Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum

Abstract

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.

Homogenized enriched model for blast wave propagation in metaconcrete with viscoelastic compliant layer

“…whereM andK denote the mass and the time-independent stiffness matrices, respectively. Note that the time-dependent damping response via Boltzmann superposition integral associated withD departs from the standard expression in the work of Sridhar et al, 23 due to our viscoelastic consideration. Recall that C e is employed as per standard elastic material via (2a), whereas C v can be determined from (4a).…”

“…Note from that a microeffect manifests itself naturally in the homogenized stress. At this juncture, it is highlighted again that (29) involves enrichments from linear viscoelastic actions at the three master nodes via terms involving , a departure from the expression in the work of Sridhar et al…”

“…We adopt a bottom‐up strategy as summarized in Figure to adequately characterize the underlying rapidly fluctuating responses within a locally resonant metamaterial, for a consistent propagation from micro to macro. The general approach follows that of Sridhar et al The key departure here is the extension of the existing approach to incorporate linear viscoelasticity. Specifically, the strategy in Figure encompasses the following: …”

“…Step ( conditions are imposed on the unit cell; (C) The Craig-Bampton dynamic substructuring procedure is carried out to achieve a reduced-order model; (D) The Hill-Mandel condition is applied to effect the micro-macro transition follows that of Sridhar et al 23 The key departure here is the extension of the existing approach to incorporate linear viscoelasticity. Specifically, the strategy in Figure 1 encompasses the following:…”

On the role of micro-inertia in enriched continuum mechanics

“…Metamaterials are often studied from both a static (those with enhanced mechanical properties with respect to traditional materials) [4][5][6] and a dynamic (those exhibiting bandgaps, negative refraction, cloaking, focusing, etc.) point of view [7][8][9][10][11][12][13][14]. The characteristic size of microstructures in such metamaterials usually ranges from micrometres to centimetres, so that they show dispersive behaviours for wavelengths which are relatively large.…”