Calculus of variations on rigid elastic structures

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PakzadDMS-0907844 This work is a part of a long-term project of theinvestigator and his collaborators to study the elasticityproperties of materials through methods of calculus ofvariations. Here, the focus is on situations when the elasticbodies exhibit some sort of rigid behavior. In this manner, thestudy of geometric properties of structures, e.g. the propertiesof various spaces of isometries and infinitesimal isometries of2-dimensional surfaces, comes to the fore. This differs fromclassical differential geometry in the weaker regularity of thegiven surface or of the deformations. A first objective is toidentify and rigorously derive variational theories for thinshells from the 3-dimensional theory of nonlinear elasticitythrough Gamma-convergence methods. The derived theoriesnaturally depend on the scaling regimes of the elastic energy orbody forces in terms of shell thickness. In this context, theinvestigator works on problems involving various spaces of weaklydifferentiable isometries. In parallel, properties of thederived theories, such as multiplicity and regularity ofsolutions, or their dependence on the geometry of the shell, areinvestigated. Another line of investigation is the study ofnon-stress-free configurations observed e.g. in the growth ofleaves. A tangentially heterogeneous 3-dimensional nonlinearelastic model is studied and the Gamma-limit approach is used toreduce the dimension and analyze the model. Quantitativerigidity estimates also are studied as a strong analytical toolin tackling these variational problems. The investigator studies different mechanical phenomenaunder the unifying umbrella of the nonlinear elasticity theory. Elastic materials exhibit qualitatively different responses todifferent kinematic boundary conditions or body forces. Thisfact has given rise to many interesting questions in themathematical theory of elasticity. The main goal of this theoryis to explain various, apparently different, phenomena in termsof some shared mathematical principles. The variational approachto the nonlinear theory has been very effective in dealing withthese questions. It has also been helpful in rigorously derivingmodels for elastic shells or plates pertaining to differentscaling regimes of the body forces. In the latter context, thestrength of this approach lies in the fact that it can predictthe appropriate model together with the response of the elasticplate for the given scaling of forces or kinematic boundaryconditions without any a priori assumptions other than thegeneral principles of the 3-dimensional nonlinear elasticity. Another feature of this approach is that it can lead to newmodels that were not previously considered.