A computational framework for uncertain locally resonant metamaterial structures

Locally resonant metamaterial structures, i.e., structures artificially engineered with periodic arrays of small resonators, are attracting a growing interest among scientists. The main reason lies in their remarkable attenuation properties of elastic waves over finite frequency ranges, named band gaps, resulting from periodicity and local resonance. Yet, very little attention has been paid to investigate whether and to which extent the behavior of these structures may be affected by uncertainties. This is indeed the purpose of our study, which proposes a comprehensive computational framework to calculate the frequency response of uncertain locally resonant structures, assuming an interval model for all the relevant parameters of the resonators: mass, stiffness and damping. The computational framework is conceived for finite-element models of the locally resonant structure and standard mass-spring-dashpot models of the resonators. The key steps are an exact dynamic condensation of the degrees of freedom within the resonators, the derivation of an exact and elegant expression for the transfer matrix of the locally resonant structure via the Sherman-Morrison-Woodbury formula, the calculation of the interval frequency response via either a sensitivity-based method or a global optimization method, the choice being driven by a preliminary monotonicity test. Considering two typical locally resonant structures, a beam and a plate, the computational framework proves easy to implement, accurate and robust as compared with the standard Monte Carlo method.