A coupled Legendre-Laguerre polynomial method with analytical integration for the Rayleigh waves in a quasicrystal layered half-space with an imperfect interface

The Laguerre polynomial method has been successfully used to investigate the dynamic responses of a half-space.However,it fails to obtain the correct stress at the interfaces in a layered half-space,especially when there are significant differences in material properties.Therefore,a coupled Legendre-Laguerre polynomial method with analytical integration is proposed.The Rayleigh waves in a one-dimensional(1D)hexagonal quasicrystal(QC)layered half-space with an imperfect interface are investigated.The correctness is validated by comparison with available results.Its computation efficiency is analyzed.The dispersion curves of the phase velocity,displacement distributions,and stress distributions are illustrated.The effects of the phonon-phason coupling and imperfect interface coefficients on the wave characteristics are investigated.Some novel findings reveal that the proposed method is highly efficient for addressing the Rayleigh waves in a QC layered half-space.It can save over 99%of the computation time.This method can be expanded to investigate waves in various layered half-spaces,including earth-layered media and surface acoustic wave(SAW)devices.

2.5D scattering of incident plane SV waves by a canyon in layered half-space

This paper presents a 2.5D scattering of incident plane SV waves by a canyon in a layered half-space by using the indirect boundary element method （IBEM）. A free field response analysis is performed to provide the displacements and stresses on the boundary of the canyon where fictitious uniform moving loads are applied to calculate the Green＇s fi.mctions for the displacements and stresses. The amplitudes of the loads are determined by the boundary conditions. The free field displacements are added to the fictitious uniform moving loads induced displacements and the total response is obtained. Numerical calculations are performed for a canyon with homogenous and in one layer over bedrock. The effects of the thickness and stiffness of the layer on the amplification are studied and discussed.

2.5D scattering of incident plane SH waves by a canyon in layered half-space

This paper presents 2.5D scattering of incident plane SH waves by a canyon in layered half-space by the indirect boundary element method (IBEM). The free field response is carried out to give the displacements and stresses on the line which forms boundary of the canyon. The fictitious uniform moving loads are applied to the same line to calculate the Green's functions for the displacements and stresses. The amplitudes of the loads are determined by the boundary conditions. The displacements due to the free field and from the fictitious uniform moving loads have to be added to obtain the whole motion. The numerical results are carried out for the cases of a canyon in homogenous and in one layer over bedrock. The results show that the 2.5D wave scattering problem is essentially different from the 2D case, and there exist distinct differences between the wave amplification by a canyon in layered half-space and that in homogeneous half-space. The reasons for the distinct difference are explored, and the effects of the thickness and stiffness of the layer on the amplification are discussed.

Green's function of multi-layered poroelastic half-space for models of ground vibration due to railway traffic

This study proposes a Green＇s function, an essential representation of water-saturated ground under moving excitation, to simulate ground borne vibration from trains. First, general solutions to the governing equations of poroelastic medium are derived by means of integral transform. Secondly, the transmission and reflection matrix approach is used to formulate the relationship between displacement and stress of the stratified ground, which results in the matrix of the Green＇s function. Then the Green＇s function is combined into a train-track-ground model, and is verified by typical examples and a field test. Additional simulations show that the computed ground vibration attenuates faster in the immediate vicinity of the track than in the surrounding area. The wavelength of wheel-rail unevenness has a notable effect on computed displacement and pore pressure. The variation of vibration intensity with the depth of ground is significantly influenced by the layering of the strata soil. When the train speed is equal to the velocity of the Rayleigh wave, the Mach cone appears in the simulated wave field. The proposed Green＇s function is an appropriate representation for a layered ground with shallow ground water table, and will be helpful to understand the dynamic responses of the ground to complicated moving excitation.

Dynamic soil-tunnel interaction in layered half-space for incident plane SH waves

The dynamic soil-tunnel interaction is studied by indirect boundary element method （IBEM）, using the model of a rigid tunnel in layered half-space, which is simplified to a single soil layer on elastic bedrock, subjected to incident plane SH waves. The accuracy of the results is verified through comparison with the analytical solution. It is shown that soil-tunnel interaction in layered half-space is larger than that in homogeneous half-space and this interaction mechanism is essentially different from that of soil-foundation-superstructure interaction.

Dynamic soil-tunnel interaction in layered half-space for incident P-and SV-waves

The dynamic soil-tunnel interaction is studied by the model of a rigid tunnel embedded in layered half-space, which is simplified as a single soil layer on elastic bedrock to the excitation of P- and SV-waves. The indirect boundary element method is used, combined with the Green＇ s function of distributed loads acting on inclined lines. It is shown that the dynamic characteristics of soil-tunnel interaction in layered half-space are different much from that in homoge- neous half-space, and that the mechanism of soil-tunnel interaction is also different much from that of soil-founda- tion-superstructure interaction. For oblique incidence, the tunnel response for in-plane incident SV-waves is com- pletely different from that for incident SH-waves, while the tunnel response for vertically incident SV-wave is very similar to that of vertically incident SH-wave.

3D scattering of obliquely incident plane SV waves by an alluvial valley embedded in a fluid-saturated, poroelastic layered half-space

The indirect boundary element method is used to study the 3D dynamic response of an infinitely long alluvial valley embedded in a saturated layered half-space for obli- quely incident SV waves. A wave-number transform is first applied along the valley＇s axis to reduce a 3D problem to a 2D plane strain problem. The problem is then solved in the section perpendicular to the axis of the valley. Finally, the 3D dynamic responses of the valley are obtained by an inverse wave-number transform. The validity of the method is con- firmed by comparison with relevant results. The differences between the responses around the valley embedded in dry and in saturated poroelastic medium are studied, and the effects of drainage conditions, porosity, soil layer stiffness, and soil layer thickness on the dynamic response are dis- cussed in detail resulting in some conclusions.

ONE-DIMENSIONAL CONSOLIDATION OF LAYERED AND VISCO-ELASTIC SOILS UNDER ARBITRARY LOADING

Based on the layered visco-elastic soil model, according to the Terzaghi's one dimensional consolidation theory, by the method of Laplace transform and matrix transfer technique, the problems about the consolidation of layered and saturated visco-elastic soils under arbitrary loading were solved. Through deductions, the general solution, in the terms of layer thickness, the modulus and the coefficients of permeability and Laplacian transform's parameters was obtained. The strain and deformation of the layered and saturated visco-elastic soils under arbitrary loading can be calculated by Laplace inversion. According to the results of several numerical examples, the consolidation of visco-elastic soils logs behind that of elastic soils. The development of effective stress and the displacement is vibrant process under cyclic loading. Finally, an engineering case is studied and the results prove that the methods are very effective.

Surface Motion of Alluvial Valley in Layered Half-Space for Incident Plane P-Waves

The indirect boundary element method (IBEM) is used to study the surface motion of an alluvial valley in layered half-space for incident plane P-waves based on Wolf’s theory. Firstly, the free field response can be solved by the direct stiffness method, and the scattering wave response is calculated by Green’s functions of distributed loads acting on inclined lines in a layered half-space. The method is verified by comparing its results with literature and numerical analyses are performed by taking the amplification of incident plane P-waves by an alluvial valley in one soil layer resting on bedrock as an example. The results show that there exist distinct differences between the wave amplification by an alluvial valley embedded in layered half-space and that in homogeneous half-space and there is interaction between the valley and the soil layer. The amplitudes are relatively large when incident frequencies are close to the soil layer’s resonant frequencies.

Time domain method for calculating free field motion of a layered half-space subjected to obliquely incident body waves

In numerical simulation of wave scattering under oblique incident body waves using the finite element method, the free field motion at the incident lateral boundary induced by the background layered half-space complicates the computational area. In order to replace the complex frequency domain method, a time-domain method to calculate the free field motion of a layered half-space subjected to oblique incident body waves is developed in this paper. The new method decouples the equations of motion used in the finite element method and offers an interpolation formula of the free field motion. This formula is based on the fact that the apparent horizontal velocity of the free field motion is constant and can be calculated exactly. Both the theoretical analysis and numerical results demonstrate that the proposed method offers a high degree of accuracy.

3-D Scattering of Obliquely Incident Plane P Waves by Alluvial Valley Embedded in Layered Half-Space

The indirect boundary element method(IBEM) was established to solve the problem of 3-D seismic responses of 2-D topographies,by calculating the free-field responses with the direct-stiffness method and simulating the scattering wave fields with the dynamic Green's functions of moving distributed loads.The proposed method yields accurate results,because the 3-D dynamic stiffness matrixes used are exact and the fictitious moving distributed loads can be acted directly on the interface between the alluvial valley and the layered half-space without singularity.The comparison with the published methods verifies the validity of the proposed method.And the numerical analyses are performed to give some beneficial conclusions.The study shows that 3-D scattering by an alluvial valley is essentially different from the 2-D case,and that the presence of soil layer affects not only the amplitude value of surface displacements but also the distribution of surface displacements.

Three-dimensional Scattering of Obliquely Incident Plane SH Waves by an Alluvial Valley in a Layered Half-space

The indirect boundary element method （IBEM） is used to study three-dimensional scattering of obliquely incident plane SH waves by an alluvial valley embedded in a layered half-space. The free-field response of the layered half-space is calculated by the direct stiffness method, and dynamic Green＇s functions of moving distributed loads acting on inclined lines in a layered half-space are calculated to simulate the scattering wave field. The presented method yields very accurate results since the three-dimensional dynamic stiffness matrix is exact and the moving distributed loads can act directly on the valley boundary without singularity. Numerical results and analyses are performed for amplification of obliquely incident plane SH waves around an alluvial valley in a uniform half-space and in single layer over half-space. The results show that the three-dimensional responses are distinctly different from the two-dimensional responses, and the displacement amplitudes around alluvial valleys in a uniform haft-space are obviously different from those in a layered half-space.

Study on Electrical Potential by Buried Source Electrode within the Horizontally Layered Half-space Model

The paper first studies the analytical expression of electrical potential by a point current source in a uniform half-space medium,and then focuses on the distribution of electrical potential in a horizontally layered half-space model by a point current source within the surface layer or the bottom layer.Finally,the electrical potential by a source electrode in any layer of a layered half-space model is presented.

Superposition model for analyzing the dynamic ground subsidence in mining area of thick loose layer

The dynamic ground subsidence due to underground mining is a complicated time-dependent and rate- dependent process. Based. on the theory of rock rheology and probability integral method, this study developed the superposltlOn model for the prediction and analysis of the ground dynamic subsidence in mining area of thick !oose layer. The model consists of two parts （the prediction of overlying bedrock and the prediction of thick loose layer）. The overlying bedrock is regarded as visco-elastic beam, of which the dynamic subsidence is predicted by the Kelvin visco-elastic rheological model. The thick loose layer is regarded as random medium, and the ground dynamic subsidence, is predicted by the probability integral model. At last, the two prediction models are vertically stacked in the same coordinate system, and the bedrock dynamic subsidence is regarded as a variable mining thickness input into the prediction model of ground dynamic subsidence. The prediction results obtained were compared w^th actual movement and deformation data from Zhao I and Zhao II mine, central China. The agreement of the prediction results with the. field measurements.show that the superposition model （SM） is more satisfactory and the formulae obtained are more effective than the classical single probability Integral model（SPIM）, and thus can be effectively used for predicting the ground dynamic subsidence in mining area of thick loose layer.

Static response of a layered magneto-electro-elastic half-space structure under circular surface loading

A cylindrical system of vector functions, the stiffness matrix method and the corresponding recursive algorithm are proposed to investigate the static response of transversely isotropic,layered magneto-electro-elastic（MEE） structures over a homogeneous half-space substrate subjected to circular surface loading. In terms of the system of vector functions, we expand the extended displacements and stresses, and deduce two sets of ordinary differential equations, which are related to the expansion coeficients. The solution to one of the two sets of these ordinary differential equations can be evaluated by using the stiffness matrix method and the corresponding recursive algorithm. These expansion coeficients are then integrated by adaptive Gaussian quadrature to obtain the displacements and stresses in the physical domain. Two types of surface loads, mechanical pressure and electric loading,are considered in the numerical examples. The calculated results show that the proposed technique is stable and effective in analyzing the layered half-space MEE structures under surface loading.

Steady-state response of a saturated half-space with an overlying dry layer subjected to a moving load

In this paper,the steady-state response of a saturated half-space with an overlying dry layer subjected to a moving rectangular load is investigated.The governing partial differential equations are solved using the Fourier transform.The solutions in time-space domain are expressed in terms of infinite Fourier type integrals,which can be evaluated only by numerical quadrature.Numerical results show that the influence of a drained or undrained interface on the response is related to the permeability of the underlying saturated soil.Moreover,the effect due to the upper dry layer is associated with the thickness of the layer.

Analytical solutions for dynamic pressures of coupling fluid-solid-porous medium due to P wave incidence

Wave reflection and refraction in layered media is a topic closely related to seismology,acoustics,geophysics and earthquake engineering.Analytical solutions for wave reflection and refraction coefficients in multi-layered media subjected to P wave incidence from the elastic half-space are derived in terms of displacement potentials.The system is composed of ideal fluid,porous medium,and underlying elastic solid.By numerical examples,the effects of porous medium and the incident wave angle on the dynamic pressures of ideal fluid are analyzed.The results show that the existence of the porous medium,especially in the partially saturated case,may significantly affect the dynamic pressures of the overlying fluid.

Identifying boundary between near field and far field in ground vibration caused by surface loading

The boundary between the near and far fields is generally defined as the distance from the vibration source beyond which ground vibrations are mainly dominated by Rayleigh waves. It is closely related to the type of vibration source and the soil properties. Based on the solutions of the Lamb＇s problem, the boundary at the surface between the near and far fields of ground vibration was investigated for a harmonic vertical concentrated load and an infinite line load at the surface of a visco-elastic half-space. Particularly, the variation of the boundary with the material damping was investigated for both cases. The results indicate that the material damping slightly contributes to the attenuation of vibrations in the near-source region, but significantly reduces the vibrations in the region that is at some distance away from the source. When taking the material damping into consideration, the boundary between the near and far fields tends to move towards the vibration source. Compared with the vibrations caused by a concentrated load, the vibrations induced by an infinite line load can affect a larger range of the surrounding environment, and they attenuate more slowly. This means the boundary between the near field and far field should move fitrther away from the source. Finally, the boundaries are defined in terms of R-wave length （2R） and Poisson ratio of the ground （o）. For the case of a point load, the boundary is located at the distance of （5.0-6.0）2R for v≤0.30 and at the distance of （2.0--3.0）2R for v≥0.35. For the case of an infinite line load, the boundary is located at the distance （5.5-6.5）2rt for v≤0.30 and at the distance （2.5--3.5）2R for v≥0.35.

NONLINEAR DYNAMICAL CHARACTERISTICS OF PILES UNDER HORIZONTAL VIBRATION

The pile-soil system is regarded as a visco-elastic half-space embedded pile. Based on the method of continuum mechanics, a nonlinear mathematical model of pile-soil interaction was established-a coupling nonlinear boundary value problem. Under the case of horizontal vibration, the nonlinearly dynamical characteristics of pile applying the axis force were studied in horizontal direction in frequency domain. The effects of parameters, especially the axis force on the stiffness were studied in detail. The numerical results suggest that it is possible that the pile applying an axis force will lose its stability. So,the effect of the axis force on the pile is considered.

3D seismic response of a 2D hill-valley staggered topography modeled by a 2.5D multi-domain IBEM

A two and a half dimensional(2.5D)multi-domain indirect boundary element method(IBEM)is developed to study the wave scattering of obliquely incident P-,SV-and SH-waves by a hill-valley staggered topography in a multi-layered half-space.The IBEM algorithm includes using 2.5D full-space and half-space Green’s functions to construct scattered fields in decomposed closed and opened half-space regions,respectively,and using the dynamic stiffness method to solve the free fields.All regions are finally integrated by introducing the compatibility conditions to obtain the total wave fields.The proposed 2.5D IBEM has the flexibility in dealing with complex boundaries by directly applying the fictitious loads on the regions’boundaries,with a less storage requirement compared to the full 3D models.Besides,by combining the specific advantages of the two kinds of Green’s functions,the method is well suitable for handling coupled topographies with high accuracy.The method is validated by comparison with published results for a single valley as well as a single hill topography.The effects of height-to-width ratio of hill and layering on dynamic responses are further parametrically investigated by numerical implementations in frequency domain.Results show that the interaction between valley and hills can lead to a more significant amplification within the valley region,and dynamic responses are deeply influenced by the height-to-width of hill,simultaneously depending on incident angle and frequency.Besides,the site effects become more complex when the stratification feature is taken into account.