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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.
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