Dynamics of prestressed elastic lattices: homogenization, instabilities, and strain localization

TL;DR

This paper investigates the dynamic behavior and instabilities of prestressed elastic lattices, using homogenization and perturbation methods to understand strain localization and design implications for architected materials under extreme loads.

Contribution

It introduces a homogenization approach for prestressed elastic lattices, linking macro- and micro-instabilities to strain localization and enabling design of materials with tailored instability features.

Findings

Homogenized elastic response derived from Floquet-Bloch asymptotics.

Loss of strong ellipticity coincides with macro-bifurcation.

Micro-instabilities exhibit explosive, non-localized modes.

Abstract

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.