Dispersive Waves in 2D Second Gradient Continuum Media

Abstract

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, relying on the discrete asymptotic method extended up to the second gradient of the displacement. The lattice beams have a viscoelastic behavior described by Kelvin-Voigt model and the homogenized second gradient viscoelasticity model reflects both the initial 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 study of the dispersion relations and damping ratio evolutions for the longitudinal and shear waves has been done for the reentrant lattice. An important increase of the natural frequency due to second order effects is observed.