The principal investigator and his colleagues study several second-gradient problems of nonlinear elasticity, with applications to multi-phase structures and solids -- especially lipid bilayer vesicles, shape-memory alloys and thin structures. A common thread running through these problems is the following mathematical structure: a non-convex potential energy in the first-gradient term and an additive second-gradient regularization, often characterized by a small positive parameter. The main goals of this work are: (1) to provide classes of rational models -- particularly in the case of two-phase lipid bilayer vesicles -- for understanding the often exotic behavior of such structures under loadings; (2) to systematically find their equilibria corresponding to local minima of the total potential energy (meta-stable solutions); (3) to obtain the existence of global energy minima as weak equilibrium solutions of those models and study their structure in the asymptotic limit as the small parameter approaches zero. Goals (2) and (3) are inextricably linked to (1). We employ rational continuum models, characterized by general constitutive functions, and study thresholds of bifurcation and the structure of global energy minimizers to compare with experiment. The proposed work is highly interdisciplinary, requiring tools and perspectives from several fields, e.g., nonlinear analysis and partial differential equations, bifurcation theory, calculus of variations, nonlinear continuum mechanics, computational methods, symmetry ideas, biophysics, and materials science.
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 -- in particular, to problems associated with bio-molecular structures and advanced materials. 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 pure, 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. Likewise for phase transitions in shape-memory alloys and wrinkling of thin films -- a fundamental understanding and 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.
By far the most important aspect of this work is its potential application to biological membranes (e.g., vesicles) and rods (e.g., DNA). We bring new modeling ideas, analytical tools and quantitative results (prediction of deformation, foces, moments, stress, energy, etc.) to this important area. Clearly, accurate modeling and quantitative prediction of mechanical phenomena related to cell function could potentially lead to breakthroughs in human health and medicine. [1] T.J. Healey, Q. Li & R.-B. Cheng, J. Nonlinear Science 23 (2013) p. 777. [2] T. J. Healey & C. Papdapoulos, Quart. Appl. Math. 71 (2013) p. 729. [3] T.J. Healey & A. Sipos, Physica D 261 (2013) p. 62. [4] T.J. Healey & H. Kielhöfer, J. Dynamics and Differential Equations (2013), electronically published.
