弹性晶格中面内波与结构化可穿透线缺陷的相互作用
Interaction of in-plane waves with a structured penetrable line defect in an elastic lattice
AI总结：
本研究通过矢量Wiener-Hopf方程和对称性简化，解析求解了弹性晶格中面内波与结构化可穿透线缺陷的散射问题，揭示了局域界面波及三类对称动态模式，并经有限元验证，展示了动态各向异性、负反射和负折射等现象。
摘要：
我们考虑了面内波与三角形晶格中所含结构化可穿透（惯性）线缺陷边缘相互作用的散射问题，该晶格由周期性放置的质量块通过无质量弹性杆相互连接而成。利用傅里叶变换，将时谐激励下的稳态问题转化为矢量Wiener-Hopf方程。该方程的矩阵Wiener-Hopf核描述了散射过程中涉及的所有动态现象，包括沿结构化缺陷可能出现的局域界面波。利用这一信息，我们确定了这些波的存在对入射波参数和惯性缺陷性质的依赖性。散射介质结构的对称性使我们能够将矢量问题转化为沿包含缺陷的晶格行建立的一对标量Wiener-Hopf方程。该解以复平面上的围道积分形式精确表示了散射场，其中包括倏逝波和传播波的贡献。该解揭示，在远处晶格中，入射场的反射和透射分量除了局域界面波之外，还伴随来自三个不同对称类的动态模式。这些类分别对应于横向作用于缺陷晶格行的拉伸模式、平行于该行的剪切模式，以及表示为这两种响应混合的波模式。提供了基准有限元计算，以验证所获得的半解析解，其中涉及围道积分的数值计算。图形示例展示了波散射过程中遇到的特殊动态响应，包括动态各向异性、负反射和负折射。
英文摘要：
We consider the scattering of in-plane waves that interact with an edge of a structured {penetrable (inertial)} line defect contained in a triangular lattice, composed of periodically placed masses interconnected by massless elastic rods. The steady state problem for time-harmonic excitation is converted into a vector Wiener-Hopf equation using Fourier transform. The matrix Wiener-Hopf kernel of this equation describes all dynamic phenomena engaged in the scattering process, which includes instances where localised interfacial waves can emerge along structured defect. This information is exploited to identify the dependency of the existence of these waves on the incident wave parameters and properties of the inertial defect. The symmetry in the structure of scattering medium allows us to convert the vectorial problem into a pair of scalar Wiener-Hopf equations posed along the lattice row containing the defect. The solution embodies the exact representation of scattered field, in terms of a contour integral in the complex plane, that includes the contributions of evanescent and propagating waves. The solution reveals that in the remote lattice, the reflected and transmitted components of incident field are {accompanied by dynamic modes from three distinct symmetry classes in addition to localised interfacial waves}. These classes correspond to tensile modes acting transverse to the defected lattice row, shear modes that act parallel to this row, and wave modes represented as a mixture of these two responses. Benchmark finite element calculations are provided to validate results against the obtained semi-analytical solution, which involve numerical computations of the contour integrals. Graphical illustrations demonstrate special dynamic responses encountered during the wave scattering process, including dynamic anisotropy, negative reflection and negative refraction.
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