  Homogenization of the incremental response of grids made up of preloaded
elastic rods leads to homogeneous effective continua which may suffer
macroscopic instability, occurring at the same time in both the grid and the
effective continuum. This instability corresponds to the loss of ellipticity in
the effective material and the formation of localized responses as, for
instance, shear bands. Using lattice models of elastic rods, loss of
ellipticity has always been found to occur for stress states involving
compression of the rods, as usually these structural elements buckle only under
compression. In this way, the locus of material stability for the effective
solid is unbounded in tension, i.e. the material is always stable for a tensile
prestress. A rigorous application of homogenization theory is proposed to show
that the inclusion of sliders (constraints imposing axial and rotational
continuity, but allowing shear jumps) in the grid of rods leads to loss of
ellipticity in tension, so that the locus for material instability becomes
bounded. This result explains (i.) how to design elastic materials passible of
localization of deformation and shear banding for all radial stress paths;
(ii.) how for all these paths a material may fail by developing strain
localization and without involving cracking.
