Curvy cuts: Programming axisymmetric kirigami shapes

Although bending a sheet of paper is an easy operation, stretching is more limited and it leads to rupture and tears. However, well-designed cuts on the sheet can induce a large effective stretchability. This kirigami technique offers a large scope of engineering applications ranging from deployable structures to compliant electronics. We are here interested in the axisymmetric configuration where cuts are designed along concentric circles. Applying an increasing transverse load at the center of the sheet results into a 3D axisymmetric structure of growing amplitude which eventually saturates. We first describe the linear response of the structure and determine the evolution of the deployed shape until its asymptotic geometrical limit. Reversing the problem in the linear regime, we propose, a design procedure for the cuts leading to a desired 3D shape. The structure can also be deployed by inflating an inner balloon. Exploring further the interplay between mechanics and geometry, we finally describe the maximum volume of inflated kirigami structures as a function of the cutting pattern.