Nonlinear Problems for Thin Elastic Structures

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Healey1312377 The principal investigator and his colleagues study several problems of nonlinear elasticity for thin structures and solids, with applications to multi-phase lipid bilayer vesicles, wrinkling of highly stretched sheets and damage/fracture of solids. The main goals of the work are: (1) provide classes of rational, accurate models for understanding the often exotic behavior of such systems under various loadings; (2) systematically find their locally stable equilibria (corresponding to local minima of the total potential energy) as loading and/or composition parameters vary, particularly as small parameters like thickness, inter-facial capillarity, etc., asymptotically approach zero. Goal (2) is inextricably linked to (1). The investigator employs rational continuum models, characterized by general constitutive functions, to study questions of existence, thresholds of bifurcation and instability, and the structure of local energy minima. The work is highly interdisciplinary, requiring tools and perspectives from several areas of mathematics as well as biophysics and materials science. The investigator undertakes fundamental modeling and mathematical analysis enabling a quantitative, predictive characterization of the behavior of certain structures and solids under applied loading: lipid-bilayer membrane vesicles, thin films and the nucleation and progression of damage/fracture zones in solids. Each of these has direct and important connections to basic science and technology -- especially biotechnology and materials and manufacturing. For example, lipid-bilayer membranes are ubiquitous in bio-molecular systems; understanding and predicting their mechanical behavior is crucial for understanding cell function. The project focuses on understanding the behavior of man-made membranes or liposomes under changes in osmotic pressure, temperature, or composition. The future promise of liposome vesicles (closed membranes) as vehicles for drug delivery demands a fundamental understanding of their multi-phase mechanical behavior under loading and change of composition. Associated with this, but also of more general interest, is the wrinkling of thin films, which also shows up in the design of many thin devices or coatings, fabric-like structures, in the behavior of human skin, etc. The investigator studies the onset and development of wrinkles in very thin structures in highly stretched environments. Finally, the fracture of solids, a well-known culprit behind sudden and catastrophic failures in structures, is also currently of great interest for purposes of harvesting natural gas. Predictive models, which are especially lacking in this field, are addressed in this project. In particular, he studies the onset and development of damage leading to fracture of solids under loading.