Second-gradient homogenized model for wave propagation in heterogeneous periodic media

Abstract

The paper is focused on a homogenization procedure for the analysis of wave propagation in materials with periodic microstructure. By a reformulation of the variational-asymptotic homogenization technique recently proposed by Bacigalupo and Gambarotta (2012a), a second-gradient continuum model\ud is derived, which provides a sufficiently accurate approximation of the lowest (acoustic) branch of the dispersion curves obtained by the Floquet–Bloch theory and may be a useful tool for the wave propagation analysis in bounded domains. The multi-scale kinematics is described through micro-fluctuation functions of the displacement field, which are derived by the solution of a recurrent sequence of cell BVPs and obtained as the superposition of a static and dynamic contribution. The latters are proportional to the even powers of the phase velocity and consequently the micro-fluctuation functions also depend on the direction of propagation. Therefore, both the higher order elastic moduli and the inertial terms result to\ud depend by the dynamic correctors. This approach is applied to the study of wave propagation in layered bi-materials with orthotropic phases, having an axis of orthotropy parallel to the direction of layering, in which case, the overall elastic and inertial constants can be determined analytically. The reliability of the proposed procedure is analysed by comparing the obtained dispersion functions with those derived by the Floquet–Bloch theory

A complete description of bi-dimensional anisotropic strain-gradient elasticity

“…This tiling was proposed by Lakes (1991) and studied in Prall and Lakes (1997). Since then this material has bee studied by numerous authors (Spadoni et al, 2009;Liu et al, 2012;Dirrenberger et al, 2011Dirrenberger et al, , 2012Dirrenberger et al, , 2013Bacigalupo and Gambarotta, 2014). The symmetry class of this pattern is [Z 6 ], and hence, in the framework of SGE, there is no coupling between first-and second-order elasticity.…”

“…The first point has rarely been addressed in the literature, and at the present time little is known, except in some specific situations (Wang and Sun, 2002;Bacigalupo and Gambarotta, 2014). Despite its interest, this subject will not be considered in the present paper in which attention will be focused on higher-order elasticity tensors.…”

“…where s ij is the effective second-order symmetric stress tensor, κ jik is a third order micro-inertia tensor and κ jkil a fourth order micro-inertia tensor (Mindlin and Eshel, 1968;Mindlin, 1964;Ben-Amoz, 1976;Askes and Aifantis, 2011;Bacigalupo and Gambarotta, 2014) . It is important to remark that the third order micro-inertia tensor κ jik is vanishing for centrosymmertic materials (see e.g.…”

“…It is important to remark that the third order micro-inertia tensor κ jik is vanishing for centrosymmertic materials (see e.g. Bacigalupo and Gambarotta (2014)). This tensor is defined as follows:…”

High-frequency parametric approximation of the Floquet-Bloch spectrum for anti-tetrachiral materials

“…As particular feature of theoretical and applied interest, the two dispersion curves related to null initial frequencies (ω 1 = ω 2 = 0 at the left B 1 -boundary) represent the acoustic branches of the Floquet-Bloch spectrum [33,51,52]. Each acoustic branch corresponds to a wave propagating with nondimensional…”