In this paper, we show that acoustoelasticity in hyperelastic materials can be understood using the framework of nonlinear wave mixing, which, when coupled with an induced static stress, leads to a change in the phase velocity of the propagating wave with no change in frequency. By performing Floquet wave eigenvalue analysis, we also show that band gaps for periodic composites, acting as 1-D phononic crystals, can be tuned using this static stress. In the presence of second-order elastic nonlinearities, the phase velocity of propagating waves in the phononic structure changes, leading to observable shifts in the band gaps. Finally, we present numerical examples as evidence that the band gaps are tuned by both the direction of the stress and its magnitude.