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，为工程师分析从建筑物圆顶到飞机机身、船体和汽车车身等各种类型部件的强度和安全性提供了方便和有效的手段。薄壳也构成了大量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。薄结构在自然界和技术中无处不在，因为它们提供了迄今为止最有效的安排，既能承受载荷，又能最大限度地减轻重量。目前关于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，以帮助制定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无法实现。该奖项支持这些理论的基本进展，这些理论不依赖于这些假设，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分析更有效的航空航天、汽车和民用结构成为可能，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，也将有利于生物力学等新兴领域。这些方法和结果将以目前正在编写的专著的形式，并通过课程讲座和专门的短期课程传播。板壳理论的现代工作强调严格的降维程序，如渐近展开或收敛。这些都是基于板或壳具有弹性的基本假设。相比之下，在较早的文献中，人们发现模型可以解释更一般的材料行为，如可塑性和纤维增强引起的各向异性，尽管基于对潜在运动学的特殊假设，这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，这些模型在特设方法中解决了这种影响，现代模型仅限于纯弹性行为，但总体上更严格。因此，这项研究有可能大大扩展结构和固体力学的范围。研究小组将根据降维方法定义壳的非弹性行为的新模型，以包括壳材料内物种的塑性和扩散。该团队还将建立模型，以解释纤维增强板和壳中纤维的固有弯曲和扭转刚度的影响。预计这将大大改进现有的理论，这些理论只考虑纤维赋予的各向异性。这些理论的进步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.