Fig. 1

REV of an l-periodic double-porosity permeo-elastic material, comprising a microporous matrix Ωm and a connected mesoscopic fluid network Ωf. Highly flexible thin elastic solids, denoted by Γ, are clamped onto a perfectly rigid frame Ωs. The solid-fluid interface is Γs, while Γm represents the interface between the pore fluid network and the microporous matrix. These multiscale metamaterials, referred to as Materials A, have a macroscopic characteristic size L (color online)"

Fig. 2

REV of a double-porosity permeo-elastic material with (a) cavities closed by thin solids (Materials B), and (b) a microporous constituent Ωm and mesopores closed by thin solids Γ leading to a non-connected mesopore fluid network Ωf (Materials C) (color online)"

Fig. 3

Microstructure (left) and 2D axisymmetric illustration (right) of the REV of multiscale metamaterials of Types (a) A, (b) B, and (c) C (color online)"

Fig. 4

(a) Real and imaginary parts and (b) normalized phase of the normalized dynamic density ρ(ω)/ρ0 of a multiscale metamaterial of Type A. The vertical dotted and dashed-dotted lines, respectively, correspond to the anti-resonance frequency fa and resonance frequency fg. The dashed lines correspond to the effective parameter of a single-porosity material with the same geometry, but with the films and microporous matrix, respectively, replaced by perfectly rigid and impermeable films and matrix. The x-axis uses a logarithmic scale"

Fig. 5

Real part (continuous lines) and negative of the imaginary (dashed lines) part of the normalized effective compressibility of a multiscale metamaterial of Type A (i.e., P0C), a microporous matrix material P0Cm, and a single-porosity material P0Cp (see the text for its description). The vertical dotted lines correspond to thermal or pressure diffusion characteristic frequencies"

Fig. 6

(a) Magnitude and (b) phase of the effective speed of sound of a multiscale metamaterial of Type A (black lines) and that of a single-porosity material (grey lines) with the same geometry but having the films and microporous matrix replaced by perfectly rigid and impermeable solid materials. The vertical dotted lines correspond to the anti-resonance frequency fa, which is matched to the pressure diffusion characteristic frequency fd, and the resonance frequency fg. The x-axis uses a logarithmic scale"

Fig. 7

(a) Normalized magnitude of the effective compressibility (i.e., |CP0|) of a double-porosity permeo-elastic material of Type B and a permeo-elastic material. The plot (b) shows the respective normalized phase of the effective compressibility, i.e., ϑC/π. The vertical dotted lines correspond to the anti-resonance frequency fa′, which is matched to the pressure diffusion characteristic frequency fd, and the resonance frequency fg′. The normalized magnitude |CP0| is bounded from below by |ϕmφmℑ(F(ω))|, as shown by the dashed black line. The x-axis uses a logarithmic scale. DPPE: double-porosity permeo-elastic; PE: permeo-elastic"

Fig. 8

(a) Normalized magnitude and (b) phase of the effective speed of sound, and (c) normalized phase constant and (d) attenuation coefficient of a double-porosity permeo-elastic material of Type B and a permeo-elastic material. The vertical dotted lines correspond to the anti-resonance frequency fa′, which is matched to the pressure diffusion characteristic frequency fd, and resonance frequency fg′. The dashed black line in plot (a) shows that |c/C0| is bounded from above by ϕp/|α∞γφmϕmℑ(F)|. The x-axis uses a logarithmic scale. DPPE: double-porosity permeo-elastic; PE: permeo-elastic"

Fig. 9

Normalized magnitudes of (a) dynamic permeability |K(ω)|/(φmKm0) and (b) density |ρ(ω)/ρ0| of a double-porosity permeo-elastic material of Type C. The dashed vertical line represents the resonance frequency ωg, and the inset plots show the normalized phase of the dynamic permeability and density. Normalized (c) magnitude of the effective speed of sound |c(ω)/c0| and (d) attenuation coefficient −ℑ(kc)c0/ω of a double-porosity permeo-elastic material of Type C. The dashed vertical line represents the resonance frequency ωg. The inset plots show the normalized (c) phase of the effective speed of sound and (d) phase constant. The x-axis uses a logarithmic scale"