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

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