Geometric Mechanics of Thin Periodic Surfaces

TL;DR

This work develops a universal geometric framework for thin periodic surfaces, revealing a duality between isometric deformations and in-plane equilibrium stresses. By introducing shape-periodic deformations described by six macroscopic parameters $E^{\alpha\beta}$ and $H^{\alpha\beta}$, the authors prove that exactly three of these modes are isometries and that these three form a Lagrangian subspace under an underlying symplectic structure. The energy-rigidity coupling is captured by a symplectic inner product, and a mode-compatibility condition shows how isometric modes constrain each other, with special cases reproducing opposite in-plane and out-of-plane Poisson ratios observed in origami patterns. The framework applies to both smooth and creased surfaces, including origami tessellations, and is validated numerically on triangulated models, offering exact, non-homogenized insight with practical implications for origami metamaterials and packaging design.

Abstract

Thin surfaces are ubiquitous in nature, from leaves to cell membranes, and in technology, from paper to corrugated containers. Structural thinness imbues them with flexibility, the ability to easily bend under light loads, even as their much higher stretching stiffness can bear substantial stresses. When surfaces have periodic patterns of either smooth hills and valleys or sharp origami-like creases this can substantially modify their mechanical response. We show that for any such surface, there is a duality between the surface rotations of an isometric deformation and the in-plane stresses of a force-balanced configuration. This duality means that of the six possible combinations of global in-plane strain and out-of-plane bending, exactly three must be isometries. We show further that stressed configurations can be expressed in terms of both the applied deformation and the isometric deformation that is dual to the pattern of stress that arises. We identify constraints rooted in symplectic geometry on the three isometries that a single surface can generate. This framework sheds new light on the fundamental limits of the mechanical response of thin periodic surfaces, while also highlighting the role that continuum differential geometry plays in even sharply creased origami surfaces.

Figures (3)

• Figure 1: We consider thin sheets that are either flat (a), singly periodic (b), doubly periodic (c). In addition to such smooth sheets, our results apply to creased sheets such as origami (d). Prototypical doubly periodic sheets are readily realized through common fabrication techniques (e) and (f).
• Figure 2: (a) Any local portion of a smooth simply-connected surface can be represented by a vector field $\mathbf x\left(u^1, u^2\right)$, where $u^1$ and $u^2$ are the local coordinates. The grid lines depicted in the figure are the corresponding coordinate lines, i.e., curves of constant $u^1$ or constant $u^2$. For every point on the surface there is a local frame composed of three vectors, including the unit surface normal vector ($\hat{\mathbf n}$) and the vectors tangent to the two coordinate lines, $\partial_1{\mathbf x}$ and $\partial_2{\mathbf x}$. In the figure we use the red-green-blue (RGB) triad of vectors to represent the local frame $\left\{\partial_1{\mathbf x}, \partial_2{\mathbf x}, \hat{\mathbf n}\right\}$ at a point. (b) When the local portion is deformed isometrically each point on the surface is displaced by a displacement vector ($\delta_\mathrm{iso}{\mathbf x}$) in such a way that infinitesimal patches of the surface are locally rotated with respect to a so-called angular velocity field ($\boldsymbol\omega$) without being stretched. In the figure the RGB triads of vectors still represent the local frames, but we rescale the lengths of the surface tangent vectors in the frames, so that their rescaled lengths are approximately equal to the side lengths of the corresponding infinitesimal patches. The local rotations of the infinitesimal patches can hence be understood as rotations of the corresponding local frames with respect to the angular velocity field. (c) While an isometric deformation---such as the one represented by the light-blue arrows in the figure---does not stress the surface it is mathematically dual to a stress field, as described in Section \ref{['subsec: duality, the duality']}. The corresponding lines of principal stress are depicted in the figure as the green and the red curves, respectively, indicating the directions along which the surface is most (the red curves) and least (the green curves) stressed. The thickness of each curve is proportional to the corresponding principal stress.
• Figure 3: Geometry of doubly periodic surfaces and their shape-periodic deformations. (a) A doubly periodic surface is generated by repeatedly translating the unit cell (denoted by $\mathrm C$), which is enclosed by the magenta curve $\partial{\mathrm C}$ in the figure, along the directions of $\boldsymbol\ell_1$ and $\boldsymbol\ell_2$. The coarse-grained geometry of the doubly periodic surface is flat. The corresponding flat plane is spanned by $\boldsymbol\ell_1$ and $\boldsymbol\ell_2$ and has the normal vector $\hat{\mathbf z}$. We choose as the zero-height level the mid-height plane of the periodic surface, which coincides with the coarse-grained flat plane. (b) Under a shape-periodic deformation, the Euclidean distance between any pair of points on the deformed surface remains invariant if both points are shifted by the same number of unit cells; e.g., $\left\lVert{\mathbf M - \mathbf N }\right\rVert = \left\lVert{\mathbf M' - \mathbf N'}\right\rVert$ in the figure. The inset illustrates the corresponding deformation of the coarse-grained flat plane. (c) The deformation of the coarse-grained flat plane can be decomposed into six deformation modes: three in-plane modes and three out-of-plane modes, as illustrated by $\mathrm{ (c, i) }$ and $\mathrm{ (c, ii) }$, respectively.