Study of Instabilities in Phase Transitions, Shell Buckling, and Inverse Problems

Energy minimization principles can often be used to explain what is observed in nature, such as instabilities - phenomena where small changes in the environment cause large quantitative or even qualitative changes. This project considers energy methods to analyze three types of instability relevant to applications. The first type concerns phase transformations in solids, which underly shape memory effects and other applications of smart materials, with a focus on stability of phase boundaries, to improve the ability of identifying critical strains at the onset of phase transitions.

The second is buckling, where the failure of a slender structure occurs abruptly, after the critical stress threshold has been crossed. Slender structures play an increasingly important role in the technological world, delivering light-weight and highly functional devices. However, a full understanding of their extreme sensitivity to imperfections is not yet available in both mechanics and engineering.

One specific goal of the project is to shed light on the buckling of axially compressed cylindrical shells, by revealing the mechanisms that allow small imperfections of shape and load to have a dramatic effect on the critical stress. The third instability analyzed is of a numerical type, where the precision of measurements of properties of materials, such as electromagnetic permittivity or electrical impedance spectrum, does not translate to equally precise prediction of their response at much higher or much lower frequencies than in the available data.

Inspired by applications to radiology and remote sensing, the aim is to quantify these instabilities to inform the creation of new algorithms that reconstruct the material response characteristics with provable optimality. The project provides research mentoring and training opportunities for graduate students.

Mathematics being developed to study stability of phase boundaries represents a contribution to Calculus of Variations, where the almost intractable concept of quasiconvexity plays a central role. The research project will develop tools needed to gain insight into the structure of quasiconvex envelopes. Notwithstanding the fact that quasiconvexity in general defies any meaningful general attack, the information this research will deliver will be of direct practical importance, permitting exact or approximate relaxations of specific energies, and will provide a provably complete set of practically accessible information.

The investigation of buckling of cylindrical shells will create a new set of theoretical predictions to be compared with experiment. The key feature of these predictions is the quantitative description of mechanisms of instability responsible for the discrepancy between the classical buckling load and the experimentally observed ones. Theoretical issues pertaining to more general shells will also be addressed.

Stieltjes functions form a special class of analytic functions virtually ubiquitous in physics. Quantitative understanding of such functions will contribute to practical algorithms for reconstructing material responses, such as complex electromagnetic permittivity of dielectric media and complex impedance of electrical circuits. A new investigation into the properties of completely monotone functions will complement and expand the current scant knowledge about this important class of functions. Applications to more efficient processing of radiology and remote sensing data are anticipated.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.