Discretization Approach for the Homogenization of Three-Dimensional Solid-Solid Phononic Crystals in the Quasi-Static Limit: Density and Elastic Moduli

With the application of a homogenization theory, based on Fourier formalism (which provides efficient and exact formulas by which to determine all components effective stiffness mass density tensors, valid in regime large wavelengths), new approach calculate quasi-static response three-dimensional solid-solid phononic crystals is reported. The derived this work for calculating elastic parameters show dependence, terms summations over vectors, reciprocal lattice discretization volume inclusion small parts (e.g., cubes), obtain system equations from we define response. In particular, present numerical results calculated several cubic lattices with solid constituents different shapes inclusions unit cell versus filling fraction, as well fixed values it. By means, analyzed effect type Bravais materials, geometry that constitute array, resulting anisotropy. Finally, our theory confirms other well-known previous theories particular case study. regard, examples shown here can be useful design metamaterials predetermined properties.