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

9623273 Hilgers提出了对非线性高弹性膜的改进模型的继续探索。这项研究与传统膜力学的不同之处在于，即使是高度柔性的膜也会对曲率的变化表现出一定的阻力。这被称为弯曲刚度，并通过在应变能密度中包含二阶导数依赖而引入改进模型。基础理论先前已经建立。在这个建议中，它是寻求将理论应用于各种实际问题，目的是了解弯曲刚度如何使膜的行为规范化。这些问题的基本主题是将膜置于一种状态，在这种状态下，某些区域试图支持压应力。我们预计薄膜会起皱。标准的非线性膜理论无法描述这种行为。其中一个目标是找出皱纹的性质。另一个是了解弯曲刚度如何稳定起皱变形。此外，零和非零弯曲刚度膜响应之间的结果比较是非常有趣的，因为它对数学和力学都有影响。要考虑的具体例子包括容易起皱的充气材料，隔膜，不可扩展的近似，以及紧张和松弛板材的振动。技术给我们带来的舒适通常要求工程师使用薄而灵活的薄片、层、层压板和涂层材料。对于设计工程师来说，这些材料在制造或操作过程中会折叠、起皱、起泡、扣扣和折痕，从而降低了它们的可靠性和质量。不幸的是，大多数数学模型无法预测这种不必要的行为，这使得工程师很难阻止它。幸运的是，近年来，在美国国家科学基金会的资助下，一种旨在解决这些缺点的材料模型得到了发展。当前提案的目的是通过使用该模型来提供详细说明各种情况下褶皱和皱纹形成的示例，从而对这些工程困难进行定性和定量的洞察，从而继续进行这项调查。该模型的通用性允许工程师和材料科学家将其应用于一系列工业问题，从薄涂层的起泡到生物医学手术充气材料的起皱。可以想象，所获得的知识可能会影响电路板制造，安全气囊技术，甚至是医疗保健中出现的生物力学细胞膜问题。＊＊＊

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. ***