Multi-displacement microstructure continuum modeling of anisotropic elastic metamaterials

An elastic metamaterial made of lead cylinders coated with elliptical rubbers in an epoxy matrix is considered, and its anisotropic effective dynamic mass density tensor is numerically determined and demonstrated. To capture both dipolar resonant motion and microstructure deformation in the composite, a new multi-displacement microstructure continuum model is proposed. In the formulation, additional displacement and kinematic variables are introduced to describe global and local deformations, respectively. The macroscopic governing equations of the two-dimensional anisotropic elastic metamaterial are explicitly derived through a simplified procedure. To verify the current model, wave dispersion curves from the current model are compared with those from the finite element simulation for both longitudinal and transverse waves. Very good agreement is observed in both the acoustic and optic wave modes. This work could provide a benchmark of continuum modeling of elastic metamaterials with nonelementary microstructures.