Identification of non-local continua for lattice-like materials

The paper is focused on the dynamic homogenization of lattice-like materials with lumped mass at the nodes to obtain energetically consistent models providing accurate descriptions of the acoustic behavior of the discrete system. The equation of motion of the lattice is transformed according to a unitary approach aimed to identify equivalent non-local continuum models of integral-differential and gradient type, the latter obtained through standard or enhanced continualization. The bilateral Z-transform of the difference equation of motion is matched to the governing integral-differential equation of the equivalent continuum in the transformed Fourier space, which has the same frequency band structure as the Lagrangian one. Firstly, it is shown that the approximation of the kernels via Taylor polynomials leads to the differential field equations of higher order continua endowed with non-local constitutive terms. The field equations derived from such approach corresponds to the ones obtained through the so called standard continualization. However, the differential problem turns out to be ill-posed because the non-positive definiteness of the potential energy density of the higher order continuum. Energetically consistent equivalent continua have been identified through a proper mapping correlating the transformed macro-displacements in the Fourier space with a new auxiliary regularizing continuum macro-displacement field in the same space. Specifically, the mapping here introduced has zeros at the edge of the first Brillouin zone. The integral-differential governing equation and the corresponding differential one has been reformulated through an enhanced continualization that is characterized by energetically consistent differential equations. The constitutive and inertial kernels of the integral-differential equation exhibit polar singularities at the edge of the first Brillouin zone.