  This study investigates the impact of sliders -- constraints acting on
elastic rods allowing for a transverse displacement jump while maintaining
axial and rotational displacement continuity -- on the dynamics of a periodic
elastic grid, including the effects of axial preload. The grid is linearly
elastic and subject to in-plane incremental deformation, involving normal and
shear forces and bending moment. The periodicity of the infinite grid permits a
Floquet-Bloch wave analysis and a rigorous dynamic homogenization, leading to
an equivalent prestressed elastic solid. The investigation is complemented by
{\it ad hoc} developed F.E. simulations and perturbations with a pulsating
Green's function. Results show that the sliders create band gaps, flat bands,
and Dirac cones in the dispersion diagrams and generate macro-instability even
for tensile prestress. The latter corresponds to the loss of ellipticity at the
parabolic boundary in the equivalent elastic solid and provides a rare example
of an almost unexplored form of material instability. Therefore, our results
offer design strategies for metamaterials and architected materials showing
reversible material instabilities and filtering properties for mechanical
signals.
