Dynamics of Prestressed Elastic Lattices: Homogenization, Instabilities, and Strain Localization

A lattice (or ‘grillage’) of elastic Rayleigh rods (possessing a distributed mass density, together with rotational inertia) 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. Including every possible structural geometry and for an arbitrary amount of axial stretching, 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. 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 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 which spread throughout the whole lattice with an ‘explosive’ character. In particular, 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 comparison between the mechanics of the lattice and its equivalent solid shows that the homogenization technique allows an almost perfect representation, except when micro-bifurcation is the first manifestation of instability. Therefore, the presented results pave the way for the design of architected cellular materials to be used in applications where extreme deformations are involved.