  Floquet-Bloch wave asymptotics is used to homogenize the in-plane mechanical
response of a periodic grillage of elastic Rayleigh rods, possessing a
distributed mass density, together with rotational inertia. The grid is subject
to incremental time-harmonic dynamic motion, superimposed to a given state of
axial loading of arbitrary magnitude. In contrast to the quasi-static energy
match (addressed in Part I of the present study), the vibrational properties of
the lattice are directly represented by the acoustic tensor of the equivalent
solid (without passing through the constitutive tensor). The acoustic tensor is
shown to be independent of the rods' rotational inertia and allows the analysis
of strong ellipticity of the equivalent continuum, evidencing coincidence with
macro-bifurcation in the lattice. On the other hand, micro-bifurcation
corresponds to a vibration of vanishing frequency of the lowest dispersion
branch of the lattice, occurring at finite wavelength. Dynamic homogenization
reveals the structure of the acoustic branches close to ellipticity loss and
the analysis of forced vibrations (both in physical space and Fourier space)
shows low-frequency wave localizations. A comparison is presented between
strain localization occurring near ellipticity loss and forced vibration of the
lattice, both corresponding to the application of a concentrated pulsating
force. The comparison shows that the homogenization technique allows an almost
perfect representation of the lattice, with the exception of cases where
micro-bifurcation occurs, which is shown to leave the equivalent solid
unaffected. Therefore, the presented results pave the way for the design of
architected cellular materials to be used in applications where extreme
deformations are involved.
