We study the anti-plane shear waves in a domain consisting of an elastic layer (plate) with a coating attached to an elastic half-space (substrate). We assume an imperfect contact between the layer and the half-space, allowing some sliding. We also assume some elastic bonds between the layer and the substrate. On the free top surface we apply the compatibility conditions within the Gurtin–Murdoch surface elasticity. We found two different solutions: (i) the transversely exponential–transversely exponential (TE–TE) regime with amplitudes decaying exponentially from the free top surface and the interface in both the plate and the half-space, and (ii) the transversely harmonic–transversely exponential (TH–TE) regime with harmonic wave behaviour in the transverse direction in the plate and exponential decay in the half-space. The TE regime of anti-plane waves in an elastic half-space with non-perfect contact is also considered as a special case. A detailed analysis of the derived dispersion relations reveals a crucial influence of the interface stiffness on the phase velocities of anti-plane waves. This effect consists in the decrease of the phase velocities when the interfacial bonds are weakened. The strongest effect of the interfacial sliding on the phase velocities was observed for the long-length waves belonging to the TE–TE regime. Based on the derived lower bounds for the wave numbers from which the TE–TE regime of anti-plane waves exists, we have developed the theoretical background and methodology for assessing the bond stiffness of thin plates imperfectly bonded to an elastic substrate.

中文翻译：

---

界面滑动对不完全连接到弹性半空间的弹性板中反平面波的影响


我们研究了由弹性层（板）和连接到弹性半空间（基体）的涂层组成的域中的反平面剪切波。我们假设层和半空间之间的接触不完美，允许一些滑动。我们还假设该层和基板之间存在一些弹性键。在自由顶面上，我们应用了 Gurtin-Murdoch 表面弹性内的相容条件。我们发现了两种不同的解：（i） 横向指数-横向指数 （TE-TE） 状态，振幅从自由顶面和板和半空间的界面呈指数衰减，以及 （ii） 横向谐波-横向指数 （TH-TE） 状态，在板的横向上具有谐波行为，在半空间中呈指数衰减。具有非完美接触的弹性半空间中反平面波的 TE 状态也被认为是一种特殊情况。对推导的色散关系的详细分析揭示了界面刚度对反平面波相位速度的关键影响。这种效应包括当界面键减弱时相速度的降低。对于属于 TE-TE 状态的长波，观察到界面滑动对相速度的最强影响。基于反平面波的 TE-TE 范围存在的波数的推导下限，我们开发了评估未完全粘合到弹性基材的薄板的粘合刚度的理论背景和方法。