  A lattice of elastic Rayleigh rods organized in a parallelepiped geometry can
be axially loaded up to an arbitrary amount without distortion and then be
subject to incremental time-harmonic dynamic motion. At certain threshold
levels of axial load, the grillage manifests instabilities and displays
non-trivial axial and flexural incremental vibrations. Floquet-Bloch wave
asymptotics is used to homogenize the in-plane mechanical response, so to
obtain an equivalent prestressed elastic solid subject to incremental
time-harmonic vibration, which includes, as a particular case, the incremental
quasi-static response. The equivalent elastic solid is obtained from its
acoustic tensor, directly derived from homogenization and shown to be
independent of the rods' rotational inertia. Loss of strong ellipticity in the
equivalent continuum coincides with macro-bifurcation in the lattice, while
micro-bifurcation remains undetected in the continuum and corresponds to a
vibration of vanishing frequency of the lowest dispersion branch of the
lattice, occurring at finite wavelength. A perturbative approach based on
dynamic Green's function is applied to both the lattice and its equivalent
continuum. This shows that only macro-instability corresponds to localization
of incremental strain, while micro-instabilities occur in modes that spread
throughout the whole lattice with an 'explosive' character. Extremely localized
mechanical responses are found both in the lattice and in the solid, with the
advantage that the former can be easily realized, for instance via 3D printing.
In this way, features such as shear band inclination, or the emergence of a
single shear band, or competition between micro and macro instabilities become
all designable features. The presented results pave the way for the design of
architected materials to be used in applications involving extreme
deformations.
