This contribution presents a novel homogenization technique for modeling heterogeneous materials with micro-inertia effects such as locally resonant acoustic metamaterials. Linear elastodynamics is used to model the micro and macro scale problems and an extended first order Computational Homogenization framework is used to establish the coupling. Craig Bampton Mode Synthesis is then applied to solve and eliminate the microscale problem, resulting in a compact closed form description of the microdynamics that accurately captures the Local Resonance phenomena. The resulting equations represent an enriched continuum in which additional kinematic degrees of freedom emerge to account for Local Resonance effects which would otherwise be absent in a classical continuum. Such an approach retains the accuracy and robustness offered by a standard Computational Homogenization implementation, whereby the problem and the computational time are reduced to the on-line solution of one scale only.
This paper presents a computational frequency-domain boundary value analysis of acoustic metamaterials and phononic crystals based on a general homogenization framework, which features a novel definition of the macro-scale fields based on the Floquet-Bloch average in combination with a family of characteristic projection functions leading to a generalized macro-scale continuum. Restricting to 1D elastodynamics and the frequency-domain response for the sake of compactness, the boundary value problem on the generalized macro-scale continuum is elaborated. Several challenges are identified, in particular the non-uniqueness in selection of the boundary conditions for the homogenized continuum and the presence of spurious short wave solutions. To this end, procedures for the determination of the homogenized boundary conditions and mitigation of the spurious solutions are proposed. The methodology is validated against the direct numerical simulation on an example periodic 2-phase composite structure.
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