  In this paper we derive, by means of a suitable least action principle, the
duality jump conditions to be imposed at surfaces of discontinuity of the
material properties in non-dissipative, linear-elastic, isotropic, Mindlin's
and relaxed micromorphic media, respectively. The introduced theoretical
framework allows the transparent set-up of different types of micro-macro
connections which are intrinsically compatible with the governing bulk
equations. To illustrate the interest of the many introduced jump conditions,
we focus on the case of an interface between a classical Cauchy continuum on
one side and a relaxed micromorphic one on the other side. As expected, we find
a complete reflection in the frequency intervals for which band-gaps are known
to occur in the relaxed micromorphic continuum and precise
microstructure-related reflective patterns are identified. We repeat a similar
study for analogous connections between a classical Cauchy continuum and a
Mindlin's micromorphic one and we show that the reflective properties of the
considered interfaces are drastically modified due to the fact that band-gaps
are not allowed in standard Mindlin's micromorphic media. The present work
opens the way towards the possibility of conceiving complex metastructures in
which band-gap metamaterials and classical materials are coupled together to
produce structures with completely new and unorthodox properties with respect
to wave propagation, transmission and reflection. Last, but not least, indirect
measurements of the material coefficients of the relaxed micromorphic model
based upon real experiments of reflection and transmission in band-gap
metamaterials are uncovered by the present work which makes them finally
realizable in the short term.
