Weak-form homogenization of one-dimensional systems with hidden degrees of freedom

Abstract

Acoustic metamaterials are synthetic structures that exhibit specific, often exotic, effective material properties for long-wavelength stimuli. One of the principal methods that exotic material properties, such as zero or negative mass density or bulk modulus, may be obtained is through hidden degrees of freedom. An example of a hidden degree of freedom is a resonator attached in shunt to a duct and modeling the system as just a duct with effective material properties. In this example, the resonator is “hidden” from the model by incorporating its (rather profound) effect on the system into the material properties. This paper demonstrates how hidden degrees of freedom may be incorporated into a recently developed homogenization scheme that utilizes the weak form of the dynamic system [Muhlestein, J. Acoust. Soc. Am. 147 (2020)]. Two specific examples will be considered: the shunt resonator, as described above, and a membrane inside a duct. The results are compared with other standard homogenization methods.