  The design of band-gap metamaterials, i.e., metamaterials with the capability
to inhibit wave propagation of a specific frequency range, has numerous
potential engineering applications, such as acoustic filters and vibration
isolation control. In order to describe the behavior of such materials, a novel
integral micromorphic elastic continuum is introduced, and its ability to
describe band gaps is studied in the one-dimensional setting. The nonlocal
formulation is based on a modification of two terms in the expression for
potential energy density. The corresponding dispersion equation is derived and
converted to a dimensionless format, so that the effect of individual
parameters can be described in the most efficient way. The results indicate
that both suggested nonlocal modifications play an important role. The original
local micromorphic model reproduces a band gap only in the special, somewhat
artificial case, when the stiffness coefficient associated with the gradient of
the micromorphic variable vanishes. On the other hand, the nonlocal formulation
can provide band gaps even for nonzero values of this coefficient, provided
that the penalty coefficient that enforces coupling between the micromorphic
variable and nonlocal strain is sufficiently high and the micromorphic
stiffness is sufficiently low.
