Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures

In order to design the microstructure of metamaterials showing high toughness in extension (property to beshared with muscles), it has been recently proposed (Dell’Isola et al. in Z Angew Math Phys 66(6):3473–3498,2015)toconsider pantographic structures. It is possible to model such structures at a suitably small length scale (resolving in detailthe interconnecting pivots/cylinders) using a standard Cauchy first gradient theory. However, the computational costs forsuch modelling choice are not allowing for the study of more complex mechanical systems including for instance manypantographic substructures. The microscopic model considered here is a quadratic isotropic Saint-Venant first gradientcontinuum including geometric nonlinearities and characterized by two Lam ́e parameters. The introduced macroscopictwo-dimensional model for pantographic sheets is characterized by a deformation energy quadratic both in the first andsecond gradient of placement. However, as underlined in Dell’Isola et al. (Proc R Soc Lond A 472(2185):20150790,2016),it is needed that the second gradient stiffness depends on the first gradient of placement if large deformations and largedisplacements configurations must be described. The numerical identification procedure presented in this paper consistsin fitting the macro-constitutive parameters using several numerical simulations performed with the micro-model. Theparameters obtained by the best fit identification in few deformation problems fit very well also in many others, showingthat the reduced proposed model is suitable to get an effective model at relevantly lower computational effort. The presentednumerical evidences suggest that a rigorous mathematical homogenization result most likely holds.