On local kirigami mechanics

Kirigami offers unexplored ways to tailor the morphology of thin elastic sheets. By exploiting the fundamental principles of this art and carefully tuning the geometry of the cuts, it unlocks a great potential to control the mechanical properties of thin sheets across multiple scales. In this work, we explore the emergent local non-linear effects in Kirigami by focusing on the basic building block of Kirigami: a thin sheet with a single radial cut. We consider its deformation following the opening of the slit by a given excess angle and the rotation of its lips. As the thickness of the disk approaches zero, there is 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 problem for a Saint Venant-Kirchhoff constitutive plate model to find the geometry of the e-cone as well as closed-form expressions for its surface stresses and couple-stresses. We further investigate the post-buckling stability of the e-cone and map out its phase space for symmetric boundary conditions. For a plate of finite thickness, one may not neglect the stretching contribution. We solve this case by proposing a stretchable creased e-cone model: a discretised e-cone made of a series of rigid panel connected by radial creases allowing for relative rotation and separation of the panels as a model for bending and stretching, respectively. Admissible equilibrium configurations are obtained by penalising these deformations using elastic springs with stiffness constants derived from compatible continuum plate deformations. We are hence able to solve for plates with finite thickness and find the full range of post-buckling behaviour as well as initial buckling instability.