Nonlinear composites, allowing large reversible geometry changes, have already been shown to have remarkable deformation-induced phononic properties, such as negative group velocity, asymmetric transmission, and distinct types of band gaps. The emergence of advanced electro- and magneto-elastic materials have also enabled control of the propagation of phonons by application of electric and magnetic fields, respectively, to induce symmetry changes that enhance the functionality spectrum of nonlinear composites. While many researchers have investigated the phononic properties of composites after the onset of elastic instability, there are very few studies analyzing the propagation of phonons in the nearly unstable (but still stable) composites. To the best of our knowledge, thus far, only layered and fiber composites on the edge of elastic instability have been investigated. We extend previous numerical studies to more complex periodic composites possessing p4mm plane group. Using 2D periodic geometries, we demonstrate that nearly unstable nonlinear composites possess an exciting potential for tuning phonon propagation. Specifically, in the vicinity of the elastic instability, the lowest phononic mode starts to "soften", i.e. the frequency of this mode tends to zero for a particular wavenumber, with increased strain. In the physics of crystals, a "soft mode" is a precursor to a second-order phase transition due to, for example, a change in temperature. Similarly, in continuum mechanics a "soft mode" is a forerunner of elastic instability, usually accompanied by ensuing symmetry changes in the geometry of the composite. Here, in the periodic composites, mode "softening" leads to a drastic decrease of the phase velocities and negative group velocities of elastic shear waves.
