Hussein Nassar (University of Missouri)

Compliant shell mechanisms are creased and corrugated thin-walled structures that can drastically change shape to move, deploy, or adapt to a changing environment. They have found use cases in the context of recent space programs and in other domains ranging from biomedical technology to architecture. Not unlike slender beams, thin shells prefer bending over stretching. Ideally, thin shells deform isometrically should isometric deformations exist. The problem of finding, or disproving the existence of, isometric deformations for various surfaces preoccupied many mathematicians and mechanicians. The most noteworthy results undoubtedly pertain to three broad categories of surfaces: developable surfaces, convex surfaces, and axisymmetric surfaces. In the modern context of computer graphics, discrete differential geometry and “Origami science,” more focus has been directed towards tri- and quad-based polyhedral surfaces.

Here, we report on recent results that characterize the isometric deformations of periodic surfaces, i.e., surfaces that are invariant by a two-dimensional lattice of translations. Inspired by the theory of homogenization, two classes of effective isometric deformations are introduced, membrane modes and bending modes, identified by whether their long-range behavior is membrane-like or bending-like. The main theorem proves an orthogonality relationship between the two classes. Thus, if a surface gains a membrane mode, it loses a bending mode, and conversely, in such a way that the total number of modes, membrane and bending combined, can never exceed 3. The main assumption of the theorem is that the periodic surface is simply connected (i.e., without holes). Beyond that, the theorem invariably handles piecewise smooth surfaces including smooth, polyhedral and “curved-crease” surfaces. This universality is anchored in the proof technique that is free of specific constructs (e.g., linkages, torsal rulings, conjugate nets) and instead uses high-level arguments (e.g., symmetry and integral theorems). The theorem succeeds in qualitatively describing how various periodic surfaces bend into domes or saddles and finite element simulations of thin elastic shells validate the quantitative findings in the limit of vanishing thickness. The work hopefully sheds new light on how thin corrugated shells behave be it in modern morphing applications or in classical structural ones and, just as hopefully, brings closer the communities of origamists, geometers, and mechanicians.