  The topological mechanics is a perfect tool that can bridge the gap between
the quantum and Newtonian physics and mechanics of materials. It requires
discrete models of the material with analogies with the topological
characteristics of quantum matter. Despite bridging this gap using continuous
models of matter would seem challenging, we demonstrate here this possibility.
We demonstrate the bridging of the quantum-continuum mechanics gap by studying
the topological mechanics of metamaterials based on the micromorphic theory. A
general micromorphic model of metamaterials is developed and used to study
their topological characteristics. The conditions of the band-gaps, band
localization, and band inversion are defined. Whereas closing the band-gap of
topological insulator is obviously due to band localization, we demonstrate a
mechanism of closing the band-gap with no band localization. In addition,
despite the fact that the band inversion of many topological insulators cannot
be completed without closing the gap, we demonstrate a case of optical band
inversion with no gap closing. We foresee that the exceptional topological
characteristics of metamaterials explored here will help in the design of
advanced topological insulators, and open new venues of the implementations of
topological insulators in mechanical applications.
