Stratified Half-space Research Articles

Conical indentation over a transversely isotropic and layered elastic half-space

We propose a novel method for solving the static response of a conical indenter on a transversely isotropic and layered elastic half-space. The newly developed Fourier-Bessel series (FBS) system of vector functions, along with the unconditionally stable dual-variable and position method, is employed to derive the Green's function in the transversely isotropic and layered elastic half-space under a vertical ring load on the surface. To calculate the response at different field points on the surface, we apply discrete love numbers within the FBS vector system. The load densities in the discretized rings within the contact radius of the conical indenter are determined using the integral least-square method, along with a self-adaptive algorithm developed in this study. Finally, the relationship between the indentation depth (vertical displacement) and the applied load is obtained through force balance between the external load and the summed contact traction. The developed scheme is validated using existing exact solutions for the reduced homogeneous half-space case. Selected numerical results clearly demonstrate the effect of anisotropic material and layering on the indentation response. It is observed that, regardless of whether the structure is a stratified half-space or a layered structure with a rigid substrate, the material properties in the top layer have the most significant influence on the indentation behavior. In the case of a layered structure with an underlying elastic half-space, the material properties in the interlayer and bottom layer could also affect the indentation behaviors.

Substrate effect on scattering by a chiral sphere

The effect of stratified substrates on scattering from a chiral sphere is analytically and numerically examined. A combination of vector Mie solution and the field transformations between vector spherical functions (VSFs) and plane waves (PWs) is used to determine the scattered fields of different orders. The generalized reflection coefficients of the stratified half-space and vector Mie solution for the chiral sphere are used to calculate successive interactions of the sphere and the interface and a series solution is obtained. Commercial software FEKO is used to numerically validate the derived expressions for the multilayered substrate. Various numerical results are presented and discussed. Specifically, a slab with exponential profile is considered as an example of an inhomogeneous substrate and its scattering coefficients are compared with the homogeneous case.

Analytic pavement modelling with a fragmented layer

ABSTRACT Layered elastic theory is a familiar and commonly accepted framework for modelling and analysis of pavement systems. In actuality however, the suitability of the theory is challenged whenever the system includes a fragmented layer – which violates the inherent layer continuity assumption. This paper addressed the modelling challenge in these situations by introducing a fragmented layer into the accepted framework. A new formulation was proposed, based on the hypothesis that the thickness of the fragmented layer can be incorporated as part of its governing parameters. Doing so allowed for treating the fragmented layer as a new kind of interface. Subsequently, the work focused on formulation development and clear step-by-step presentation of new derived expressions applicable to the case of a stratified half-space containing a fragmented layer. Several synthetic cases were generated and interrogated to verify the correctness of underlying equations and to demonstrate capabilities. Based on this interrogation the proposed formulation is deemed rich of prospective utility, especially for modelling pavements surfaced with concrete paving blocks, pavement systems that contain cracked-and-seated (or rubblised) Portland cement concrete, and aged or rehabilitated pavement systems that contain fatigued (multi-fractured) asphalt concrete layers.

Frequency-dependent impedance and surface waves on the boundary of a stratified dielectric medium.

We analyse waves propagating along the interface between half-spaces filled with a perfect dielectric and a Lorentz material. We show that the corresponding interface condition leads to a generalization of the classical Leontovich condition on the boundary of a dielectric half-space. We study when this condition supports propagation of (dispersive) surface waves. We derive the related dispersion relation for waves along the boundary of a stratified half-space and determine the relationship between the loss parameter, frequency and wavenumber for which interfacial waves exist.This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

Wave fields in the stratified fluid-saturated porous half-space induced by dislocation sources

In this study, the wave fields induced by both tensile and shear displacement dislocations in stratified fluid-saturated porous media are computed by using the reflection and transmission matrix method. The components of the source discontinuity vector across the source plane to describe those tensile and shear faults are explicitly displayed by using the surface vector harmonics. Numerical examples for a two layer model subjected to tensile and shear dislocations are provided. From the waveforms of surface displacements, the arrivals of transmitted and converted PS and SP waves at the interface of the two layer model can be clearly observed.

The effect of track load correlation on ground-borne vibration from railways

In predictions of ground-borne vibration from railways, it is generally assumed that the unevenness profile of the wheel and rail is fully correlated between the two rails and the two wheels of an axle. This leads to identical contact forces at the two rails and can allow further simplifications of the vehicle model, the track model and the track/ground interface conditions. In the present paper, the level of correlation of the track loading at the wheel/rail interface due to rail unevenness and its influence on predictions of ground vibration is investigated. The extent to which the unevenness of the two rails is correlated has been estimated from measurements of track geometry obtained with track recording vehicles for four different tracks. It was found that for wavelengths longer than about 3m the unevenness of the two rails can be considered to be strongly correlated and in phase. To investigate the effect of this on ground vibration, an existing model expressed in the wavenumber-frequency domain is extended to include separate inputs on the two rails. The track is modelled as an infinite invariant linear structure resting on an elastic stratified half-space. This is excited by the gravitational loading of a passing train and the irregularity of the contact surfaces between the wheels and the rails. The railway model is developed in this work to be versatile so that it can account or discard the effect of load correlations on the two rails beside the effects of variation of the tractions across the width of the track-ground interface and the vehicle sprung mass, as well as the roll motion of the sleepers and the axle. A comparative analysis is carried out on the influence of these factors on the response predictions using numerical simulations. It is shown that, when determining the vibration in the free field, it is important to include in the model the traction variation across the track-ground interface and the non-symmetrical loading at the two rails that occurs for unevenness wavelengths shorter than about 3m.

Image Solution for the 3-D Static Green’s Function of a Vertically Stratified Two-Layer Half-Space

The general form of 3-D static Green’s function for a general structure consisting of ${N}$ arbitrary adjacent dielectric wedges is derived using the Kontorovich-Lebedev and Fourier transforms. It consists of several point sources in physical space and a line image source in complex space for each region of the problem geometry. While the point sources can be treated conventionally, the line image distributions are obtained by extracting a proper asymptotic behavior of the solution, which can be numerically computed. As an example, the case of a vertically stratified two-layer half-space is investigated and the respective point and line image sources are obtained. The efficiency of the proposed solution method is demonstrated by comparing the computational time for a given structure with that required by a commercially available numerical code using the finite integration technique.

Propagation characteristic of laser-generated visco-elastic Rayleigh-like waves in stratified half-space

This paper reports on a study of the propagation characteristics of visco-elastic, Rayleigh-like waves in stratified half-space structures. Beginning with the Kelvin model, the characterization equation and the normal displacement of visco-elastic Rayleigh waves in stratified half-space structures are derived and the influence of the visco-elastic modulus on dispersion and attenuation is discussed. Theoretical calculations show that the attenuation-frequency curves perfectly match the phase-frequency curves. The effect of visco-elasticity on the attenuation of the Rayleigh-like wave is larger than its effect on dispersion. For “weak viscosity,” the attenuation is directly proportional to the viscosity modulus and the shear viscosity has a greater impact on the dispersion curves than does the bulk viscosity. The transient response of a visco-elastic Rayleigh wave is also simulated by means of Laplace and Hankel inversion transforms. The results are in good agreement with the theoretic predictions. It is believed the paper’s results and conclusions will provide insights and guidance for estimating visco-elastic parameters and for assessing adhesive quality.

On some problems of SH wave propagation in inhomogeneous elastic bodies

The paper deals with the propagation of shear horizontal (SH) waves in an nonhomogeneous elastic half-space composed of a layer whose shear modulus and mass density have a power dependence on the distance from the lower plane and the periodically stratified half-space. The equation which relates the wave speed to the wave-number and functions of the shear modulus and mass density is derived. The wave velocity is analyzed numerically. Especially, the influence of mechanical properties of the coating layer and the stratified foundation on the wave velocity is presented in the form of figures.

Solutions for Triangularly Distributed Harmonic Loadings on Axially Symmetric Area in Layered Half-Space

ABSTRACTThe paper is to solve the problem of the response of stratified half-space subjected to triangularly distributed time-harmonic loadings on an axial symmetric area in the stratified half-space. Since the distributed area can be very small, the solution can be employed to deal with the problems of elastodynamics using the fundamental solution method. The advantage of the presented solution over Green function is that no singularity will occur in the presented solution. Some numerical results are given and some conclusions have been made. The presented solution is very efficient in computation.

3D dynamics of a prestressed stratified half-space under the action of an oscillating moving load

This paper investigates the three-dimensional (3D) dynamical response to a time-harmonic oscillating moving load of a system comprising a prestressed covering layer and a prestressed half-space, by utilizing the three-dimensional linearized theory of elastic waves in initially stressed bodies (TDLTEWISB).The materials of the covering layer and half-space are assumed to be homogeneous and isotropic. It is also assumed that perfect contact conditions between the constituents of the system under consideration are satisfied. The analytical solution method, employing double exponential Fourier integral transformation with respect to the space coordinates, and the algorithm for obtaining the numerical results for the originals of the sought values are developed and employed. Numerical results for the critical velocity and for the stress distribution on the interface plane are presented and discussed. In particular, it is established that the minimal value of the critical velocity obtained for the 3D problem coincides with the critical velocity obtained for the corresponding 2D (plane-strain state) problem.

Inverse problem of electromagnetic-wave reflection for dispersion-free stratified half-space with a real permittivity

We propose a method for approximate reconstruction of the profile of the refractive index from the frequency dependence of the reflection coefficient for the cases where a discontinuity of the first kind exists at the interface of the media. The method is based on using accurate analytical solutions of the inverse problem for the model reflection coefficients represented as fractional rational functions. Numerical results of using the proposed method to process the plots of the real and imaginary parts of the reflection coefficient in the specified frequency range are presented.

On the characteristics of high-frequency Rayleigh waves in stratified half-space

SUMMARY The Rayleigh wave is dispersive in a stratified half-space. The modes of Rayleigh waves in a stratified half-space are usually defined by the order of the dispersion curves, such as fundamental mode, first higher mode, etc. However, the propagation characteristics of the same mode can be essentially different (e.g. surface wave and trapped wave) at different frequencies, and there is no rule to explain which parts of dispersion curves correspond to the surface wave or trapped wave. Actually, when the frequency is high, by decomposition of dispersion equation, three basic modes (R mode, R-period mode and S-period mode) of Rayleigh waves can be defined, and the mode can be further named by the phase velocities such as R1 mode, R1∼2 mode, etc. By the new definition, the dispersion curves can be divided into several velocity zones, and the modes can then be determined. In this paper, with generalized R/T coefficient method, the eigendisplacements of the three basic modes at high frequencies are studied. The study shows the physical advantage of the new definition of modes: Rayleigh waves of the same mode share the similar characteristics, and the main eigendisplacement characteristics of one mode remain the same when the medium is changed. With the definition of modes, the study makes it possible to know the basic eigendisplacement characteristics of high-frequency Rayleigh waves at different parts of the dispersion curves immediately. Especially, it presents a rule to explain which parts of dispersion curves correspond to the surface wave or trapped wave. Besides, based on the new definition, the modes in the vicinity of the osculation points are analysed. The ‘coupled modes’ that show the characteristics of two different modes simultaneously are found. It is found that the two neighbouring dispersion curves can exchange their corresponding modes of Rayleigh waves sometimes: one mode changes into another one gradually via coupled modes on each dispersion curve. The mode conversion can happen not only between surface wave mode and trapped wave mode, but also between two different trapped wave modes. However, when there is only one mode in the vicinity of the osculation points, there will be no mode conversion.

3D Analysis of Vibration Isolation by Wave Impeding Block in Non-Uniform Layered Ground under Horizontal-Rocking Coupled Loading

The thin-layered method (TLM) is very efficient to the wave propagation in layered ground and boundary element method (BEM) is very precise to solve infinite domain problems. In order to study the isolation vibration effectiveness of non-uniform three-dimensional (3D) layered ground wave impeding block (WIB), this paper is accomplished with the aid of a 3D semi-analytical BEM based on TLM, on which the Green’s function is regarded as fundamental solution of a stratified half-space. Then the active isolation effectiveness by WIB is studied under horizontal-rocking coupled loading in elastic half-space in the upper stiff-layer and lower soft-layer. The results show that the foundation of the stratification parameter and the characteristics of non-uniform have significant influence on the vibration isolation effect of the WIB, and the isolation vibration effectiveness of lower soft-layer is better than the upper stiff-layer.

Rigid Foundations Resting on an Elastic Layered Soil

A rigid foundation of arbitrary shape resting on a stratified half space (soil), with specific geometrical and elastic properties for its horizontal layers, is examined under the action of a vertical load placed anywhere on the foundation area. On the basis of a purely analytical treatment of the deformability of the soil surface, i.e. without using a finite element discretization of the layered soil mass, a general method is developed in order to determine the contact soil pressures and the resulting settlements through an easily implemented numerical procedure. The possible presence of “inactive” foundation parts not being in contact with the soil surface is taken into account. Parametric studies are performed showing the influence of the variability of the soil layers, as well as of the vertical load location, on the contact pressures developed. In this context, for the cases of a rectangular and a circular foundation the area borders for the location of a vertical load are determined, either for the development of compressive soil pressures all over the footing (core), or for allowing the existence of some inactive part without contact with the soil (zero contact pressure), not exceeding the half of the total foundation surface (limit core).

Impedance of surface footings on layered ground

Traditionally only the static bearing capacity and stiffness of the ground is considered in the design of wind turbine foundations. However, modern wind turbines are flexible structures with resonance frequencies as low as 0.2 Hz. Unfortunately, environmental loads and the passage of blades past the tower may lead to excitation with frequencies of the same order of magnitude. Therefore, dynamic soil–structure interaction has to be accounted for in order to get an accurate prediction of the structural response. In this paper the particular problem of a rigid foundation on a layered subsoil is discussed. Based on the Green’s function for a stratified half-space, the impedance of a surface footing with arbitrary shape is computed. A wind turbine foundation is analysed in the frequency range 0–3 Hz. Analyses show that soil stratification may lead to significant changes in the impedance related to both rocking and translation at frequencies close to the first resonance frequency of an offshore wind turbine.

Pressure field around a well in a stratified inhomogeneous bed

The steady axisymmetric flow of an incompressible fluid into a vertical well hydrodynamically perfectly drilled into a stratified inhomogeneous half-space consisting of three layers with different permeabilities is considered. The boundaries of the layers are assumed to be horizontal planes and the roof of the upper layer is assumed to be impermeable. The flow obeys a linear Darcy’s law. The pressure distribution on the well is assumed to be given, which is the main obstacle to finding an exact solution of the problem. Beginning with the classical studies of Muskat and Charnyi [1, 2], approximate solutions of such problems have been constructed as a superposition of flows generated by point sources with given intensities, distributed along the well axis in accordance with a fairly simple law. In the present study, this approach is used to obtain an integral equation for the source density distribution, which is then solved numerically. Comparison with the known exact solution for a thin elongated ellipsoid (“needle”) shows that this approach makes it possible to ensure an accuracy which at any rate is sufficient for applications.

Random wave propagation in a viscoelastic layered half space

An extremely efficient and accurate solution method is presented for the propagation of stationary random waves in a viscoelastic, transversely isotropic and stratified half space. The efficiency and accuracy are obtained by using the pseudo excitation method (PEM) with the precise integration method (PIM). The solid is multi-layered and located above a semi-infinite space. The excitation sources form a random field which is stationary in the time domain. PEM is used to transform the random wave equation into deterministic equations. In the frequency-wavenumber domain, these equations are ordinary differential equations which can be solved precisely by using PIM. The power spectral densities (PSDs) and the variances of the ground responses can then be computed. The paper presents the full theory and gives results for instructive examples. The comparison between the analytical solutions and the numerical results confirms that the algorithm presented in this paper has exceptionally high precision. In addition, the numerical results presented show that: surface waves are very important for the wave propagation problem discussed; the ground displacement PSDs and variances are significant over bigger regions in the spatial domain when surface waves exist; and as the depth of the source increases the ground displacement PSDs decrease and the regions over which they have significant effect become progressively more restricted to low frequencies while becoming more widely distributed in the spatial domain.

A precise numerical method for Rayleigh waves in a stratified half space

AbstractThis paper presents theory and results for Rayleigh waves propagating in a transversely isotropic stratified solid resting on an elastic semi‐infinite space. It uses the precise integration method (PIM) and the extended Wittrick–Williams (W–W) algorithm. This problem can be reduced into the eigenvalue problem of ordinary differential equations (ODEs) in the frequency–wavenumber domain. The PIM used here is a precise method for solving the ODEs with two‐point boundary conditions, and the eigenvalue counting method of the extended W–W algorithm is used to ensure that all eigenvalues are found without the possibility of any being missed. Hence the theory presented in this paper is exact, in the sense that its precision is limited only by the precision of the computer used. Copyright © 2006 John Wiley & Sons, Ltd.

An analytical model for ground vibrations from accelerating trains

An analytical approach is used to investigate ground vibrations due to accelerating and decelerating trains. The ground is modelled as a stratified half-space with linearly viscoelastic layers. On top of the ground, a rectangular embankment is placed, supporting the rails and the sleepers. The rails are modelled as Euler–Bernoulli beams where the propagating forces (wheel loads) are acting and the sleepers are modelled with an anisotropic Kirchhoff plate. The solution is based on Fourier transforms in time and along the track. In the transverse direction the field in the embankment is developed in Fourier series and the fields in the ground with Fourier transforms. The resulting numerical scheme is efficient and displacements for a wide frequency spectrum can be considered. Numerical examples are given for an X2 train that operates at the site Ledsgard in Sweden. In particular, the effects of the wheel traction from the driving wheel pairs or the braking wheels (all wheels) are accounted for. The results at some instantaneous train speeds are compared to corresponding constant train speeds.