Mechanics of longitudinal and flexural locally resonant elastic metamaterials using a structural power flow approach

Elastic metamaterials are sub-wavelength structures with locally resonant components that contribute to the rise of tunable stop bands, i.e. frequency ranges within which waves do not propagate. A new approach is presented here to model and quantify this stop band behavior by evaluating structural vibrating power flowing in the different constituents of locally resonant metamaterials. It is shown that the patterns of power propagation resemble, to a great extent, steady-state wave profiles derived from displacement fields, and can thus be used to develop an algorithm that numerically predicts stop band frequencies for any given realization with a finite length and a known number of repeating cells. The approach is presented here in the context of one-dimensional metamaterials with single and multiple internal resonators and is applied to two traditional examples constituting both longitudinal and flexural type structures. The presence of dissipative elements is taken into consideration since the active component of vibrational power is shown to depend on the damping matrix of the finite element description. The presented approach can be further extended to complex metamaterials with multi-dimensional locally resonant configurations to locate critical energy transmission paths within the media of such structures.