The goal of this study is to better understand the mathematical structure and ramifications of the secondorder homogenization of low-frequency wave motion in periodic solids. To this end, multiple-scales asymptotic approach is applied to the scalar wave equation (describing anti-plane shear motion) in one and two spatial dimensions. In contrast to previous studies where the second-order homogenization has lead to the introduction of a single fourth-order derivative in the governing equation, present investigation demonstrates that such (asymptotic) approach results in a family of field equations uniting spatial, temporal, and mixed fourth-order derivatives-that jointly control incipient wave dispersion. Given the consequent freedom in selecting the affiliated lengthscale parameters, the notion of an optimal asymptotic model is next considered in a onedimensional setting via its ability to capture the salient features of wave propagation within the first Brillouin zone, including the onset and magnitude of the phononic band gap. In the context of two-dimensional wave propagation, on the other hand, the asymptotic analysis is first established in a general setting, exposing the constant shear modulus as sufficient condition under which the second-order approximation of a bi-periodic elastic solid is both isotropic and limited to even-order derivatives. On adopting a chessboard-like periodic structure (with contrasts in both modulus and mass density) as a testbed for in-depth analytical treatment, it is next shown that the second-order approximation of germane wave motion is governed by a family fourthorder differential equations that: i) entail exclusively even-order derivatives and homogenization coefficients that depend explicitly on the contrast in mass density; ii) describe anisotropic wave dispersion characterized by the "sin 4 θ + cos 4 θ" term, and iii) include the asymptotic model for a square lattice of circular inclusions as degenerate case. For completeness, the analysis is illustrated by a set of numerical results highlighting the effects of periodic structure on the long-wavelength material response in terms of wave dispersion, phononic band gap, and second-order anisotropy.

The full 3‐D macroscopic mechanical behavior of snow is investigated by solving kinematically uniform boundary condition problems derived from homogenization theories over 3‐D images obtained by X‐ray tomography. Snow is modeled as a porous cohesive material, and its mechanical stiffness tensor is computed within the framework of the elastic behavior of ice. The size of the optimal representative elementary volume, expressed in terms of correlation lengths, is determined through a convergence analysis of the computed effective properties. A wide range of snow densities is explored, and power laws with high regression coefficients are proposed to link the Young's and shear moduli of snow to its density. The degree of anisotropy of these properties is quantified, and Poisson's ratios are also provided. Finally, the influence of the main types of metamorphism (isothermal, temperature gradient, and wet snow metamorphism) on the elastic properties of snow and on their anisotropy is reported.