Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices

In the present work, we show that the linearized homogenized model for a pantographic lattice must necessarily be a second gradient continuum, as defined in Germain (1973). Indeed, we compute the effective mechanical properties of pantographic lattices following two routes both based in the heuristic homogenization procedure already used by Piola (see Mindlin, 1965; dell'Isola et al., 2015a): (i) an analytical method based on an evaluation at micro-level of the strain energy density and (ii) the extension of the asymptotic expansion method up to the second order. Both identification procedures lead to the construction of the same second gradient linear continuum. Indeed, its effective mechanical properties can be obtained by means of either (i) the identification of the homogenized macro strain energy density in terms of the corresponding micro-discrete energy or (ii) the homogenization of the equilibrium conditions expressed by means of the principle of virtual power: actually the two methods produce the same results. Some numerical simulations are finally shown, to illustrate some peculiarities of the obtained continuum models especially the occurrence of bounday layers and transition zones. One has to remark that available well-posedness results do not apply immediately to second gradient continua considered here.