In this investigation, three-dimensional nonreciprocal transmission is realized by the combination of structural asymmetry and material nonlinearity. Analytical solutions of the coupled quasi-longitudinal (qP) and quasi-shear (qS₁ and qS₂) waves are derived. According to Bloch’s law and the stiffness matrix method, we obtain the band gaps and the transmission coefficients of the fundamental and double frequency waves. We also discuss the effects of the incident azimuthal angle on the propagation behaviors in nonlinear phononic crystals and elastic wave metamaterials. This present work is expected to help design practical devices of nonreciprocal transmission with vector characteristics.

在这项研究中，通过结构非对称性和材料非线性的组合实现了三维不可逆传输。推导了准纵向（qP）波和准剪切（qS ₁和qS ₂）波的解析解。根据布洛赫定律和刚度矩阵法，我们获得了基频和双频波的带隙和传输系数。我们还讨论了入射方位角对非线性声子晶体和弹性波超材料中传播行为的影响。预期该当前工作将有助于设计具有矢量特性的不可逆传输的实用装置。