This work is a part of a long-term project of the investigator and his collaborators to study the elasticity properties of materials through methods of calculus of variations. Here, the focus is on situations when the elastic bodies exhibit some sort of rigid behavior. In this manner, the study of geometric properties of structures, e.g. the properties of various spaces of isometries and infinitesimal isometries of 2-dimensional surfaces, comes to the fore. This differs from classical differential geometry in the weaker regularity of the given surface or of the deformations. A first objective is to identify and rigorously derive variational theories for thin shells from the 3-dimensional theory of nonlinear elasticity through Gamma-convergence methods. The derived theories naturally depend on the scaling regimes of the elastic energy or body forces in terms of shell thickness. In this context, the investigator works on problems involving various spaces of weakly differentiable isometries. In parallel, properties of the derived theories, such as multiplicity and regularity of solutions, or their dependence on the geometry of the shell, are investigated. Another line of investigation is the study of non-stress-free configurations observed e.g. in the growth of leaves. A tangentially heterogeneous 3-dimensional nonlinear elastic model is studied and the Gamma-limit approach is used to reduce the dimension and analyze the model. Quantitative rigidity estimates also are studied as a strong analytical tool in tackling these variational problems.
The investigator studies different mechanical phenomena under the unifying umbrella of the nonlinear elasticity theory. Elastic materials exhibit qualitatively different responses to different kinematic boundary conditions or body forces. This fact has given rise to many interesting questions in the mathematical theory of elasticity. The main goal of this theory is to explain various, apparently different, phenomena in terms of some shared mathematical principles. The variational approach to the nonlinear theory has been very effective in dealing with these questions. It has also been helpful in rigorously deriving models for elastic shells or plates pertaining to different scaling regimes of the body forces. In the latter context, the strength of this approach lies in the fact that it can predict the appropriate model together with the response of the elastic plate for the given scaling of forces or kinematic boundary conditions without any a priori assumptions other than the general principles of the 3-dimensional nonlinear elasticity. Another feature of this approach is that it can lead to new models that were not previously considered.
