Elastic metamaterials may exhibit band gaps due to the presence of local resonance frequencies associated with embedded or branching substructures. In a branched metamaterial, such as a pillared plate, waves propagating in the base medium–e.g., the plate portion–experience attenuation within the band-gap frequency range. A periodic unit cell comprising a base mass-spring chain connected to a multi-degree-of-freedom, mono-coupled branch is considered in this work. Bloch’s theorem is utilized to obtain the unit-cell dispersion curves. In addition, a sub-structuring approach is used to incorporate the vibrational effect of the resonating branch, which is analyzed separately and condensed into its effective dynamic stiffness. The effective dynamic stiffness can be easily obtained analytically or by using numerical finite-element analysis or experimental frequency response measurement. Thus, the results are generally applicable to an arbitrary resonating branch regardless of its size and properties. Moreover, the concepts developed may also be potentially applied to problems with higher dimensions as will be shown. The mass-spring model analysis identifies fundamental bounds for the local resonance band-gap edges in terms of the dynamical characteristics of the branching substructure, e.g., its resonance and antiresonance frequencies. Analytical sensitivity functions are also derived for the dependence of the band-gap edges on the degree of coupling between the base and the branch. The degree of coupling between the base and the resonating branch is classified into either mass or stiffness coupling. The derived sensitivity functions reveal the critical role of the frequency derivative of the resonating-branch effective dynamic stiffness.

 

Finally, the applicability of the derived bounds and sensitivity functions to two-dimensional continuum models is investigated by conducting a finite-element numerical study on a pillared unit cell. In the sub-Bragg low-frequency regime, band-gap edges are found to follow the predicted trends of the derived bounds and sensitivity functions for the lumped parameter system. It is also theoretically shown that tighter bounds can be obtained for the sub-Bragg band-gap edges in a continuum-base branched metamaterial unit cell. These tighter bounds are also found to hold in the numerical finite-element results.

Therefore, using the branch’s effective dynamic stiffness to describe its influence reduces the design parameter space in locally resonant branched metamaterials design problems. In addition to the theoretical results obtained using the proposed approach, it proves to be effective in revealing the underlying mechanisms of local resonance band gaps from the point of view of the resonating branch. Therefore, it serves to simplify the resonating branch design problem in application-specific optimization problems.


Branched Metamaterials Theoretical Band-Gap Edge Bounds and Sensitivity to Branch Coupling Properties

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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