On kirigami mechanics

Kirigami, the Japanese art of paper cutting, offers unexplored ways to tailor the morphology of thin elastic sheets. By exploiting the fundamental principles of this art and carefully tuning the geometry and the topology of the cuts, it unlocks a great potential to control the structural and mechanical properties of thin sheets across multiple scales. In order to generate predictable and adaptable patterns or motion control as well as regulate the multi-stable nature of these structures, one would need to understand the fundamental behaviour of such structures. In this work, we explore the emergent local non-linear effects in Kirigami by focusing our attention on the study of the deformation of a thin sheet with a single cut and the inherent interplay of geometry and mechanics. It is indeed the most basic and fundamental geometric building block of Kirigami. We consider its deformation following the opening of the slit by a given excess angle. In the isometric limit—as the thickness of the disk approaches zero—the elastic energy has no stretching contribution and the shape of the disk is governed by the bending energy as it approaches that of an e-cone: a conical solution where all the generators remain straight and intersect at a singularity on its apex. We solve the geometrically nonlinear boundary value problem for a Saint Venant-Kirchhoff constitutive plate model to find the geometry of the e-cone as well as the surface stresses and couple-stresses (moments). We further investigate the stability of the e-cone and map its phase plane including the post-buckling non-equilibrium branches.

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