域，工作的主要目标是提供关于加载和界面能量存在的亚稳定平衡的新工具和结果。对于壳问题，主要目的是了解名义上的球壳的非轴对称平衡，这已经成为分子生物结构中普遍存在的脂双层膜或生物壳的一个重要问题。从分析的角度来看，支配这些模型的非线性偏微分方程式基本上没有受到影响。这项工作是跨学科的，具有挑战性，需要来自多个不同领域的工具和视角，如非线性分析和偏微分方程、分叉理论、非线性连续介质物理、计算方法、对称性思想、材料科学和生物物理学。当然，这两种现象的详细数学模型是不同的，但该项目的总体方法和理念是相同的。实验证明，存在一种“完美”或同质的状态，这种状态会失去稳定性，之后会出现更多奇特的图案。这项工作的目标是提供新的数学工具，从定量的角度预测和理解此类现象的出现和持续。这项工作与技术有直接的联系：形状记忆合金在作为“智能材料”的应用中很重要，并有可能在微观尺度上提供驱动能力；生物膜的可变形性和机械行为与细胞功能密切相关，而细胞功能在生物学中具有核心重要性。

Proposal: DMS-0406161PI: Timothy J HealeyInstitution: Cornell UniversityTitle: Nonlinear Problems of Elasticity for Multiphase Solids and ShellsABSTRACTT. Healey plans to study various problems from nonlinear elasticity, with applications to multi-phase, shape-memory solids and nonlinearly elastic shells - especially bio-membranes or bio-shells. For the first class of problems, coming from a field dominated by global-energy minimization techniques, the main goal of the work is provide new tools and results concerning meta-stable equilibria in the presence of loading and interfacial energy. For shell problems the main goal is to understand non-axisymmetric equilibria of nominally spherical shells, which has recently become an important question for lipid-bilayer membranes or "bio-shells", which are ubiquitous in molecular-biological structures. The nonlinear partial differential equations governing these models are essentially untouched - from the point of view of analysis. The proposed work is interdisciplinary and challenging, requiring tools and perspectives from several different fields, e.g., nonlinear analysis and partial differential equations, bifurcation theory, nonlinear continuum physics, computational methods, symmetry ideas, materials science and bio-physics.The proposed work will provide new mathematical models and predictive analysis for two seemingly different physical phenomena - phase transitions in shape-memory solids and in bio-membranes. Of course the detailed mathematical models for the two phenomena are distinct, but the overall approach and philosophy of the project is the same. As verified in experiments, there is a "perfect" or homogeneous state, which loses stability, after which more exotic patterns emerge. The goal of the work is to provide new mathematical tools to predict and understand the emergence and persistence of such phenomena from a quantitative point of view. The work has direct connections to technology: Shape-memory alloys are important in applications as "smart materials" and can potentially provide actuation capabilities on the micro scale; the deformability and mechanical behavior of bio-membranes is closely related to cell function, which is of central importance in biology.