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In this work, the three-dimensional (3D) propagation behaviors in the nonlinear phononic crystal and elastic wave metamaterial with initial stresses are investigated. The analytical solutions of the fundamental wave and second harmonic with the quasi-longitudinal (qP) and quasi-shear (qS 1  and qS 2 ) modes are derived. Based on the transfer and stiffness matrices, band gaps with initial stresses are obtained by the Bloch theorem. The transmission coefficients are calculated to support the band gap property, and the tunability of the nonreciprocal transmission by the initial stress is discussed. This work is expected to provide a way to tune the nonreciprocal transmission with vector characteristics.

Tunable three-dimensional nonreciprocal transmission in a layered nonlinear elastic wave metamaterial by initial stresses