Biaxial bias extension test for pantographic sheets

Pantographic fabrics are emerging in the literature as promising metamaterials. They are made up of two orthogonal families of fibers connected at their intersection points by pivots. Fibers are supposed to be straight in the reference configuration and to store energy while undergoing extension and bending deformations only. Pivots, which oppose to relative rotations of fibers, are supposed to store energy when undergoing torsion, thus to confer shear stiffness at macro level. In the paper we present some new numerical simulations in which a 2D planar non linear second gradient continuum model is employed, derived by a heuristic homogenization procedure (dell’Isola et al. 2016) and aimed at describing the mechanical behaviour inherited by its micro-structure. A bi-axial extension test has been studied. We show that, similarly to the uni-axial bias extension test, the internal stored energy is localized at the corners of the clamping areas, whereas in the central area the internal stored energy distributes almost uniformly. Unlike to the uni-axial test, due to symmetry reasons, shear deformation in the central area of the domain vanishes.