Colloquium

The unfolding of a ladybird's wings, the trapping mechanism used by a flytrap, the design of self-deployable space shades, and the constructions of curved origami are diverse examples where strategically placed material defects are leveraged to generate large and robust deformations.

With these applications in mind, we derive thin structure models incorporating the possibly of curved folds as the limit of thin three-dimensional hyper-elastic materials with defects. This results in a fourth order geometric partial differential equation for the plate deformations further restricted to be isometries. The latter nonconvex constraint encodes the plates inability to undergo shear nor stretch and is critical to justify large deformations.

In passing, we explore the rigidity of the folding process by taking advantage of the natural moving frames induced by piecewise isometries along the creases. We then deduce relations between the crease geodesic curvature, normal curvature, torsion, and folding angle.

The numerical algorithms for the approximation of the plate deformations are based on local discontinuous Galerkin methods, where high order derivatives in the continuous models are replaced by weakly converging discrete reconstructions. This talk put little emphasis on the numerical methods. Rather, we briefly describe the algorithms along with their analysis and explore numerically their capabilities.