Continuum modeling of a composite material with internal resonators

Abstract

Two continuum methods were investigated for modeling the dynamic behavior of an acoustic metamaterial in the form of a composite material with internal resonators. First, an effective homogeneous classical continuum model was proposed. The effective elastic constants for this continuum are obtained by taking the static equivalence between the continuum and the composite, while the effective mass density adopts the form of a second order tensor. The second model is also a continuum model that is described by two displacement variables. In addition to the usual displacement vector, a displacement vector for the motion of the resonator mass is included, thus making this multi-displacement model quite different from the classical model for elastic solids. It was shown that the dispersion relations predicted by the proposed two approaches were practically the same. The accuracy of the dispersion curves was verified by finite element analyses. Simplicity is the main advantage of the first approach. However, it has to adopt an unusual frequency-dependent effective mass density which may become negative in certain frequency range. On the other hand, the multi-displacement model can be constructed based on the actual material properties of the composite and, in general, is more versatile for further extensions to complex microstructures.