Elastica of pantographic curves led by conic sections: A discrete point of view

Pantographic unit cells, sometimes called scissor-like mechanisms, are largely used to design metamaterials with exotic mechanical properties. The latter derive from the intrinsic property of pantographic metamaterials to be naturally multiscale. In addition, such metamaterials can be used hierarchically to improve their mechanical behaviour. Mainly, these interesting properties are due to the existence of a zero energy mode (often called floppy mode), i.e. a deformation mode having zero strain energy. Based on a two-dimensional unit cells, pantographic beams (1D), sheets (2D) and blocks (3D), are largely studied by means of discrete and continuum models and also by using the capability of the 3D printing processes. Up to now, the pantographic unit cell mainly used in the field of metamaterials is the classic pantographic unit cell: two rectilinear beams connected by a pivot which split the beams in four arms of equal length. Differently, in the field of deployable structures there are remarkable examples of pantographic unit cells, also three-dimensional ones, specifically designed to cover large surfaces when unfolded and to be very compact when folded. Here we try to define the concept of pantographic curves led by conic sections and to discuss their behaviours, i.e. the existence of a zero energy mode and the mechanical response when the zero energy mode is prevented. Starting from a well-defined conic section, an algorithm for building the associated pantographic curve is developed, utilising unit cells inspired by Spanish architect Pi & ntilde;ero's deployable theater roof. Having in mind to look at such pantographic curves as unit bricks for the design of novel metamaterials, a discrete element model made only by springs and angle springs is briefly discussed in the framework of a self-adaptive step-by-step algorithm for large displacements. Our goal is to verify the deployability of the considered pantographic curves and their mechanical behaviour for some selected load conditions. Several numerical experiments concerning pantographic curves shaped as conic sections-ellipses, parabolas and hyperbolas-are presented and discussed for some classic and special load conditions. Finally, we try to illustrate with a numerical experiment the differences between curved beams and curved pantographic beams.