A two-dimensional continuum model of pantographic sheets moving in a 3D space and accounting for the offset and relative rotations of the fibers

Recently growing attention has been paid to the particular class of metamaterials which has been called pantographic. Pantographic metamaterials have the following peculiar features: (i) their continuum model, at the macroscale, has to include a term of the deformation energy depending on the second gradient of placement, (ii) they can show an elastic behavior in large deformation regimes, and (iii) they are resilient and tough during rupture phenomena (dell'Isola et al. 2015). In order to predict pantographic metamaterials' mechanical behavior, it is possible to introduce a three-dimensional continuum micromodel, in which their internal geometrical microstructure is described in detail. However, the computational costs of this choice are presently prohibitive. In this paper, we introduce a reduced order model for pantographic sheets-which are an example of an elastic surface-whose kinematics include, for each of the two constituting families of fibers fully independent three-dimensional (i) placement and (ii) rotation fields. In this way it is possible to include, also in the reduced order model, (i) the initial and the actual offset between the fibers and (ii) the deformation energy of the interconnecting pivots. By postulating a simplified expression for the deformation energy we prove that also a reduced order model can describe some experimental observed buckling and postbuckling phenomena. The promising results which we present here motivate the quest of more general expressions for deformation energy capable of capturing the fully nonlinear behavior exhibited by pantographic sheets.