  Mechanical metamaterials have recently gathered increasing attention for
their uncommon mechanical responses enabling unprecedented applications for
elastic wave control. To model the mechanical response of large metamaterials'
samples made up of 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 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 metamaterials' response 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.
More than being counter-intuitive, 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 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.
