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.

在过去的十年中，kirigami（日本的剪纸艺术）在新兴的机械超材料和无数其他机械应用领域中发挥着越来越重要的作用。但是，尚未获得对折纸结构的数学和力学的深刻理解，以释放它们的全部潜力，以开拓该领域的更高级应用程序。在这项工作中，我们研究了kirigami的最基本的几何构造块：单切的薄板。我们考虑具有径向狭缝的圆形薄盘的简化二维平板模型，并研究狭缝的打开和其唇缘旋转后其变形。在等轴测极限中-当圆盘的厚度接近零时-弹性能没有拉伸作用，并且薄片呈圆锥形，称为e锥。我们假设采用Saint Venant-Kirchhoff本构板模型，在几何非线性环境中解决了电子锥的后屈曲问题。我们找到了应力场的闭式表达式，并显示了由球弹性系数问题决定的电子锥的几何形状。这使我们能够完整地描绘出解的空间，并假设在开口缝上唇部旋转时镜面对称边界条件成立，从而研究了后屈曲e-锥问题的稳定性。我们找到了应力场的闭式表达式，并显示了由球弹性系数问题决定的电子锥的几何形状。这使我们能够完整地描绘出解的空间，并假设在开口缝上唇部旋转时镜面对称边界条件成立，从而研究了后屈曲e-锥问题的稳定性。我们找到了应力场的闭式表达式，并显示了由球弹性系数问题决定的电子锥的几何形状。这使我们能够完整地描绘出解的空间，并假设在开口缝上唇部旋转时镜面对称边界条件成立，从而研究了后屈曲e-锥问题的稳定性。