Abstract
We aim to detect defects or perturbations of periodic media, e.g. due to a defective manufacturing process. To this end, we consider scalar waves in such media through the lens of a second-order macroscopic description, and we compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coefficients in the governing mean-field equation – including both the leading order (i.e. quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. Then, we present a method that permits sub-wavelength sensing of periodic media, given the (anisotropic) phase velocity of plane waves illuminating the considered medium for several angles and wavenumbers.
Highlights
The sensitivity of an homogeneous material to the nucleation of a periodic array of inhomogeneities, and the computation of the corresponding homogenized coefficient, are welladdressed issues, see e.g. [1] and the references therein
Summary and discussion We derived the topological sensitivities of the coefficients of the wave equation obtained by second-order homogenization of a periodic material, with respect to the size of an inhomogeneity perturbing the unit cell that defines such material
These sensitivities are expressed in terms of (i) three unit-cell solutions used to formulate the unperturbed macroscopic model, (ii) two adjoint-field solutions driven by the mass density variation inside the unperturbed unit cell, and (iii) the polarization tensor, that synthesizes the geometric and constitutive features of the perturbation
Summary
The sensitivity of an homogeneous material to the nucleation of a periodic array of inhomogeneities, and the computation of the corresponding homogenized coefficient, are welladdressed issues, see e.g. [1] and the references therein. We assume time-harmonic conditions, and aim at performing sub-wavelength sensing of periodic media, i.e. using low-frequency probing waves to detect defects or perturbations of the medium. This goal is achieved by relying on the anisotropic dispersive properties of such medium. To capture these properties, we chose to deploy the second-order model obtained by two-scale asymptotic homogenization [5], and we compute the sensitivities of the (tensor-valued) coefficients of the corresponding wave equation. The leading-order expansion of the perturbed coefficients (μ0a, 0a, μ2a, 2a) w.r.t. a are computed, following previous studies e.g. [4] for in-plane elastostatics
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