The incorporation of nonlinearity in phononic materials enables complex wave interactions in both space and time enriching the dynamic response of the underlying linear media. In this talk, we discuss the strongly nonlinear wave response of continuum phononic material with periodic and discrete nonlinearity. The studied phononic material is a periodic architecture of contact interfaces with rough surfaces, connecting linear elastic layers. These contacts exhibit strong nonlinearity, stemming from variable contact areas under compressive loads and their inability to support the tensile loads. We reveal the evolution of propagating waves using finite element time-domain simulations. The interplay of strong nonlinearity, and dispersion, in the presence of elastic layers, generate traveling localized waves, referred to as “stegotons.” Unlike classical solitons, these stagotons exhibit evolving spatial wave profiles and local variations in wave speed. Moreover, the elastodynamic effects arising from the nonlinearly coupled elastic layers enable strongly nonlinear energy transfer in the frequency domain by activating acoustic resonances of the layers. This study sheds light on the role of the strongly nonlinear coupling of linear media on the formation of traveling localized waves, and could open opportunities for enhanced dynamic response not possible with pure discrete or continuum phononic materials.