Layered Half-space Research Articles

Nonlinear Rayleigh wave propagation in a layered half-space in dual-phase-lag

We investigate nonlinear Rayleigh wave propagation in a layered thermoelastic medium composed of a slab rigidly bonded to the surface of a half-space under prescribed external thermal boundary conditions within the dual-phase-lag theory. The heat conduction coefficient for both the slab and the matrix have a linear dependence on temperature. Our aim is to assess the effect of temperature dependence of the heat conductivity, as well as the thermal relaxation times, on the process of wave propagation in the layered medium. Poincaré expansion of the solution in a small parameter and the generation of higher harmonics allow to evaluate the coefficient of this nonlinear coupling in the slab through heat wave propagation measurement. For the used numerical values, the results show that some characteristics of the problem, e.g. the temperature, heat flux and one stress component suffer jumps at the interface, while the other stress components are continuous there. The jump in the heat flux is noticeable only in the first order of nonlinearity. The existence of jumps at the interface may be of interest for measurements. Comparison with the case of the half-space showed that the presence of the slab contributes to faster damping of the solution with depth in the half-space.

The influence of the modes of surrounding buildings on the ground vibration from railways

To improve the accuracy of railway induced ground vibration predictions, the influence of surrounding buildings should be considered. For this reason, a fast semi-analytical model is developed to investigate the vibration of the ground and the influence of a nearby building. The model contains two sub models: a finite element model which can simulate the building, and a semi-analytical model which can simulate the infinite free-field ground. The dynamic stiffness matrix method is used to simulate the ground as a homogeneous or layered half-space. The ground model works in the frequency-wavenumber domain and is based on the Green's functions developed for a half-space. Focusing on a building with a piled foundation, the coupling forces due to soil-structure interaction at coupled pile nodes are calculated assuming that the soil and pile displacements are identical. The results from the proposed model show that the vertical modes of the building, especially those that involve the vertical movement of the floors, can significantly influence the vibration of the surrounding ground. The findings suggest that when predicting the ground surface response induced by the railway, the surrounding buildings and their vertical modes should be included in the calculations.

Application of the Unscented Kalman Filter to Estimate the Material Properties of a Layered Half-Space

Full waveform inversion (FWI) to estimate physical properties of a system is one of the major research topics in science and engineering. This study proposes a probabilistic approach toward these solutions by applying the unscented Kalman filter (UKF). The responses of the vertical displacements of a layered half-space subjected to a harmonic vertical disk load on the surface are calculated from an estimated profile of shear-wave velocities and compared with measurements or observations. In the calculation of the dynamic responses, the thin-layer method (TLM), which is efficient for layered media, is employed. In order to improve the solutions to the considered inverse problem, regularization terms are included in the observations so that the differences in the material properties between two consecutive layers vanish. The proposed UKF method is demonstrated with a variety of FWI problems in a layered half-space. The results show that the proposed method can estimate the material properties of a layered half-space accurately.

Tabakalı Zeminlerde Tekerlek Temas Gerilmesi ve Deplasman Dağılımlarının Üç Boyutlu Sınır Eleman Yöntemi ile Belirlenmesi

In this study, the Boundary Element Method (BEM) was employed for the numerical determination of the response of a layered half-space. The material behaviour of the soil was assumed to be isotropic and linear elastic. The BEM was used in the Fourier transform space (FTS). The focus of this paper is to determine the stress and deflection distributions of interior points of a layered half-space. To achieve this aim, in this study, a computer program is developed for three-dimensional elastic or visco-elastic problems. The results of stress and deflection distributions in a layered half-space determined using boundary element formulation are presented in figures.

Forest trees as a natural metamaterial for surface wave attenuation in stratified soils

Forest trees have been shown to act as a natural metamaterial to attenuate elastic waves. However, the existing studies assumed that forest trees are located on homogeneous soils and did not consider the soil stratification that widely exists in natural strata. This study investigates the forest trees as a natural metamaterial to manipulate the propagation of elastic waves in stratified soils. A semi-analytical method is proposed to calculate the dynamic responses of a layered half-space coupled with arbitrary number of trees. A close-form Green’s function for a harmonic line load on a layered half-space is derived by the thin layer method, and subsequently used to determine the multiple scattered wavefield induced by forest trees as surface oscillators. The attenuation performance of forest trees on stratified soils is systematically investigated. Trees with a graded array of heights can provide a wide attenuation zone at frequencies below 80 Hz. Since the higher surface wave modes in a layered half-space can propagate forward without being affected by trees, the attenuation efficiency of trees on stratified soils is significantly lower than that of trees on corresponding homogeneous soils. The performance uncertainty induced by the randomness of the height, sizes, and spacing of trees is subsequently investigated through a Monte Carlo simulation. The forest trees can still obtain a significant attenuation efficiency at frequencies below 80 Hz even the randomness of tree properties being considered. The randomness in height and radius leads to significant uncertainty in attenuation performance, while the randomness in spacing results in reasonable uncertainty.

Exact artificial boundary condition and 1D time-domain method for seismic response analysis of layered saturated poroelastic half-space

This study proposed a one-dimensional (1D) time-domain finite element method (FEM) for seismic response analysis of a layered saturated soil half-space under obliquely incident plane waves, based on Biot's u-w wave equations. The spatially two-dimensional problem was transformed into an equivalent 1D time-domain problem along the depth according to Snell's law. An exact artificial boundary condition for saturated poroelastic media was developed to simulate the wave absorption and input effects of the truncated half-space. Using the displacement and stress continuity conditions between the adjacent saturated poroelastic-medium layers and combining all finite elements and the exact artificial boundary condition, the total dynamic equation for the spatially 1D problem was obtained. The 1D dynamic equation was solved using the FEM with explicit time integration. The proposed method was verified by analyzing two engineering sites in the time domain and a comparison with the frequency-domain transfer matrix method with fast Fourier transform.

Probabilistic Full-Waveform Inversion Considering the Discontinuities of the Material Properties of a Layered Half-Space

The goal of full-waveform inversion (FWI) is to extract quantitative information pertaining to the material and geometrical properties from measured wave fields with their full contents considered. As an inverse problem, FWI itself is associated with ill-posedness, which means instability and nonuniqueness of the solution, because the fields are measured only at some locations. Owing to these unfavorable conditions, several regularization techniques have been utilized for stable estimations, but it is difficult when using a regularization technique to select a regularization factor, which has no physical meaning. In this paper, the FWI approach is applied to estimate the material properties of a layered half-space. The thin-layered method, an efficient analysis method for wave propagation in a layered medium, is employed for this purpose. It is formulated in the maximum a posteriori (MAP) form, and Tikhonov regularization is expressed as a prior distribution, whose standard deviation replaces the role of the regularization factor to ensure an intuitive meaning. Additionally, a technique to remove outliers is newly introduced for the regularization technique; it captures discontinuities accurately in the material properties of a layered half-space. Through this additional technique, the limitation of the smoothness effect in existing regularization techniques is alleviated. The proposed MAP approach including the regularization-based prior and outlier-removal technique is applied to several numerical examples to estimate the profiles of the shear velocities of a layered half-space. The numerical investigations conducted here confirm the validity of the proposed approach with satisfactory accuracy.

Propagation of Love Waves in a Plane Layered Half-Space with Nonrigid Contact at the Boundaries

Propagation of Love Waves in a Plane Layered Half-Space with Nonrigid Contact at the Boundaries

Transient Green's functions of dislocations in transversely isotropic and layered poroelastic half-spaces

We derive, for the first time, the transient response (or Green's function, GF) induced by a general point dislocation in a transversely isotropic and layered poroelastic half-space where the contributions from the fluid dislocation and fluid-phase coupling effect are considered. The GF is expressed in terms of the powerful Fourier-Bessel series system of vector functions recently introduced (with the expansion coefficients being the novel dislocation Love numbers). The corresponding source functions, i.e. discontinuities across the source level, are derived, which show that 1) the fluid-phase creates a new type of source functions called fluid dislocation, and 2) it further contributes directly to the traditional solid horizontal tensile dislocation (or tensile-fracture) via the Biot effective stress coefficient. While the dual-variable and position (DVP) method is applied to take care of multilayers, the Talbot's method is employed to carry out the inverse Laplace transform, both showing excellent numerical stability, efficiency, and accuracy. Key features of these GFs are analyzed numerically. It is shown that 1) the poroelastic process is featured by some transient statuses, including drained and undrained limits; 2) while these two limits are sharply different when the dislocation source is a vertical strike-slip or horizontal tensile-fracture, they are the same when a vertical dip-slip or vertical tensile-fracture is in a homogeneous half-space; 3) in all the cases, there are temporal poroelastic deformations which are significantly different from these two limits; and 4) the fluid dislocation alone could significantly contribute to the poroelastic deformation at the earlier time history. These GFs provide the kernel functions in the corresponding boundary element formulation and the method of fundamental solutions, with potential applications in geomechanics and biomedical engineering.

Elastodynamic response of an orthotropic layered elastic half-space to time-harmonic loading.

The elastodynamics of an orthotropic half-space coated by a thin orthotropic layer is theoretically investigated in this article. We newly propose explicit expressions of free Rayleigh waves in a layered half-space that are dependent on only one unknown constant representing amplitude. The main contribution is on deriving, in a simple manner, the theoretical predictions of far-field Rayleigh wave motion arising from time-harmonic loads using elastodynamic reciprocity theorems. These are the very first closed-form exact solutions found for the forced motion of Rayleigh waves in a layered half-space of orthotropic materials. To demonstrate the theoretical results, computation of Rayleigh wave motion in a jointed rock, including a layer of quartz-schist and a half-space of soil, is considered. We present the phase and group dispersion curves superimposed with the amplitude spectra that provide useful information on wave modes, frequencies, and displacement amplitudes. The inclusion of the amplitude spectra in the dispersion curves is a significant improvement over other dispersion curves currently available in the literature. The analytical predictions are compared with numerical results found by finite element analysis, and they show excellent agreement for the cases of a uniform distributed load and a varying distributed load both applied over a strip on the layer surface. The calculations obtained in the current study could generally be very useful for applications in seismology and materials characterization of coated structures.

Pyprop8: A lightweight code to simulate seismic observables in a layered half-space

pyprop8: A lightweight code to simulate seismic observables in a layered half-space

Ground-borne vibration screening in layered dry and saturated grounds using optimal inclined wave barriers

This paper aims to design optimal inclined wave barriers for screening vibrations in homogeneous and layered grounds with different groundwater table levels. The wave propagation phenomenon in the dry and saturated layered half-space is modeled using finite element numerical simulations considering elastodynamic and Biot's poroelastodynamic theories in dry and saturated layers, respectively. The inclined barriers' geometry and position are optimized using Covariance Matrix Adaption Evolution Strategy (CMA-ES), in which the finite element simulations are incorporated to find the fitness function. The studied grounds are subjected to dynamic loadings with different frequencies, and they are considered to have single or multiple layers with different groundwater levels. It is observed that the barriers perform better at larger frequencies and tend to locate close to the loading area to screen the waves actively. Also, the optimal barriers alleviate the environmental nuisance caused by the ground-borne vibrations significantly.

A numerical method based on Polonsky-Keer's conjugate gradient iterative scheme for solving axisymmetric contact of layered half-space

Axisymmetric contact analysis of layered half-space is of considerable interest in many tribological applications including protective hard coating, metal-matrix composites, etc. Polonsky-Keer's iterative scheme based on conjugate gradient method (CGM) is powerful for a general three-dimensional contact problem using rectangular meshes, but would inevitably lead to excessive computational burden and inaccurate boundary discretization for axisymmetric contact. In this paper, Polonsky-Keer's CG-based iterative scheme is adapted for axisymmetric indentation of a layered elastic half-space by taking advantage of annular ring discretization, where the elementary solutions for the surface deflection and subsurface stresses can be determined with assistance of the Hankel transforms. The present method can not only effectively circumvent the discretization error, but is also versatile and flexible to solve the contact between the layered materials and an arbitrary axisymmetric indenter. Parametric studies are provided to validate the present solutions, and the comparisons of the average relative errors for different types of elements are conducted to demonstrate the excellent convergence and accuracy of the proposed numerical scheme.

An efficient full-wavefield computational approach for seismic testing in a layered half-space

Inversion of subsurface shear-wave velocity profile based on effective dispersion curve or full wavefield is still rare due to the lack of efficient forward computation kernel for the full wavefield in a layered medium. To facilitate exploiting the complete signal content in seismic testing, an efficient new computational framework for modeling synthetic full wavefields in horizontally layered systems is proposed. Dynamic response of an isotropic elastic layered system subjected to vertical loading is implemented and thoroughly validated. The new procedure is based the Fourier-Bessel series expansion (not the traditional Fourier-Bessel transform), and it is at least two-orders faster than the current full wavefield computation method based on integral transformation. It considers survey parameters and models all wave phenomena, including near-field effect and leaky waves, which cannot be captured by the modal summation method. Examples of full wavefield simulation are presented to demonstrate different levels of information provided by the experimentally available fundamental mode, broadband effective dispersion curve, and the full velocity spectrum. Different earth models may have similar fundamental modes or even effective dispersion curves; however, their difference manifests in different higher-mode contributions and can only be exploited by utilizing the entire velocity spectra including the vertical and radial components. The coupling between the vertical and radial components in terms of their spectral ratio and phase difference can provide additional constraints (data) in seismic testing.

Study of the SH-wave propagation in an MEFR layer bounded by heterogeneous viscoelastic layer and elastic half-space

PurposeThe purpose of this study is stated regarding the impact of the horizontally polarized shear wave vibration on a composite medium in the terms of phase and damped velocity.Design/methodology/approachThe assumed composite is composed of magneto-elastic fiber-reinforced (MEFR) layer constrained between heterogeneous viscoelastic layer and heterogeneous elastic half-space. The considered heterogeneity is associated with the directional rigidity and mass density in the uppermost layer and half-space of quadratic and trigonometric types, respectively. The coupled field equations related to the respective medium are solved analytically by employing the method of separation of variables.FindingsThe dispersion relation of the stated problem is secured by using the continuity assumptions, imposed at the stress-free surface and the interfaces of the expressed medium. The adopted numerical examples are used to compute the dispersion relation and plot the graphs between phase/damped velocity and wave number. Parametric studies on the phase and damped velocity yield five main conclusions: (1) Phase velocity decreases with increasing value of wave number and damped velocity increases up to a certain number and then starts falling simultaneously with increasing magnitude of wave number while keeping the rest parametric values fixed. (2) The presence of heterogeneity in the upper layer enhances the phase velocity and diminishes the damped velocity, but the presence of heterogeneity in the half-space enhances both the phase and damped velocity. (3) The appearance of reinforced parameters enhances the phase velocity for the considered crystalline graphite material and diminishes the phase velocity for the rest materials (carbon fiber-epoxy resin and steel) of the MEFR layer. Similarly, damped velocity decreases for the assumed crystalline graphite material of the MEFR layer and increases for the rest materials of the MEFR layer. (4) The induced dissipation factor due to viscoelastic property shows reversal decreasing and increasing effect on phase and damped velocity of SH-wave. (5) Ascending values of the angle at which the wave crosses the magnetic field increase the phase velocity and decrease the damped velocity for all the considered MEFR examples.Originality/valueTill date, the mathematical modeling as well as vibrational analysis of wave propagation through the composite structure consisting of MEFR layer constrained between viscoelastic media and elastic half-space under the effect of different varying properties with depth remains a new challenging issue for the researchers around the globe. The current analysis is an approach to move ahead in the era of wave propagation in different realistic models based on their parametric studies. Also, these studies are very helpful to find their applications in the field of mechanical, construction, aerospace, automobile, biomedical, marine, manufacturing industries and many branches of science and technology where magnetic fields induced in elastic deformation occur.

3D structure-soil-structure interaction in an arbitrary layered half-space

The aim of the study is to evaluate the 3D dynamic response of a finite geological region containing two structures and rested on a semi-infinite elastic layered half-space with a dynamic source radiating transient waves. The hybrid modelling approach is applied. It is based on the decomposition of the whole domain under consideration into two sub-regions: a finite-sized near-field elastic isotropic zone with two containment structures and the open semi-infinite far-field layered region. The far-field zone is a semi-infinite elastic isotropic arbitrary layered medium where the near-field finite geological region is located on. The 3D hybrid computational tool is based on the boundary element method (BEM) for the far-field layered zone and the finite element method (FEM) for the finite near-field domain. The model for the semi-infinite layered zone is further extended by the incorporation of a new condensation algorithm which makes it possible to handle 3D wave propagation through arbitrary layered half-space. The condensation algorithm is developed to avoid high computational memory cost while retaining the compatibility with the hybrid FEM-hosted procedure which facilitates the useful solution for the practical three-dimensional engineering problems. The BEM model of the dynamically active far-field zone is inserted as a macro-finite element (MFE) in the FEM commercial program ABAQUS. The accuracy and convergence study of the hybrid numerical scheme is presented. Numerical simulations convincingly illustrate that the dynamic response of structure-soil-structure system depends on different key factors and their mutual interplay. These factors are arbitrary layering of the far-field geological zone, the characteristics of the dynamic source, the site effects phenomena, the structure-soil-structure dynamic interaction, the type and geometrical disposition of foundations and structures and the 3D features of the dynamic motion.

Deformation induced by distributions of single forces in a layered half-space: [formula omitted

In the present paper we introduce a numerical model for the representation of displacement, strain and stress due to single forces embedded in a layered elastic half-space. The code EFGRN/EFCMP (Elastic Forces GReeN functions/Elastic Forces CoMPutation) is able to represent the mechanical effects due to pre-assigned distributions of single forces. Even if internal deformation sources can be described by distributions of equivalent body forces with vanishing resultant and moment, single forces are employed in geophysics to represent hydraulic and/or lithostatic loads, effects of internal density anomalies, and even some kind of seismic events. A distribution of single forces is also used to describe the effects of an inelastic inclusion located inside an elastic medium. In fact, the recent literature shows that poro-elastic and thermo-elastic inclusions can be represented using single forces distributed on their boundaries. EFGRN/EFCMP shares the benefits of rapid and semi-analytical calculation offered by the parent code, EDGRN/EDCMP, which is instead suitable for the representation of extended dislocation sources, as seismic faults. The present code also provides an option for computing the effects of a distribution of single forces embedded in a homogeneous half-space, by using the analytical solutions of Mindlin. Accordingly, EFGRN/EFCMP can be a valid support both for the representation of forward models of deformation sources and for the procedures of inversion of geodetic data in a layered medium. We show some applications of the code and we provide several scripts in MATLAB language which help the user to quickly start using EFGRN/EFCMP.

Rayleigh wave scattering at shallow cavities in layered half spaces

For layered half spaces in which the shear wave (S-wave) velocity of layer increases with the layer depth, the subsurface seismic wave field generated by surface sources is dominated by the fundamental mode Rayleigh waves (R-waves). When a cavity full of air is present in the underlying half space, the forward (or incident) R-waves are scattered. The scattered wave field in front of the cavity is dominated by the back-scattered R-waves and the interference fringes are formed. Under the conditions that the length (i.e. horizontal extent) of cavity is enough large relative to the wavelength and the burial depth, the majority of the diffracted waves over the cavity approach the fundamental mode plate waves. The displacement structures (variations of displacement with depth) of the diffracted waves are different from those of R-waves within the burial depth. The differences could result in the decrease of the spectral amplitude over the cavity relative to that in front of the cavity. In the surface wave-field behind the cavity, the diffracted waves under the cavity could cause the increase of the spectral amplitude relative to that over the cavity at some low frequencies. By transforming the spectrum into the wavelength domain, a method is proposed to estimate the horizontal extent and burial depth of a cavity from the spectral changes in the offset-wavelength domain. The feasibility is verified by a preliminary experiment. ∙ The normalized depth of energy distribution in a layered half space is smaller than that in a half space with the Poisson’s ratio equal to that of the first layer. ∙The majority of the diffracted waves over a cavity behave as the fundamental mode plate waves under conditions that the length of cavity is large compared to the wavelength and the burial depth. ∙The spectral changes over a cavity are explained using displacement structure differences between R-waves and the diffracted waves. ∙The spectral density over a cavity is weaker than that in front of the cavity. ∙In the wave field behind a cavity, the frequency content relatively increases for the waves with the wavelengths larger than the burial depth. ∙The location and burial depth of cavity can be estimated from the spectral changes in front of and behind a cavity in the offset-wavelength domain.

Waves in a “Thin Layer–Half Space” System with Mixed Boundary Conditions

Waves in a “Thin Layer–Half Space” System with Mixed Boundary Conditions

Investigation of saturation effects on vibrations of nearly saturated ground due to moving train loads using 2.5D FEM

An improved two-and-a-half dimensional finite element method (2.5D FEM) model is developed to predict ground-borne vibrations in coastal areas from passing high-speed trains (HSTs). The model treats the ground as nearly saturated layered half-space, and describes the pore water/air mixed fluid by the equivalent fluid approach. Reliability of the proposed model is verified by comparing with the field measurements for X-2000 train and published analytical solutions. On this basis, the ground vibrations and excess pore water pressure under different train speeds and on various soil cases are analyzed in detail. In particular, three kinds of water saturation are discussed to show the sensitivity of vibration responses to the minor change of saturation. Results show that the saturation would cause pronounced influence on vibration displacements and excess pore pressure, and a minor decrease in water saturation would lead to a much higher vibration displacement level and a big difference of excess pore pressure distribution along depth. The influence of saturation is dependent both on soil configurations and on train speeds. It is highlighted that the vibration level on coastal ground would be underestimated by using fully saturated model instead of the nearly saturated model.