Mathematical Sciences: An Improved Nonlinear Elastic Membrane Model: Regularizing Using Bending Stiffness

小行星9623273 目前正在继续探索一种改进的非线性高弹性膜模型。本研究与传统膜力学的出发点是观察到即使是高度柔性的膜也对曲率的变化表现出一定的阻力。这被称为弯曲刚度，并通过以下方式引入改进的模型中： 包括应变能密度的二阶导数依赖性。基本理论已经建立。 在这个建议中，它试图将理论应用于各种各样的 具有实际意义的问题，目标是了解弯曲如何 刚度使膜的行为规则化。这些问题中的基本主题是将膜置于一种状态，在该状态中，一些区域试图支撑压应力。我们预计 膜会起皱标准的非线性膜理论不能描述这种行为。一个目标是找到定性的 皱纹的性质。另一个是了解弯曲 刚度使弯曲变形稳定。此外，比较 零和非零弯曲刚度薄膜响应之间的结果是非常有趣的，因为它对数学和 力学需要考虑的具体例子包括容易起皱的充气物、隔膜、不可伸展的近似物以及紧张和松弛的片材的振动。 %%% 舒适性技术给我们带来的往往需要工程师与薄，灵活的片材，层，层压材料和涂层的工作， 材料对于设计工程师来说，这些材料将 在制造或操作过程中出现折叠、起皱、起泡、弯曲和折痕 从而降低了它们的可靠性和质量。 不幸的是大多数 数学模型不能预测这种不想要的行为， 这使得工程师们很难预防。幸运的是， 近年来，在美国国家科学基金会的资助下，开发了一种旨在解决这些缺陷的模型。是 目前提议继续进行调查的目的是， 使用这个模型提供的例子，详细介绍了形成的褶皱和皱纹，在各种情况下，导致定性和定量的洞察这些工程 困难 模型的通用性 工程师和材料的许可证申请 科学家们都对一系列工业问题感兴趣，从 薄涂层的起泡对生物医学 外科用充气袋可以想象，获得的知识可能会影响电路板制造，安全气囊安全技术，甚至 生物力学细胞膜的问题，出现在医疗保健。 ***

9623273 Hilgers The continued exploration of an improved model for nonlinear highly elastic membranes is being proposed. The point of departure of this investigation from traditional membrane mechanics is the observation that even highly flexible membranes exhibit some resistance to a change of curvature. This is known as bending stiffness and is introduced in the improved model by including a second order derivative dependence in the strain energy density. The foundational theory has been previously established. In this proposal, it is sought to apply the theory to a variety of problems of practical interest with the goal of understanding how bending stiffness regularizes the behavior of membranes. The basic theme among these problems is placing a membrane into a state in which some region is attempting to support a compressive stress. We anticipate that the membrane will wrinkle. Standard nonlinear membrane theory fails to describe this behavior. One objective is to find the qualitative nature of the wrinkles. Another is to understand how the bending stiffness stablizes wrinkling deformations. Also the comparison of results between the zero and nonzero bending stiffness membrane responses is of extreme interest as it has implications for both mathematics and mechanics. Specific examples to be considered include inflatables which are prone to wrinkle, diaphragms, inextensible approximations, and vibrations of tense and slack sheets. %%% The comfort technology has given us often requires engineers to work with thin, flexible sheets, layers, laminates, and coatings of material. It is problematic for design engineers that these materials will fold, wrinkle, blister, buckle, and crease during manufacture or operation thereby reducing their reliability and quality. Unfortunately, most mathematical models cannot predict this unwanted behavior, which makes it difficult for engineers to prevent it. Fortunately, a material model seeking to address these shortcomings was developed in recent years under funding from the National Science Foundation. It is the purpose of this current proposal to continue this investigation by using this model to provide examples detailing the formations of folds and wrinkles in a wide variety of situations leading to a qualitative and quantitative insight into these engineering difficulties. The generality of the model permits application by engineers and material scientists alike to a spectrum of industrial problems ranging from the blistering of thin coatings to the wrinkling of biomedical surgical inflatables. Conceivably, the knowledge gained could impact circuit board manufacture, air bag safety technology, and even biomechanical cellular membrane questions which arise in health care. ***