  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.
