Developing nonlinear resonant metamaterial plates requires an effective wave simulation method to analyze the wave interactions between multiple nonlinear scatterers. The multiple scattering method has the advantages of high computational efficiency and closed-form displacement solutions in modeling the finitely periodic scatterer array. However, the multiple scattering method of nonlinear scatterers is absent in the available studies of plate structures. In this paper, a nonlinear multiple scattering approach is formulated and analyzed for thin plates with nonlinear resonant scatterers. The resonant scatterer consists of an inner plate with a local resonator modeled by the nonlinear mass-spring system. The motion equation of resonant scatterers is solved using the perturbation technology and wave expansion method. The T-matrix of the resonant scatterer of each order is then derived at the arbitrary harmonic frequency from continuous interfacial conditions of resonant scatterers. Finally, the multiple scattering is formulated for a finitely periodic array of resonant scatterers on a thin plate. Numerical simulations capture the scattering characteristics and stop-band formations of fundamental and second-harmonic waves. Furthermore, the finitely periodic array of nonlinear and linear resonant scatterers achieves the nonreciprocal transmission of flexural waves. These numerical results demonstrate the potential applications of the proposed method in designing metamaterial-based plates with complex transmission properties.

中文翻译：

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薄板上局部非线性谐振器的多重散射


开发非线性谐振超材料板需要有效的波模拟方法来分析多个非线性散射体之间的波相互作用。多重散射法在有限周期散射体阵列建模中具有计算效率高、位移解闭式等优点。然而，现有的板结构研究中缺乏非线性散射体的多重散射方法。本文针对具有非线性共振散射体的薄板制定并分析了非线性多重散射方法。谐振散射体由一个内板和一个由非线性质量弹簧系统建模的局部谐振器组成。利用微扰技术和波展开法求解共振散射体的运动方程。然后，根据共振散射体的连续界面条件，在任意谐波频率下导出各阶共振散射体的 T 矩阵。最后，为薄板上的有限周期共振散射体阵列制定了多重散射。数值模拟捕获基波和二次谐波的散射特性和阻带结构。此外，非线性和线性谐振散射体的有限周期阵列实现了弯曲波的不可逆传输。这些数值结果证明了所提出的方法在设计具有复杂传输特性的基于超材料的板中的潜在应用。