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

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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. ***