On the fractional homogenization of one-dimensional elastic metamaterials with viscoelastic foundation
On the fractional homogenization of one-dimensional elastic metamaterials with viscoelastic foundation
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DOI：
10.1007/s00419-022-02170-w
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发表时间：
2022-05
影响因子：
2.8
通讯作者：
Weipeng Ding;John P. Hollkamp;Sansit Patnaik;F. Semperlotti
中科院分区：
文献类型：
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作者：
Weipeng Ding;John P. Hollkamp;Sansit Patnaik;F. Semperlotti
本文研究了分数阶时空算子在粘弹性地基上一维周期结构中线性弹性波传播模拟中的应用。更具体地说，本研究的重点是分数阶数学的基础上开发有效的降阶模型，能够捕获周期性的，粘弹性一维超材料的波动动力学的可能应用。通过利用空间-时间分数形式的波动方程，我们开发了一个均匀化的模型，能够捕获材料或几何不均匀性和粘弹性行为。首先，我们推导了一个一维无限周期杆的纵向粘弹性基础上的色散关系，作为参考点，在这项工作中使用整数阶的提法。然后，我们得到了与两个不同的分数公式相关的色散关系。第一个配方依赖于时间分数阶导数的使用，并专注于捕捉粘弹性基础引起的耗散。第二个配方依赖于使用的空间-时间分数阶导数，以导致均匀的一维模型的周期性酒吧。为了实现实值分数阶，分数阶和整数阶微分方程的色散关系之间的匹配方法。数值模拟表明，时空分数阶波动方程是一种有效的均匀化模型，可以很好地描述粘弹性地基上一维周期杆中的波动传播。结果还表明，使用空间分数阶导数允许建模（低阶）频带间隙内的动态，结果通常无法实现与经典的均匀化技术。
This work investigates the application of space–time fractional-order operators to the simulation of linear elastic waves propagating in 1D periodic structures resting on a viscoelastic foundation. More specifically, this study focuses on the possible application of fractional-order mathematics as the foundation to develop efficient reduced-order models capable of capturing the wave dynamics in periodic, viscoelastic one-dimensional metamaterials. By leveraging a space–time fractional formulation of the wave equation, we develop a homogenized model capable of capturing either material or geometric inhomogeneity and viscoelastic behavior. First, we derive the dispersion relation for a 1D infinite periodic bar resting on a longitudinal viscoelastic foundation using integer order formulation, which serves as a reference point in this work. Then, we obtain the dispersion relationships associated with two different fractional formulations. The first formulation relies on the use of time-fractional derivatives and focuses on capturing the dissipation induced by the viscoelastic foundation. The second formulation relies on the use of space–time fractional derivatives in order to lead to a homogenized one-dimensional model of the periodic bar. In order to achieve real-valued fractional orders, a matching approach between the dispersion relations of the fractional- and integer-order differential equations is used. Numerical simulations show that the space–time fractional wave equation serves as an effective homogenized model that well represents the wave propagation in a 1D periodic bar on a viscoelastic foundation. The results also illustrate that the use of space-fractional derivatives allows modeling the dynamics within (low order) frequency band gaps, a result typically not achievable with classical homogenization techniques.