Nonlinear Problems for Highly Deformable Elastic Solids and Structures

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This research project concerns the modeling, analysis, and numerical exploration of deformations of mechanical and biomechanical systems, in particular highly deformable thin-surface structures and solids. These occur naturally in bio-molecular systems and also in man-made thin films and elastomers. Lipid-bilayer membranes are ubiquitous in bio-molecular systems, and the accurate modeling and prediction of their mechanical response under external stimuli is crucial for understanding the behavior of cell function and also that of liposomes, the latter of which can be used as vehicles for nutrient and drug delivery. Likewise, a fundamental understanding of the nonlinear response of highly deformable structures and materials is important, for example, in the design of sensors as well as for many other engineering applications. The research aims to provide new classes of continuum-mechanical models and novel approaches to their mathematical analysis, leading to a quantitative understanding of the behavior of such systems.This project centers on the modeling, computation, and analysis of highly deformable, thin elastic structures and solids. In particular, classes of problems for incompressible solids, thin elastic surfaces, two-phase lipid-bilayer vesicles, and generalized rod models will be addressed. The main goals of the work are: (i) to provide new classes of continuum-mechanics-based models, (ii) to systematically find global equilibria as loading and other parameters vary and assess their stability (local energy minima), and (iii) to identify new phenomena. Goal (ii) entails rigorous existence results as well as systematic numerical computation. In particular, the former will entail addressing new questions in both partial differential equations and the calculus of variations. Goals (i) and (ii) inform and enrich the other; goal (iii) is enabled by goals (i) and (ii). The proposed work is highly interdisciplinary, requiring tools and perspectives from several fields, including nonlinear continuum mechanics, biophysics, materials science, nonlinear elliptic partial differential equations, bifurcation theory, calculus of variations, numerical methods, and symmetry ideas.