  In this paper we substantiate the claim implicitly made in previous works
that the relaxed micromorphic model is the only linear, isotropic, reversibly
elastic, nonlocal generalized continuum model able to describe complete
band-gaps on a phenomenological level. To this end, we recapitulate the
response of the standard Mindlin-Eringen micromorphic model with the full
micro-distortion gradient of P, the relaxed micromorphic model depending only
on the Curl P of the micro-distortion P, and a variant of the standard
micromorphic model in which the curvature depends only on the divergence Div P
of the micro distortion. The Div-model has size-effects but the dispersion
analysis for plane waves shows the incapability of that model to even produce a
partial band gap. Combining the curvature to depend quadratically on Div P and
Curl P shows that such a model is similar to the standard Mindlin-Eringen model
which can eventually show only a partial band gap.
