Non-locality of the Willis coupling in fluid laminates

The closed form expressions of the effective properties in periodic fluid laminates are derived thanks to the Padé approximation of the transfer matrix. A second-order Taylor expansion of the transfer matrix elements exhibits Willis coupling. This coupling is the sum of a local term and a nonlocal term. The nonlocal term arises from the apparent bulk modulus in quasi one-dimensional problems. The nonlocality directly impacts the governing equations modeling the acoustic wave propagation in these Willis materials, which then involve convolution products in space. As an example, a two-orthotropic porous material laminate is considered. The theoretically derived effective properties and scattering coefficients are found in excellent agreement with those numerically calculated. The Willis coupling widens the frequency range of validity and accuracy of the effective properties and thus of the calculated scattering coefficients when compared to classical homogenization results for which the Willis coupling is absent. This widening mostly relies on the effect of Willis coupling on the impedance of the fluid laminate. The effective properties are finally derived for a general laminate.

Non-locality of the Willis coupling in fluid laminates Articles