  Over the past decade, kirigami--the Japanese art of paper cutting--has been
playing an increasing role in the emerging field of mechanical metamaterials
and a myriad of other mechanical applications. Nonetheless, a deep
understanding of the mathematics and mechanics of kirigami structures is yet to
be achieved in order to unlock their full potential to pioneer more advanced
applications in the field. In this work, we study the most fundamental
geometric building block of kirigami: a thin sheet with a single cut. We
consider a reduced two-dimensional plate model of a circular thin disk with a
radial slit and investigate its deformation following the opening of the slit
and the rotation of its lips. In the isometric limit--as the thickness of the
disk approaches zero--the elastic energy has no stretching contribution and the
thin sheet takes a conical shape known as the e-cone. We solve the
post-buckling problem for the e-cone in the geometrically nonlinear setting
assuming a Saint Venant-Kirchhoff constitutive plate model; we find closed-form
expressions for the stress fields and show the geometry of the e-cone to be
governed by the spherical elastica problem. This allows us to fully map out the
space of solutions and investigate the stability of the post-buckled e-cone
problem assuming mirror symmetric boundary conditions on the rotation of the
lips on the open slit.
