Exact closed-form solutions for Lamb's problem (II): a moving point load

TL;DR

The paper addresses the displacement field in a homogeneous elastic half-space due to a downward vertical point load moving at constant speed along the surface, extending the classical Lamb's problem. It adopts the Bakker–Verweij 1999 integral framework and the Feng–Zhang approach to transform these integrals into exact closed-form expressions involving elementary functions and elliptic integrals. The authors validate the results against BVK99 numerics and demonstrate the ability to isolate Rayleigh-wave contributions, offering a robust analytic tool for moving-load problems. The work provides a theoretically solid, computationally efficient basis for modeling dynamic responses to moving loads such as high-speed trains and informs potential extensions to dipole sources.

Abstract

In this article, 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 \textit{et al.}, we followed the method developed in 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 \textit{et al.}, which confirms the correctness of our formulas. The solution obtained in this article may lay a solid foundation for further consideration of the response of an actual physical moving load, such as a high-speed rail train.

Figures (7)

• Figure 1: The geometry of the problem. The Cartesian coordinate $x_1 x_2 x_3$ is used in this paper. The free surface of the homogeneous half-space is loaded with a vertical point force at $t=0$, which starts from the origin $O$ and propagates with a constant velocity $c$ along the $x_1$ axis. The coordinate of the receiver is ($x_{1}$, $x_{2}$, $x_3$). The distance between the origin and the receiver is $r$, and $R$ is the projection of $r$ on the place $x_1 x_2$. $\theta$ is the angle between $r$ and $R$, and $\phi$ between $R$ and $x_1$ axis.
• Figure 2: The geometry of the numerical instances. $R_1$ denotes the receivers in Fig. 3 and Fig. 7, $R_2$ in Figs. 4--6 and $R_3$ in Fig. 8, respectively.
• Figure 3: (a) Comparison of displacements between Johnson's (1974) solutions and those in BVK99 after our reorganization, when velocity comes to zero. The receiver is at the $(10, 1, 2)$ (km). (a) $u_{1}$; (b) $u_{3}$.
• Figure 4: (a) Comparison of $u_{1}$ between our solutions and those in BVK99. The receiver is at $(1, 10, 0.2)$ (km) and $c=2.00$ km/s. The depth of the receiver is much smaller than the epicentral distance such that the Rayleigh wave is obvious. The three vertical dotted lines mark the estimated arrival times of direct P, direct S and Rayleigh waves. (b) Decompose the displacement into Rayleigh part and other part. In the vicinity of the arrival time of the Rayleigh wave, Rayleigh part fluctuates and other part is smooth.
• Figure 5: The same as Fig. 4, except for $u_{2}$ here.