The investigator focuses on the understanding and systematic study of instabilities that can be explained by variational principles of minimum energy. One such instability is buckling. Buckling is ubiquitous and very important in engineering and mechanics. Modern engineering models of buckling, though very successful, are not related to 3D hyperelasticity, which is supposed to contain a description of all elastic phenomena, including buckling. The project initiates a systematic study of all elastic instabilities, putting buckling in its proper setting within a general theory. Another interesting and important instability studied by the project is nucleation of a new phase in martensitic phase transitions. This study is also related to rate-independent hysteresis -- an unexpected phenomenon of history dependence, that does not yet have a universally agreed-upon explanation. The energy-based model of martensitic phase transformations was very successful at predicting many aspects of behavior of shape memory alloys. Hysteresis, however, could be explained by this model only if one assumes that the material "gets stuck" in metastable states, modeled as local energy minima. The project provides a hitherto missing general tool for identifying metastable states, allowing one to construct and analyze continuum mechanical models with hysteresis quantitatively on the basis of the energy-minimization principle.
This project aims to advance our understanding of nonlinearly elastic materials, such as polymers and rubbers and materials with shape memory effect. When loads are applied to these materials, they deform in such a way as to minimize the total energy stored in the deformed system. When the loads increase, the nonlinear nature of these materials often manifests itself in the appearance of new ways to decrease the energy. When this happens, physicists talk of instabilities. One of the most common instabilities is buckling, which occurs when the load on a slender column exceeds a certain critical value. Recent work of the investigator with Lev Truskinovsky produced a new understanding of buckling that is used in this project to explore new methods of computing buckling loads for structures with complex geometries, such as elastic shells and composite materials. The systematic study of elastic instabilities that is performed in this project also sheds some light on rate-independent hysteresis -- a still poorly understood phenomenon, whereby a shape memory alloy follows different deformation paths upon loading and unloading. The project's contribution towards better understanding of instabilities permits a quantitative analysis of one of the proposed explanations for hysteresis, that the material "gets stuck" in local minima. The project also presents educational and training opportunities for graduate and undergraduate students. Parts of this project form the cores of doctoral dissertations of the investigator's graduate students, while other parts inform the content of advanced undergraduate classes and independent studies.
