Analytical characterization of the dynamic response of viscoelastic metamaterials

The band-gap frequencies of elastic metamaterials are ideally determined by a metamaterial architecture; yet, in practical situations, are often dependent on the material damping in their constituent(s). The analysis of viscoelastic metamaterials requires however substantial computational resources and, except for oversimplified cases, is solely done numerically. Here, we propose an analytical procedure based on the spectral element method (SEM) to analyze bulk metamaterials with viscoelastic damping as continuous systems. Due to intrinsic limitations of the SEM to deal with complex geometries, we develop a procedure to build an approximate model based on SEM frame elements. The viscoelastic behavior is included by means of complex viscoelasticity moduli expressed by the generalized Maxwell mechanical model. We validate this approach by analyzing metamaterial plates and verify the findings experimentally. We demonstrate that our SEM-based analytical model can accurately capture wave transmission around the first band-gap frequencies. Therefore, our extension of the SEM approach to analyze three-dimensional meta-structures is promising to characterize wave propagation in realistic viscoelastic structures (with any type of linear viscoelastic behavior) in an accurate and computationally efficient way.