Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations

In this paper, we determine numerically a large class of equilibrium configurations of an elastic two-dimensional continuous pantographic sheet in three-dimensional deformation consisting of two families of fibers which are parabolic prior to deformation. The fibers are assumed: i) to be continuously distributed over the sample, ii) to be endowed of bending and torsional sti↵nesses and iii) tied together at their points of intersection to avoid relative slipping by means of internal (elastic) pivots. This last condition characterizes the system as a pantographic lattice [1, 2, 34, 35]. The model that we employ here, developed by Steigmann and dell’Isola [108] and first investigated in [55], is applicable to fiber lattices in which three dimensional bending, twisting and stretching are significant as well as a resistance to shear distortion, i.e. to the angle change between the fibers. Some relevant numerical examples are exhibited in order to highlight the main features of the model adopted: in particular buckling and post-buckling behavior of pantographic parabolic lattices is investigated. The fabric of the metamaterial presented in this paper has been conceived to resist more e↵ectively in the extensional bias tests by storing more elastic bending energy and less energy in the deformation of elastic pivots: a comparison with a fabric constituted by beams which are straight in the reference configuration shows that the proposed concept is promising.