Seminars & Colloquia

Frustrating Geometry: Elastic Theory of non-Euclidean Plates

Efi Efrati, University of Massachusetts Amherst

Natural growth processes tend to result in bodies which possess no stress free configuration. We formulate a hyper-elastic theory for such bodies in which strain is measured with respect to a reference metric rather than a reference configuration. In this formulation, the residual stress arises from the geometrical frustration involved in the attempted isometric embedding of the non-Euclidean 3D metric in Euclidean space. Applying this formalism to thin sheets, we derive a reduced 2D elastic theory enabling us to treat thin bodies which are neither plates nor shells, which we term non-Euclidean plates. In this talk I will present some of the phenomena exhibited by non-Euclidean plates such as, spontaneous buckling and the convergence to the Willmore energy minimizing isometry in the limit of vanishing thickness. I will also discuss the existence of such minimizers of the Willmore energy and their relation to minimal surfaces.