From frequency-dependent models to frequency-independent enriched continua for mechanical metamaterials

Mechanical metamaterials have recently gathered increasing attention for their uncommon mechanical responses enabling unprecedented applications for elastic wave control. Many research efforts are driven towards the conception of always new metamaterials’ unit cells that, due to local resonance or Bragg-Scattering phenomena, may produce unorthodox macroscopic responses such as band-gaps, cloaking, focusing, channeling, negative refraction, etc. To model the mechanical response of large samples made up of these base unit cells, so-called homogenization or upscaling techniques come into play trying to establish an equivalent continuum model describing these macroscopic metamaterials’ characteristics. A rather common approach is to assume a priori that the target continuum model is a classical linear Cauchy continuum featuring the macroscopic displacement as the only kinematical field. This implies that the parameters of such continuum models (density and/or elasticity tensors) must be considered to be frequency-dependent to capture the complex response of the considered mechanical systems in the frequency domain. These frequency-dependent models can be useful to describe some of the aforementioned macroscopic metamaterials’ properties, yet, they suffer some drawbacks such as featuring negative masses and/or elastic coefficients in some frequency ranges which are close to resonance frequencies of the underlying microstructure. This implies that the considered Cauchy continuum is not positive-definite for all the considered frequencies. In this paper, we present a procedure, based on the definition of extra kinematical variables (with respect to displacement alone) and through the use of the inverse Fourier transform in time, to convert a frequency-dependent model into an enriched continuum model of the micromorphic type. All the parameters of the associated enriched model are constant (i.e., frequency-independent) and the model itself remains positive-definite for all the considered frequency ranges. The response of the frequency-dependent model and the associated micromorphic model coincide in the frequency domain, in particular when looking at the dispersion curves. Moreover, the micromorphic (frequency-independent) model results to be well defined both in time- and in the frequency-domain, while the Cauchy (frequency-dependent) model can only exist in the frequency domain This paper aims to build a bridge between the upscaling techniques usually found in the literature and our persuasion that macroscopic continua of the micromorphic type should be used to model metamaterials’ response at the macroscopic scale.