Geometric Problems in Elasticity of Thin Films, Kirigami, and the Monge-Ampere System

The investigator pursues projects that combine questions in mathematical analysis, differential geometry, calculus of variations, materials science and engineering design. The key components are: (i) seeking to determine mechanical theories of thin multi-dimensional films with nonzero stored energy due to shape-formation processes such as growth or plasticity; (ii) the quest for regularity of solutions to a class of partial differential equations arising when the aforementioned prestrained films deform in order to release their energies; (iii) describing properties of “kirigamized” sheets, namely thin films with cuts of different geometries and distributions. Some of these projects are accessible to graduate students and contribute to their training. The related analytical projects include: (i) dimension reduction in nonlinear elasticity of prestrained materials, in function of the general prestrain given by a Riemannian metric, Gamma-convergence and rigidity estimates; (ii) convex integration and flexibility in the Holder regularity classes for the Monge-Ampere system and the k-Hessian system; and (iii) investigating structure and rectifiablity of geodesics in the kirigamized sheets in relation to the sheet’s deployment trajectory. 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.