Nonlinear geometric mechanics of spatially periodic surfaces

Thin periodic sheets---ranging from corrugated containers and origami metamaterials to natural structures such as leaves---exhibit a remarkable balance of strength and mechanical adaptability. Our recent work demonstrates that any spatially periodic surface possesses exactly three uniform linear isometric modes, each pair of which satisfies a symplectic mode compatibility condition. In this talk, we further show that, as previously established for triangulated origami tessellations, only two of these three modes can extend nonlinearly. 

The necessary and sufficient condition for such nonlinear extensions is that the macroscopic Gaussian curvatures associated with the surviving linear modes vanish. Geometrically, this implies that, under large deformations, a spatially periodic surface can only stretch macroscopically or fold into a cylindrical shape. Linearizing these nonlinear deformations around a cylindrical corrugated reference state reveals that the symplectic structure of the isometric subspace persists, though its dimension is reduced from three to two. This framework offers a new perspective for identifying fundamental constraints on the nonlinear mechanics of thin periodic and cylindrically periodic surfaces, thereby informing the design principles of mechanical metamaterials with programmable flexibility and rigidity.