Linear pantographic sheets. Asymptotic micro-macro models identification

In this paper, we consider linear pantographic sheets which in their natural configuration are con- stituted by two orthogonal arrays of straight fibers interconnected by internal pivots. We introduce a continuous model by means of a micro-macro identification procedure based on the asymptotic homogenization method of discrete media. The rescaling of the mechanical properties and of the deformation measures is calibrated in order to complies with the specific kinematics imposed by the quasi-inextensibility of the fibers together with the large pantographic deformability. The obtained high order continuum model shows interesting and exotic features, related to its extreme anisotropy but also to the sub-coercivity of its deformation energy. Some first numerical simulations are presented, showing that the model can account for experimental uncommon phenomena occuring in pantographic sheets. The paper focuses on the precise analysis and the understanding of the effective behaviour based on a well-calibration of the extension and bending phenomena arising at the local scale. In an upcoming work the analysis will be extended to oblique arrays, to some analytical solutions to proposed equations and to some further applications.