In this paper, we investigate the response of a cavity embedded in an elastic half-plane (2D) subjected to a harmonic SH wave. In previous work, the method of conformal mapping and the indirect boundary element method (indirect BEM) were employed to solve the 3D wave scattering from a cylindrical tunnel embedded in a half-space. Inaccurate results were obtained particularly at high frequencies (method of conformal mapping). Therefore, in this study we focus on a comparison of the two methods with the method of images, which serves as a benchmark solution. Through a systematic evaluation, we confirm that the two methods accurately work within the complete considered ranges of the dimensionless frequency and the embedded cavity depth. This suggests that representing the waves scattered from the free surface by cylindrical waves in the method of conformal mapping is the cause of the inaccuracies at high frequency in the 3D problem; the cylindrical waves are probably not able to fully capture all wave conversions taking place at the free surface. The presented results reveal significant effects of the system parameters on the responses. The system's response curves display nearly equally spaced resonances, which is in line with those of the 1D shear layer subject to bedrock motion, while similar response curves for the 3D case do not have this feature.

This study develops a comprehensive analytical solution for predicting three-dimensional ground motions induced by the vibration of large-diameter end-bearing piles subjected to vertically distributed uniform loads. Both the pile and the surrounding soil are treated as elastic continuum media to capture the coupled effects of P-SV and Rayleigh waves accurately. General solutions for wave potentials, displacements, and stresses are derived using small-strain theory and continuum elasticity. Modal wave numbers are determined through a root-searching approach employing the argument principle and subdivision method. Bi-orthogonality relationships are reorganized using Betti's theorem. Soil–pile interactions are rigorously modeled through continuity conditions at the soil–pile interface. Mode-matching method is used to solve the unknown coefficients. The boundary-value problem is reduced to a system of linear algebraic equations with series truncation to ensure convergence and computational efficiency. Parametric studies reveal that excitation frequencies significantly influence the distribution of soil and pile responses. Shear waves corresponding to pile frictions dominate near-field responses and Rayleigh waves resulting from surface load propagate at larger distances. Transient displacement responses show the significant influence of complex Rayleigh wave propagation and the secondary subsurface scattering on the ground motions. The particle motions reveal the Rayleigh waves are generating and propagating through the ground surface induced by pile vibrating. This study contributes to the accurate prediction of ground motions and supports the design of vibrating grounds which ensures the safety of pile-supported structures in urban and displacement-sensitive environments.

An analytical solution for the scattering of harmonic P1 and SV waves in a poroelastic half-plane with a shallow lined tunnel is obtained using the plane complex theory in elastodynamics. In light of the wave function expansions, the wave fields of the poroelastic medium and the liner with unknown coefficients are obtained based on Biot's theory and Helmholtz decomposition. Complex-valued expressions of the effective stresses, the fluid stress, and the displacements of the poroelastic medium and the liner are expressed by the complex variable function method and the conformal transformation technique. With the boundary conditions and the continuity of the medium-liner interface, the boundary value problem results in a series of algebraic equations. The unknown coefficients in the infinite set of algebraic equations can be solved numerically by truncating the series number. A parametric study for the incident SV waves is performed to investigate dynamic stress concentrations and fluid stress of the medium and the liner. Numerical results show that the embedment depth of the tunnel, the incident angle of the excitations, and the porosity of the medium have considerable influence on the dynamic responses of the medium and the liner. The shielding effect of the tunnel on the incident SV waves is obvious. For the big embedment depth of the tunnel, the scattered waves contribute little to the displacements and dynamic stress concentration of the medium and the liner. For a high porosity close to the critical value, the response of the medium-liner system to the incident waves is great.

This paper investigates the instability of vertical vibrations of an object moving uniformly through a tunnel embedded in soft soil. Using the indirect Boundary Element Method in the frequency domain, the equivalent dynamic stiffness of the tunnel-soil system at the point of contact with the moving object, modelled as a mass-spring system or as the limiting case of a single mass, is computed numerically. Using the equivalent stiffness, the original 2.5D model is reduced to an equivalent discrete model, whose parameters depend on the vibration frequency and the object's velocity. The critical velocity beyond which the instability of the object vibration may occur is found, and it is the same for both the oscillator and the single mass. This critical velocity turns out to be much larger than the operational velocity of high-speed trains and ultra-high-speed transportation vehicles. This means that the model adopted in this paper does not predict the vibrations of Maglev and Hyperloop vehicles to become unstable. Furthermore, the critical velocity for resonance of the system is found to be slightly smaller than the velocity of Rayleigh waves, which is very similar to that for the model of a half-space with a regular track placed on top (with damping). However, for that model, the critical velocity for instability is only slightly larger than the critical velocity for resonance (of the undamped system), while for the current model the critical velocity for instability is much larger than the critical velocity for resonance due to the large stiffness of the tunnel and the radiation damping of the waves excited in the tunnel. A parametric study shows that the thickness and material damping ratio of the tunnel, the stiffness of the soil and the burial depth have a stabilising effect, while the damping of the soil may have a slightly destabilising effect (i.e., lower critical velocity for instability). In order to investigate the instability of the moving object for velocities larger than the identified critical velocity for instability, we employ the D-decomposition method and find instability domains in the space of system parameters. In addition, the dependency of the critical mass and stiffness on the velocity is found. We conclude that the higher the velocity, the smaller the mass of the object should be to ensure stability (single mass case); moreover, the higher the velocity, the larger the stiffness of the spring should be when a spring is added (oscillator case). Finally, in view of the stability assessment of Maglev and Hyperloop vehicles, the approach presented in this paper can be applied to more advanced models with more points of contact between the moving object and the tunnel, which resembles reality even better.

A wave function expansion method for out-of-plane scattering of SH waves by two symmetrical circular cavities in two bonded exponentially graded half spaces is presented by using the plane elastic complex variable theory. The medium is composed of a semi-infinite homogeneous space with a circular cavity and a bonded exponentially graded half-space with a symmetrical circular cavity along the boundary interface. In terms of Helmholtz decomposition to the wave propagation equation, the stresses and displacements of the medium are expressed by the displacement potentials. The scattered waves by the boundary interface are assumed to transmit from the images of the origins of the respective two cavities. A conformal mapping function is introduced to convert the physical plane of two bonded semi-infinite spaces into two jointed concentric annular regions. The boundary value problem is formulated as a series of infinite algebraic equations. The convergence of the present solution is examined by investigating the variations of the solution results with the truncation of the series number. Parametric study shows that the interface displacements are almost independent to the inhomogeneous coefficient βR of the right medium. The amplification of the interface displacement becomes great for high incident frequency and with the decreasing of the distance-to-radius ratio h/R. The distribution of stress concentration of the right cavity depends significantly on the inhomogeneous coefficient βR, the distance-radius ratio h/R, the incident angle and frequency of the excitations.