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A lattice of elastic rods organized in a parallelepiped geometry can beaxially loaded up to an arbitrary amount without distortion and then be subjectto incremental displacements. Using quasi-static homogenization theory, thislattice can be made equivalent to a prestressed elastic solid subject toincremental deformation, in such a way to obtain extremely localized mechanicalresponses. These responses can be analyzed with reference to a mechanical modelwhich can, in principle, be realized, so that features such as for instanceshear bands inclination, or emergence of a single shear band, or competitionbetween micro (occurring in the lattice but not in the equivalent solid) andmacro (present in both the lattice and the equivalent continuum) instabilitiesbecome all designable features. The analysis of localizations is performedusing a Green's function-based perturbative approach to highlight thecorrespondence between micromechanics of the composite and homogenized responseof the equivalent solid. The presented results, limited to quasi-staticbehaviour, provide a new understanding of strain localization in a continuumand open new possibilities for the realization and experimentation of materialsexhibiting these extreme mechanical behaviours. Dynamic homogenization andvibrational localization are deferred to Part II of this study.

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