Goldsberry Benjamin M, Wallen Samuel P, Haberman Michael R
Walker Department of Mechanical Engineering and Applied Research Laboratories, The University of Texas at Austin, Austin, Texas 78713-8029, USA.
Applied Research Laboratories, 10000 Burnet Road, Austin, Texas 78758, USA.
J Acoust Soc Am. 2019 Jul;146(1):782. doi: 10.1121/1.5115019.
Acoustic and elastic metamaterials with time- and space-dependent effective material properties have recently received significant attention as a means to induce non-reciprocal wave propagation. Recent analytical models of spring-mass chains have shown that external application of a nonlinear mechanical deformation, when applied on time scales that are slow compared to the characteristic times of propagating linear elastic waves, may induce non-reciprocity via changes in the apparent elastic modulus for perturbations around that deformation. Unfortunately, it is rarely possible to derive analogous analytical models for continuous elastic metamaterials due to complex unit cell geometry. The present work derives and implements a finite element approach to simulate elastic wave propagation in a mechanically-modulated metamaterial. This approach is implemented on a metamaterial supercell to account for the modulation wavelength. The small-on-large approximation is utilized to separate the nonlinear mechanical deformation (the "large" wave) from superimposed linear elastic waves (the "small" waves), which are then analyzed via Bloch wave analysis with a Fourier expansion in the harmonics of the modulation frequency. Results on non-reciprocal wave propagation in a negative stiffness chain, a structure exhibiting large stiffness modulations due to the presence of mechanical instabilities, are then shown as a case example.
具有时空相关有效材料特性的声学和弹性超材料，作为一种诱导非互易波传播的手段，近来受到了广泛关注。最近的弹簧 - 质量链分析模型表明，当在比传播线性弹性波的特征时间慢的时间尺度上施加非线性机械变形时，对于围绕该变形的扰动，通过表观弹性模量的变化可能会诱导出非互易性。不幸的是，由于单元胞几何形状复杂，对于连续弹性超材料很少能推导出类似的分析模型。本文推导并实现了一种有限元方法，用于模拟机械调制超材料中的弹性波传播。该方法在超材料超级单元上实现，以考虑调制波长。采用小量对大量近似，将非线性机械变形（“大”波）与叠加的线性弹性波（“小”波）分离，然后通过在调制频率谐波中进行傅里叶展开的布洛赫波分析来分析这些小波。作为案例示例，展示了在负刚度链中的非互易波传播结果，负刚度链是一种由于存在机械不稳定性而表现出大刚度调制的结构。
