The investigator studies several multi-phase problems of nonlinear elasticity, with applications to lipid-bilayer membranes, shape-memory alloys and thin films. A common thread running through these problems is the following mathematical structure: a non-convex potential energy in the lower-order terms, which includes the loading, in addition to a small, higher-order regularization (or singular perturbation), characterized by a small parameter e > 0. The main goals of this work are: (1) to provide classes of rational models -- particularly in the case of multi-phase lipid-bilayer membranes -- for understanding the often exotic behavior of such structures; (2) to systematically find equilibria corresponding to local minima of the total potential energy or meta-stable solutions. In addressing (1), observe that mechanical experiments on real materials often begin with a homogeneous specimen that progressively develops increasingly complex patterns or micro-structure under steady, quasi-static changes in loading. The investigator employs rational continuum models, characterized by general constitutive functions, and looks for thresholds of bifurcation and exchange of stability to compare with experiment. In particular, meta-stable solutions are the ones presumably observed in quasi-static experiments. Accordingly for (2) he considers a new methodology based upon: (a) rigorous existence of solutions for arbitrarily small e > 0 via symmetry-breaking global bifurcation methods and a-priori bounds; (b) efficient and reliable global numerical path-following to find meta-stable states.
The project focuses on fundamental modeling and predictive mathematical analysis for the quantitative characterization of shape and deformation patterns of certain micron-scale structures under applied loading, namely, lipid-bilayer membrane vesicles, shape-memory alloys and thin films. Each of these has direct and important connections to basic science and technology. 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 simple, man-made membranes or liposomes. The future promise of liposome vesicles (closed membranes) as vehicles for drug delivery demands a fundamental understanding of their mechanical behavior under loading. Likewise for phase transitions in shape-memory alloys and wrinkling/blistering of thin films -- a fundamental understanding and the mathematical prediction of their behavior are important for characterizing the mechanical properties of novel materials and for potential sensing and actuation at the micron scale.
