On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets

Recently it has been proposed the study of a new class of bidimensional metamaterials, which have been called extensible pantographic sheets. Such bidimensional continua are the generalization of the continua introduced in as they take simultaneously into account the elastic energy in the extensional deformation and geodesic bending of constituting fibers. In the present paper we consider in the deformation energy a term accounting for the effect of shear deformation. The phenomena which we highlight can be of relevance to model the mechanical behavior of composite fiber reinforcements and of some lattices of beams constrained by internal pivots. We compare the effect of different linear or cubic shear stiffnesses i) on the deformation of pairs of pantographic sheets suitably interconnected and ii) on the deformation of pantographic sheets deformed under the action of forces concentrated on points.