Variational formulation of dynamical homogenization towards nonlocal effective media

Abstract

The dynamical homogenization of heterogeneous materials is performed in the low and high frequency regimes using a variational formulation of dynamical equilibrium combined with the Laplace or Fourier-Floquet-Bloch decomposition of the mechanical fields (displacement, velocity, momentum, stress). The assumed periodicity of the initial heterogeneous structure entails that the homogenization process can be carried out at the scale of a reference unit cell over which periodicity conditions apply. A dynamical Hill type macrohomogeneity condition is formulated that plays a central role in this work. The weak form of the unit cell dynamical equilibrium is formulated as the minimization of the mean field action integral expressed versus the Floquet components of the displacement and velocity. The effective dynamical moduli are evaluated in reciprocal space as unit cell averages of the microscopic properties and localization operators relating the local velocity and strain to their macroscopic counterpart. More specific dynamical models involving locality in space (thus valid for long wavelengths) but nonlocality in time (thus describing both low and high frequency regimes) are formulated, considering Cauchy type and strain gradient effective models. The range of validity of the developed dynamical generalized continuum theories is assessed by comparing the phase velocity-wavenumber plots for inclusion based composites with reference plots obtained from Bloch's theorem. The benchmark of dynamical models evidences that the nonlocal dynamical model proves able to predict with a good accuracy the dynamical behavior of 2D periodic inclusion based composites.