Recent experiments have observed the emergence of standing waves at the free surface of elastic bodies attached to a rigid oscillating substrate and subjected to critical values of forcing frequency and amplitude. This phenomenon, known as Faraday instability, is now well understood for viscous fluids but surprisingly eluded any theoretical explanation for soft solids. Here, we characterize Faraday waves in soft incompressible slabs using the Floquet theory to study the onset of harmonic and subharmonic resonance eigenmodes. We consider a ground state corresponding to a finite homogeneous deformation of the elastic slab. We transform the incremental boundary value problem into an algebraic eigenvalue problem characterized by the three dimensionless parameters, that characterize the interplay of gravity, capillary and elastic waves. Remarkably, we found that Faraday instability in soft solids is characterized by a harmonic resonance in the physical range of the material parameters. This seminal result is in contrast to the subharmonic resonance that is known to characterize viscous fluids, and opens the path for using Faraday waves for a precise and robust experimental method that is able to distinguish solid-like from fluid-like responses of soft matter at different scales.

中文翻译：

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软弹性固体中的法拉第波

最近的实验观察到在弹性体的自由表面上出现驻波，这些弹性体连接到刚性振荡基板并受到强迫频率和振幅的临界值。这种被称为法拉第不稳定性的现象现在对于粘性流体已经很好理解，但令人惊讶的是，对于软固体却没有任何理论解释。在这里，我们使用 Floquet 理论来表征软不可压缩平板中的法拉第波，以研究谐波和次谐波共振本征模式的开始。我们考虑对应于弹性板有限均匀变形的基态。我们将增量边值问题转化为由三个无量纲参数表征的代数特征值问题，表征重力波、毛细波和弹性波的相互作用。值得注意的是，我们发现软固体中的法拉第不稳定性的特点是材料参数物理范围内的谐波共振。这一开创性结果与已知表征粘性流体的次谐波共振形成对比，并为使用法拉第波进行精确而稳健的实验方法开辟了道路，该方法能够区分软物质在不同的尺度。