Non-Standard Plate and Shell Models in Solid Mechanics

板壳理论是结构力学领域的基础。它们本质上是薄结构的二维表示，为工程师提供了一种方便有效的手段来分析不同类型部件的强度和安全性，从建筑物的圆顶到飞机机身、船体和汽车车身。薄壳还构成了一系列生物结构的基本组成部分，包括细胞、骨骼和眼睛的部分，仅举几例。薄壁结构在自然界和技术上无处不在，因为它们提供了迄今为止最有效的结构来承受载荷，同时将重量降至最低。目前板壳结构的工程理论是基于关于板的材料如何在三维中变形的合理假设，以帮助形成易于处理的二维模型。然而，众所周知，这些假设有时在实践中并未实现。该奖项支持在这些不依赖于此类假设的理论中取得根本性进展，同时考虑到组成材料中的塑性和纤维增强的影响。这项研究的结果将使更有效的航空航天、汽车和民用结构的分析成为可能，因此将使美国经济和社会总体受益。这里建立的技术将为工程师提供改进的分析工具，也将使生物力学等新兴领域受益。这些方法和结果将以目前正在进行的专著的形式传播，并通过课程讲座和专门的短期课程进行传播。现代板壳理论工作强调严格的降维程序，如渐近展开或伽马收敛。它们总是基于板或壳的弹性行为这一基本假设。相比之下，在较早的文献中，人们发现一些模型可以解释更一般的材料行为，如塑性和纤维增强引起的各向异性，尽管这些模型基于关于潜在运动学的特殊假设，而这些假设在一般应用中可能无法实现。目前的研究旨在填补现有模型和现代模型之间的空白，现有模型以特别方法解决这种影响，现代模型仅限于纯粹的弹性行为，但总体上更严格。因此，这项研究有可能显著扩大结构力学和固体力学的研究范围。研究小组将基于降维方法，为壳体中的非弹性行为定义新的模型，以包括壳体材料中物种的可塑性和扩散。该团队还将建立模型，解释纤维增强板壳中纤维固有的弯曲和扭转刚度的影响。预计这将大大改进现有的理论，这些理论只解释了纤维产生的各向异性。这些理论进展将得到广泛的数值模拟的支持。

Plate and shell theories are fundamental to the field of structural mechanics. They are essentially two dimensional representations of thin structures and afford a convenient and effective means for engineers to analyze the strength and safety of diverse types of components, ranging from the domes of buildings to aircraft fuselages, ship hulls and automobile bodies. Thin shells also constitute the basic components of an enormous range of biological structures, including cells, bones and parts of the eye, to name just a few. Thin structures are ubiquitous in nature and technology because they afford by far the most efficient arrangements for sustaining loads while minimizing weight. Current engineering theories for plate and shell structures are based on plausible assumptions about how the material of the plate deforms in three dimensions, to aid in the formulation of a tractable two-dimensional model. However, it is known that these assumptions are sometimes not realized in practice. This award supports fundamental advances in these theories that do not rely on such assumptions, while taking into account the effects of plasticity and fiber reinforcement in the constituent materials. Results from this research will enable the analysis of more efficient aerospace, automotive and civil structures, and will therefore benefit the U.S. economy and society in general. The techniques established here will provide engineers with improved analysis tools and will also benefit emerging fields such as biomechanics. These methods and results will be disseminated in the form of a monograph currently in progress, and through course lectures and specialized short courses.Modern work on theories of plates and shells emphasizes rigorous dimension reduction procedures such as asymptotic expansions or gamma convergence. These are invariably based on the underlying assumption that the plate or shell behaves elastically. In contrast, in the older literature one finds models that account for more general material behavior, such as plasticity and the anisotropy induced by fiber reinforcement, albeit based on ad hoc assumptions about the underlying kinematics which may not be realized in general applications. The present research is designed to fill the gap between existing models that address such effects in ad-hoc approaches, and modern models that are confined to purely elastic behavior but are overall more rigorous. The research therefore has the potential to significantly expand the reach of structural and solid mechanics. The research team will define new models for the inelastic behavior in shells, based on dimension reduction approaches, to include plasticity and diffusion of species within the shell material. The team will also establish models that account for the effects of intrinsic flexural and torsional stiffness of the fibers in fiber-reinforced plates and shells. This is expected to yield a substantial improvement over existing theories that account only for the anisotropy conferred by the fibers. These theoretical advances will be supported by extensive numerical simulations.