Abstract In this work, Snell's law of elastic waves on the moving property interface (MPI) of time-varying materials has been studied. First, additional constraints of the elastic wave propagation on MPI are proposed to ensure self-consistence of Snell's law. Second, based on Huygens principle, a geometric approach is developed which can be used to geometrically reveal the wave propagation directions on MPI with clear geometric meanings for the additional constraints. Two criteria are proposed to determine the existence of reflected or transmitted wave on MPI. Finally, based on the two criteria, the conditions for the existence of reflected or transmitted waves are expressed by moving velocity of MPI and wave propagation velocities on both sides of MPI. Then, according to the number of reflected and transmitted waves as well as their polarization types, the propagation of elastic wave on MPI can be classified into 6 cases for SH wave incidence, 12 cases for SV wave incidence, and 15 cases for P wave incidence determined by moving velocity of MPI and wave propagation velocities on both sides of MPI. While, the traditional case of elastic wave propagation on a static material interface, i.e. 2 emitted waves for SH wave and 4 emitted waves for SV or P waves are special cases of elastic wave propagation on MPI. It is noted that for most propagation cases presented in this work, the propagation coefficients are unable to be solved with only the continuous conditions on MPI, i.e. the continuity of displacement, stress and momentum on MPI. This work indicates the elastic wave propagation on MPI can have multiple cases of emitted wave which may provide useful insights for further studying elastodynamics of time-varying materials.

中文翻译：

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时变材料运动特性界面上弹性波传播的斯涅尔定律

摘要 在这项工作中，研究了时变材料运动特性界面（MPI）上的斯涅尔弹性波定律。首先，提出了对 MPI 上弹性波传播的附加约束，以确保斯涅尔定律的自洽性。其次，基于惠更斯原理，开发了一种几何方法，可用于几何地揭示 MPI 上的波传播方向，并为附加约束提供清晰的几何意义。提出了两个标准来确定 MPI 上是否存在反射波或透射波。最后，基于这两个判据，反射波或透射波存在的条件用MPI的移动速度和MPI两侧的波传播速度来表示。然后，根据反射波和透射波的数量及其偏振类型，弹性波在 MPI 上的传播可分为 SH 波入射 6 种，SV 波入射 12 种，P 波入射 15 种，由下式确定MPI 的移动速度和 MPI 两侧的波传播速度。而弹性波在静态材料界面上传播的传统情况，即 SH 波发射 2 个波，SV 或 P 波发射 4 个波，是 MPI 上弹性波传播的特例。需要注意的是，对于本文提出的大多数传播情况，仅在 MPI 上的连续条件下，即 MPI 上位移、应力和动量的连续性，无法求解传播系数。