A sphere impacting a plate is different from impacting a half-space bulk, because energy in the former case can be dissipated in a form of flexural wave propagating on the plate. Such energy loss by the elastic wave can be explained by the Zener model; however, this model is only valid for low-velocity impact with elastic Hertzian contact law. For high-velocity impact with contact plasticity, the Zener model is not valid any ore because energy dissipation by plasticity may take a large fraction, so that the Zener model, which ignores plasticity, will overestimate the coefficient of restitution (COR). In order to study how much the plasticity affects the sphere impacting the plate, we developed a modified Zener model by employing the contact plasticity into the controlling equation. Using the developed model, we firstly presented a thickness-dependent yielding velocity (yielding impact velocity depends on the thickness of the plate), above which the contact plasticity should be taken into account. What is more, we presented a semi-analytical solution to predict the COR over a large range of plate thickness. Meanwhile, finite element simulations were carried out to prove that the present model can give enough accuracy for the high velocity impact.

撞击板的球体不同于撞击半空间的球体，因为在前一种情况下，能量可以以在板上传播的弯曲波的形式消散。弹性波造成的这种能量损失可以用齐纳模型来解释。但是，该模型仅适用于具有弹性赫兹接触定律的低速冲击。对于具有接触可塑性的高速冲击，齐纳模型不适用于任何矿石，因为可塑性消耗的能量可能会占很大一部分，因此忽略可塑性的齐纳模型会高估恢复系数（COR）。为了研究可塑性对撞击板的球体有多大影响，我们通过将接触可塑性应用于控制方程，开发了一种改进的齐纳模型。使用开发的模型，我们首先提出了取决于厚度的屈服速度（屈服冲击速度取决于板的厚度），在此之上应考虑接触塑性。此外，我们提出了一种半解析解决方案，可以预测大范围板厚的COR。同时，进行了有限元模拟，证明了该模型对于高速冲击具有足够的精度。