Different from elastic waves in linear periodic structures, those in phononic crystals with nonlinear properties can exhibit more interesting phenomena. Linear dispersion relations cannot predict band-gap variations due to intensity of wave motion; creating nonlinear phononic crystals remains challenging and few examples have been studied. Recent studies in the literature mainly focus on discrete chain-like structures and consider weak nonlinear regimes; they cannot accurately obtain some relations between wave propagation characteristics and nonlinearities. Our models are based on exact Green-Lagrange strain relations for a structure using the B-spline wavelet on the interval (BSWI) finite element method. Numerical examples show that the proposed method performs well for band structure problems with general nonlinearities. This study can provide good support for engineering applications, such as sound and vibration control using tunable band gaps of nonlinear phononic crystals.

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