In this paper, we report on an exact closed-form solution for the displacement in an elastic homogeneous half-space elicited by a downward vertical point source moving with constant velocity over the surface of the medium. The problem considered here is an extension to Lamb’s problem. Starting with the integral solutions of Bakker et al., we followed the method developed by Feng and Zhang, which focuses on the displacement triggered by a fixed point source observed on the free surface, to obtain the final solution in terms of elementary algebraic functions as well as elliptic integrals of the first, second and third kind. Our closed-form results agree perfectly with the numerical results of Bakker et al., which confirms the correctness of our formulae. The solution obtained in this paper may lay a solid foundation for further consideration of the response of an actual physical moving load, such as a high-speed rail train.

中文翻译：

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Lamb问题的精确封闭式解决方案-II：移动点载荷

在本文中，我们报告了由向下垂直的点源在介质表面上以恒定速度移动引起的弹性均匀半空间中位移的精确封闭形式解。这里考虑的问题是Lamb问题的扩展。从Bakker等人的积分解开始，我们遵循了Feng和Zhang所开发的方法，该方法着重于在自由表面上观察到的由固定点源触发的位移，从而获得了基本代数函数的最终解，如以及第一，第二和第三类的椭圆积分。我们的闭式结果与Bakker等人的数值结果完全吻合，这证实了我们公式的正确性。本文获得的解决方案可以为进一步考虑实际物理移动负载（例如高速铁路）的响应奠定坚实的基础。