We analyze the dispersion of elastic waves in periodic beam networks based on second order gradient models obtained by the homogenization of the initially discrete network. The lattice beams have a vis-coelastic behavior described by... more
We analyze the dispersion of elastic waves in periodic beam networks based on second order gradient models obtained by the homogenization of the initially discrete network. The lattice beams have a vis-coelastic behavior described by Kelvin–Voigt model and the homogenized second gradient viscoelasticity model reflects both the lattice topology, anisotropy and microstructural features in terms of its geometrical and micromechanical parameters. The continuum models enriched with the higher-order gradients of the displacement and velocity introduce characteristic lengths parameters which account for microstructural effects at the mesoscopic level. A comparative study of the dispersion relations and damping ratio evolutions for the longitudinal and shear waves has been done for four lattices (the chiral diamond lattice, the classical and reentrant lattices, and the pantograph). The developed model allows analyzing both the effects of damping and internal length scale through the second order displacement gradients on the wave propagation characteristics. An important increase of the natural frequency due to second order effects is observed. For the pantograph lattice, the phase velocity for the longitudinal and shear modes is identical and is not influenced by the direction of wave propagation. The obtained results show overall that the pantograph lattice present the best acoustic characteristics.
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... more
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.
