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We employ the Baker–Campbell–Hausdorff formula to derive closed-form expressions for the effective properties, including emergent Willis coupling, of a one-dimensional heterogeneous poroelastic medium consisting of a periodically repeating two-layer unit-cell. In contrast to the elastic and fluidic analogs, the Willis coupling of this periodic poroelastic medium does not vanish in the low-frequency limit. However, the effective wavenumber and impedance of this asymmetric lamellar material demonstrate symmetric reflection and absorption behavior that is indicative of symmetric structures in the low-frequency limit due to the effect of Darcy’s law on the dynamic effective density, which is inversely proportional to frequency. These closed-form expressions are validated against results obtained by direct numerical evaluation. The scattering coefficients, particularly the two reflection coefficients for incidence from either side of a finite length asymmetric structure, are different at non-zero frequencies but still in the metamaterial limit and are correct when the Willis coupling is included. The results show that asymmetry in poroelastic layers results in direction-dependent absorption coefficients, one of which could be optimized based on layer properties and asymmetry factors. As a consequence, the frequency range of validity of these scattering coefficients is wider when the Willis coupling matrix is accounted for than in its absence. This work paves the way for better control of elastic and acoustic waves in multiphase materials by considering poroelastic behavior.
We employ the Baker–Campbell–Hausdorff formula to derive closed-form expressions for the effective properties, including emergent Willis coupling, of a one-dimensional heterogeneous poroelastic medium consisting of a periodically repeating two-layer unit-cell. In contrast to the elastic and fluidic analogs, the Willis coupling of this periodic poroelastic medium does not vanish in the low-frequency limit. However, the effective wavenumber and impedance of this asymmetric lamellar material demonstrate symmetric reflection and absorption behavior that is indicative of symmetric structures in the low-frequency limit due to the effect of Darcy’s law on the dynamic effective density, which is inversely proportional to frequency. These closed-form expressions are validated against results obtained by direct numerical evaluation. The scattering coefficients, particularly the two reflection coefficients for incidence from either side of a finite length asymmetric structure, are different at non-zero frequencies but still in the metamaterial limit and are correct when the Willis coupling is included. The results show that asymmetry in poroelastic layers results in direction-dependent absorption coefficients, one of which could be optimized based on layer properties and asymmetry factors. As a consequence, the frequency range of validity of these scattering coefficients is wider when the Willis coupling matrix is accounted for than in its absence. This work paves the way for better control of elastic and acoustic waves in multiphase materials by considering poroelastic behavior.
Acoustic wave propagation in a one-dimensional periodic and asymmetric duct is studied theoretically and numerically to derive the effective properties. Closed form expressions for these effective properties, including the asymmetric Willis coupling, are derived through truncation of the Peano–Baker series expansion of the matricant (which links the state vectors at the two sides of the unit-cell) and Padé's approximation of the matrix exponential. The results of the first-order and second-order homogenization (with Willis coupling) procedures are compared with the numerical results. The second-order homogenization procedure provides scattering coefficients that are valid over a much larger frequency range than the usual first-order procedure. The frequency well below which the effective description is valid is compared with the lower bound of the first Bragg bandgap when the profile is approximated by a two-step function of identical indicator function, i.e., two different cross-sectional areas over the same length. This validity limit is then questioned, particularly with a focus on impedance modeling. This article attempts to facilitate the engineering use of Willis materials.
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