Nonlinear Problems of Elasticity for Multiphase Solids and Shells

项目申请：dms -0406161项目负责人：Timothy J healey；机构：康奈尔大学；题目：多相固体和壳的非线性弹性问题。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——尤其是生物膜或生物壳。对于第一类问题，来自一个由全局能量最小化技术主导的领域，工作的主要目标是提供关于存在载荷和界面能的亚稳定平衡的新工具和结果。对于壳问题，主要目标是理解名义上的球形壳的非轴对称平衡，这最近已成为分子生物结构中普遍存在的脂质双层膜或“生物壳”的重要问题。从分析的角度来看，控制这些模型的非线性偏微分方程基本上没有受到影响。所提出的工作是跨学科和具有挑战性的，需要来自几个不同领域的工具和观点，例如非线性分析和偏微分方程，分岔理论，非线性连续介质物理，计算方法，对称思想，材料科学和生物物理学。提出的工作将为两种看似不同的物理现象-形状记忆固体和生物膜中的相变-提供新的数学模型和预测分析。当然，这两种现象的详细数学模型是不同的，但项目的总体方法和哲学是相同的。正如实验所证实的那样，存在一个“完美”或均匀的状态，它失去了稳定性，之后出现了更多的奇异模式。这项工作的目标是提供新的数学工具，从定量的角度来预测和理解这种现象的出现和持续。这项工作与技术有直接的联系：形状记忆合金作为“智能材料”在应用中很重要，可以在微观尺度上提供驱动能力；生物膜的可变形性和力学行为与细胞功能密切相关，在生物学中具有重要意义。

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.