Incremental constitutive tensors and strain localization for prestressed elastic lattices: Part I -- quasi-static response

Abstract

A lattice of elastic rods organized in a parallelepiped geometry can be axially loaded up to an arbitrary amount without distortion and then be subject to incremental displacements. Using quasi-static homogenization theory, this lattice can be made equivalent to a prestressed elastic solid subject to incremental deformation, in such a way to obtain extremely localized mechanical responses. These responses can be analyzed with reference to a mechanical model which can, in principle, be realized, so that features such as for instance shear bands inclination, or emergence of a single shear band, or competition between micro (occurring in the lattice but not in the equivalent solid) and macro (present in both the lattice and the equivalent continuum) instabilities become all designable features. The analysis of localizations is performed using a Green's function-based perturbative approach to highlight the correspondence between micromechanics of the composite and homogenized response of the equivalent solid. The presented results, limited to quasi-static behaviour, provide a new understanding of strain localization in a continuum and open new possibilities for the realization and experimentation of materials exhibiting these extreme mechanical behaviours. Dynamic homogenization and vibrational localization are deferred to Part II of this study.