非欧弹性动力学隐身理论及其在厚板顶柱控制地表地震波中的应用
Non-Euclidean elastodynamic cloaking theory and application to control of surface seismic waves with pillars atop a thick plate
摘要：
在[AIP Advances 6，121707（2016）]中，由于转换弹性动力学和有效介质理论的结合，在两个专门设计的地震斗篷中分布有混凝土柱的土壤结构被证明可以绕过圆柱形区域周围频率低于10 Hz的瑞利波。上述研究激发了我们对表面弹性波在厚板（具有土壤参数）中传播的相互作用的探索。厚板上有混凝土柱，柱高40 m，板厚100 m，因此研究中的典型频率低于1 Hz，这是地震工程中特别感兴趣的频率范围。我们证明，三个地震斗篷允许一个前所未有的流动的弹性动力学能量。这些设计是通过首先计算从Navier方程中的几何变换推导出的理想斗篷参数来实现的，该几何变换导致几乎各向同性和对称的弹性（4阶）和密度（2阶）张量。为了做到这一点，我们扩展了理论物理学家Leonhardt和Tyc提出的光的非欧隐形理论。在第二步中，理想的非均匀近各向同性斗篷的参数近似的双极坐标网格，这是一个重要的组成部分，在我们的非欧隐形理论的弹性动力波的节点放置的支柱套的平均弹性性能。研究了固支障碍物的隐身效应（降低固支障碍物后的波前及其振幅的扰动）。这些结果代表了非欧地震斗篷设计的第一步，表面（瑞利和爱）波在半无限弹性介质中传播的支柱结构。
英文摘要：
In [AIP Advances 6, 121707 (2016)], a soil structured with concrete columns distributed within two specially designed seismic cloaks thanks to a combination of transformational elastodynamics and effective medium theory was shown to detour Rayleigh waves of frequencies lower than 10 Hz around a cylindrical region. The aforementioned studies motivate our exploration of interactions of surface elastic waves propagating in a thick plate (with soil parameters) structured with concrete pillars above it. Pillars are 40 m in height and the plate is 100 m in thickness, so that typical frequencies under study are below 1 Hz, a frequency range of particular interest in earthquake engineering. We demonstrate that three seismic cloaks allow for an unprecedented flow of elastodynamic energy. These designs are achieved by first computing ideal cloaks' parameters deduced from a geometric transform in the Navier equations that leads to almost isotropic and symmetric elasticity (4th order) and density (2nd order) tensors. To do this we extend the theory of Non-Euclidean cloaking for light as proposed by the theoretical physicists Leonhardt and Tyc. In a second step, ideal heterogeneous nearly isotropic cloak's parameters are approximated by averaging elastic properties of sets of pillars placed at the nodes of a bipolar coordinate grid, which is an essential ingredient in our Non-Euclidean cloaking theory for elastodynamic waves. Cloaking effects are studied for a clamped obstacle (reduction of the disturbance of the wave wavefront and its amplitude behind a clamped obstacle). Protection is achieved through reduction of the wave amplitude within the center of the cloak.These results represent a first step towards designs of Non-Euclidean seismic cloaks for surface (Rayleigh and Love) waves propagating in semi-infinite elastic media structured with pillars.
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