Dynamic homogenization of viscoelastic phononic metasolids

The effects of dissipation in metamaterials is a sensitive issue and, although experiments show that they are more than relevant, their theoretical study and modeling has received less attention. In this work, we study the effects of viscosity on the dissipation of elastic metamaterials. It is found that these metasolids present effective constitutive parameters that are in general complex, in contrast with common elastic materials where the mass density is a real valued scalar quantity and dissipation enters only through the stiffness tensor. It is also found that, while in the low frequency limit the dissipation is higher as the viscoelastic coefficient is also higher, near a resonance of the metamaterial this condition does not hold, since the imaginary part of the constitutive parameters is higher as the viscosity is smaller. Finally, the effects of viscosity are studied on the non-local properties of the effective parameters, and it is found that this property is attenuated with dissipation although still has to be considered.