Layered Half-space Research Articles

Rayleigh waves in a layered orthotropic elastic half-space with sliding contact

The presented paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary thickness. The layer and the half-space may be compressible or incompressible and they are in sliding contact with each other. The main aim of the paper is to derive explicit exact secular equations of the wave for four possible combinations: both the layer and the half-space are compressible or incompressible, one is compressible and the other is incompressible. When the layer and the half-space are both compressible, the explicit secular equation is derived by using the effective boundary condition method. For the three remaining cases, the explicit secular equations are deduced directly from this secular equation by using the incompressible limit technique. Based on the obtained secular equations, the effect of incompressibility and the sliding contact on the Raleigh wave propagation is considered through some numerical examples. It is shown that the incompressibility (of half-spaces and coating layers) and the sliding contact strongly affects the Raleigh wave velocity.

In-plane soil-structure interaction in layered, fluid-saturated, poroelastic half-space II: Pore pressure and volumetric strain

Parts I and II of this paper present a study of linear in-plane soil-structure interaction in fluid-saturated, poroelastic, layered half-space using the Indirect Boundary Element Method (IBEM). The structure is a shear wall supported by a rigid embedded foundation. Part I presented the methodology and results for the foundation and structure responses. Part II (this paper) presents a parametric study in the frequency domain of the pore pressure, and of the volumetric strain in the solid frame and the pore fluid (water in this case) along the soil-foundation interface and the ground surface. The presented results are for the special case of a single soil layer over bedrock, semi-circular foundation and zero seepage force. The analysis of peaks in the pore pressure response versus dimensionless frequency suggests that they occur near the system frequencies and near the soil layer resonant frequencies, to a degree that depends on the incident wave. For example, for vertically incident SV wave, they occur mostly near the system frequencies, while, for incident P wave, they occur also near the soil layer resonant frequencies. The results of this linear model may be useful for better understanding of how the soil-structure interaction contributes to pore pressure buildup near foundations of structures, which may eventually lead to soil liquefaction and severe consequences in the stability of the structure.

A fast and accurate method to compute dispersion spectra for layered media using a modified Kausel-Roësset stiffness matrix approach

This brief article presents a simple modification to the widely-used Kausel-Roësset Stiffness Matrix Method (SMM), and in particular to its implementation in the context of the Thin-Layer Method (TLM). This modification allows making fast and accurate computations of wavenumber spectra even for layered media underlain by infinitely deep half-spaces. As is well known, the TLM uses a finite element expansion in the depth direction, which in principle disallows exact representations of infinitely deep media other than through Paraxial Approximations or Perfectly Matched Layers. However, with the modification presented herein, that obstacle is removed. The very simple method is first presented and then demonstrated by means of examples involving layered half-spaces.

Harmonic response of layered halfspace using reduced finite element model with perfectly-matched layer boundaries

The current paper investigates the use of perfectly-matched layers (PML) as absorbing elements for a finite element (FE) model simulating a semi-infinite medium. Due to the formulation of the FE-PML model, it is also possible to use a Craig-Bampton reduction to significantly reduce the number of degrees-of-freedom in the model, in an attempt to improve the computational efficiency of the simulation. The perfectly-matched layers use a complex-valued function to numerically stretch the apparent size of PML elements while allowing them to remain nominally small. The results from this investigation suggest the PML elements worked seamlessly with the FE elements to approximate the elastodynamic response of a 3D halfspace subjected to a surface load; the wave energy is completely absorbed by the PMLs regardless of incident angle or wavelength. Furthermore, the size of the model was reduced by approximately 32% using a Craig-Bampton reduction (CBR). The CBR transforms the system into a mixed set of coordinates, including both modal and spatial coordinates. The model reduction is accomplished by neglecting modal frequencies for the system above two and a half times the maximum forcing frequency of interest. By only transforming the frequency-independent FE section into modal coordinates, and leaving the frequency-dependent PML elements as spatial degrees-of-freedom, the mode-shapes must only be solved once and can then be reused for different forcing frequencies. The results from this investigation suggest this could provide computational benefits if a number of cases are being computed for different frequencies.

Hybrid Asynchronous Perfectly Matched Layer for seismic wave propagation in unbounded domains

Perfectly Matched Layer (PML) is recognized as a very effective tool for modeling unbounded domains. Nonetheless, the computation time required by the PML may be large, especially when an explicit time integration scheme is adopted for dealing with the wave propagation problem both in the domain of interest and in the PML medium. In this paper, it is proposed to investigate subdomain strategies enabling the appropriate time integration scheme in the PML with its own time step to be chosen, independently of the choice of the time scheme in the domain of interest. We focus on explicit time integrator in the physical subdomain (Central Difference scheme) associated with a fine time step satisfying the CFL stability criterion. The PML formulation proposed by Basu and Chopra (2004) [1] for 2D transient dynamics, has been coupled with the interior physical subdomain using the dual Schur approach proposed by Gravouil and Combescure (2001) [2]. Hybrid (implicit time integrator for the PML) asynchronous (multi time steps) PMLs have been derived. Their very good accuracy has been shown by considering the following numerical examples: Lamb׳s test, loaded rigid strip footing on an half space and a layered half space.

Correction of an effect of using a faulty model for the retrieval of a bulk wave slowness of a layered half space

This study concerns the retrieval of a single, real constitutive parameter (the bulk shear wave inverse velocity) of a simple, although representative, geophysical configuration involving two homogeneous, non-dissipative media, from its simulated response to pulsed plane wave probe radiation. This nonlinear inverse problem is solved exactly, at all frequencies, by equating the simulated frequency domain response incorporating the true real inverse velocity to the assumed response incorporating a trial complex inverse velocity. Due to discordance, caused by prior (a parameter of a model that is not retrieved, but rather assumed to be known, although its value may be wrong) uncertainty, between the models associated with the assumed and trial responses (the model giving rise to the latter is therefore qualified as being ‘faulty’), the imaginary part of the retrieved inverse velocity turns out to be non-nil, and, in fact, to be all the greater, the larger is the prior uncertainty. Moreover, the retrieved inverse velocity is found to be dispersive and its real part nearly equal to the true inverse velocity at all frequencies. The reconstructed signals (obtained by inserting the retrieved complex parameter into the faulty response model) are found to coincide exactly with the true signals for three types of probe pulses, even when the prior uncertainty is large.

Efficient numerical model for the computation of impedance functions of inclined pile groups in layered soils

This paper introduces a numerical model to obtain the time-harmonic dynamic response of pile foundations in non-homogeneous soils. The model is based on the integral formulation of the elastic problem and the use of Green’s functions for the layered halfspace, and considers the soil as a group of zoned homogeneous, linear, isotropic, viscoelastic layers. The piles, on the other hand, are treated by finite elements as Timoshenko beams. Both formulations are coupled through the required compatibility and equilibrium equations along each pile. After being validated against previous results from the literature, the model is used to study the effects of soil non-homogeneity on the impedance functions of inclined piles and pile groups by considering different soil profiles whose properties vary with depth following a generalized power law. The impedance functions for three representative non-homogeneous soils are presented and compared with the ones of related homogeneous soils. Significant differences appear between the two situations for all studied rake angles. The magnitude of these differences strongly depends on the frequency range considered, specially for the case of pile groups, which shows the necessity of analyzing the problem using soil profiles as close as possible to the actual depth-varying ones.

Wave propagation of buried spherical SH-, P1-, P2- and SV-waves in a layered poroelastic half-space

Few studies have investigated the wave propagation of spherical sources in a layered half-space. In this paper, based on Biot's theory of poroelastic media, the exact anti-axisymmetric (cylindrical SH-waves) and axisymmetric (cylindrical P1-P2-SV waves) stiffness matrices for a layered poroelastic half-space are derived. Then the dynamic responses due to buried spherical SH-, P1-, P2- and SV-waves in a layered poroelastic half-space are studied by using the direct stiffness method combined with the Hankel transform. The present solutions are in good agreements with those in a uniform pure elastic half-space as well as a uniform poroelastic half-space. These solutions have the advantages that all of the parameters in the solutions have explicit physical meanings and the thickness of discrete layers does not affect the precision of calculation; thus, the presented formulations are very convenient for engineering applications. Numerical calculations are performed in both the frequency and time domains by taking buried spherical SH-, P1- and SV- waves in a uniform poroelastic half-space and in a single poroelastic layer over a poroelastic half-space as examples. The numerical results show that wave propagation of spherical sources in a layered half-space can be significantly different from that in a uniform half-space; the dynamic responses are highly dependent on the saturated parameters, vibration frequency and the surface drained condition; the presence of the underlying half-space makes the time histories of the dynamic responses in a single layered half-space much more complicated with much longer duration.

Dynamic Response Analysis of Periodic Alluvial Valleys under Incident Plane SH-Waves

Combined with the region-matching technique, a periodic indirect boundary element method (PIBEM) is developed to investigate the scattering and diffraction of plane SH-waves by periodic alluvial valleys embedded in a layered half-space. By introducing Green’s functions for equivalent uniformly distributed loads acting on an inclined line to simulate the scattered fields outside the periodic valleys, the discretization effort is reduced to a single valley. And by using only a single irregularity as the computation cell to solve wave scattering problem of periodic irregularities (an infinite number of irregularities), the presented new method has the merits of high accuracy, low computational cost in CPU time, and low memory requirement. The implementation of the method is described in detail, and the accuracy of the method is verified by comparing its degenerated results with the analytical results of a single semi-elliptic valley, the semi-analytical results of two valleys and the IBEM results of multiple valleys. Numerical simulations are first performed in the frequency domain and then in the time domain using the inverse FFT. In addition, the time domain results reinforce the validity and reliability of the new method. The presented PIBEM can be used to study the various wave scattering problems of plane SH-waves by periodic topographies in a layered half-space.

Scattering and diffraction of plane SH-waves by periodically distributed canyons

A new method is presented to study the scattering and diffraction of plane SH-waves by periodically distributed canyons in a layered half-space. This method uses the indirect boundary element method combined with Green’s functions of uniformly distributed loads acting on periodically distributed inclined lines. The periodicity feature of the canyons is exploited to limit the discretization effort to a single canyon, which avoids errors induced by the truncation of the infinite boundary, and the computational complexity and the demand on memory can be significantly reduced. Furthermore, the total wave fields are decomposed into the free field and scattered field in the process of calculation, which means that the method has definite physical meaning. The implementation of the method is described in detail and its accuracy is verified. Parametric studies are performed in the frequency domain by taking periodically distributed canyons of semi-circular and semi-elliptic cross-sections as examples. Numerical results show that the dynamic responses of periodically distributed canyons can be quite different from those for a single canyon and significant dynamic interactions exist between the canyons.

Propagation of SH waves in a visco-elastic layer overlying an inhomogeneous isotropic half-space

This paper is concerned with the propagation of SH-waves in an inhomogeneous viscoelastic layer overlying an inhomogeneous isotropic half-space. For the study of viscoelastic layer Kelvin–Voigt type model has been considered. The exponential type inhomogeneity is considered in viscoelastic layer with different parameters for viscoelastic part and for density of medium whereas for the half-space, it is quadratic type with single inhomogeneity parameter for rigidity and density. The solutions are obtained analytically for both the layer and half-space. Dispersion relation is obtained and subjected to continuity conditions at interfaces of viscoelastic layer and half-space and the upper boundary of layer as free surface. The effects of the inhomogeneity parameter are studied for both viscoelastic layer and inhomogeneous isotropic half-space. The numerical results are shown by plotting the graph between phase velocity and wave number for different values of inhomogeneity parameter and for the variation of depth of the layer.

An Improved Formula of Fundamental Resonance Frequency of a Layered Half-Space Model Used in H/V Ratio Technique

The resonance frequency of the transmission response in layered half-space model is important in the study of site effect because it is the frequency where the shake-ability of the ground is enhanced significantly. In practice, it is often determined by the H/V ratio technique in which the peak frequency of recorded H/V spectral ratio is interpreted as the resonance frequency. Despite of its importance, there has not been any formula of the resonance frequency of the layered half-space structure. In this paper, a simple approximate formula of the fundamental resonance frequency is presented after an exact formula in explicit form of the response function of vertically SH incident wave is obtained. The formula is in similar form with the one used in H/V ratio technique but it reflects several major effects of the model to the resonance frequency such as the arrangement of layers, the impedance contrast between layers and the half-space. Therefore, it could be considered as an improved formula used in H/V ratio technique. The formula also reflects the consistency between two approaches of the H/V ratio technique based on SH body waves or Rayleigh surface waves on the peak frequency under high impedance contrast condition. This formula is in explicit form and, therefore, may be used in the direct and inverse problem efficiently. A numerical illustration of the improved formula for an actual layered half-space model already investigated by H/V ratio technique is presented to demonstrate its new features and its improvement to the currently used formula.

Reduction of train-induced vibrations by using a trench in a layered half-space

A coupled model of a track-layered ground-in-filled trench system is developed to investigate the isolation effects of an in-filled trench on reducing vibrations generated by moving train loads. By using the substructure method, the whole model is decomposed into two parts: the layered ground containing the in-filled trench and the track. Firstly, the flexibility coefficient for moving uniformly distributed loads applied on the layered ground containing the trench is obtained by using the 2.5D indirect boundary element method. Then, the dynamic equilibrium equation of the track under the moving train loads and uniformly distributed loads are established. Finally, the layered ground and the track are coupled according to the equivalence between the deformation of the track and the vertical displacement of the layered ground. The validity of the method is confirmed by comparing its results with the published ones. Numerical calculations are performed by embedding an in-filled trench in a homogenous ground, in a single layered ground and also in the real site at Ledsgard as examples. The results show that the isolation effects are different for different ground conditions and for different geometric parameters such as the depth, width and location of the in-filled trench.

Axisymmetric frictionless contact of a functionally graded piezoelectric layered half-space under a conducting punch

This paper studies the axisymmetric frictionless contact of a functionally graded piezoelectric layered half-space subjected to three typical rigid punches, i.e., flat circular punch, spherical punch and conical punch . The punch is assumed as a perfect conductor with a constant electric potential. The electro-mechanical properties of the functionally graded piezoelectric materials (FGPMs) vary exponentially along the thickness direction. By employing the Hankel integral transform technique, the problem is reduced to a set of singular integral equations of the first kind. These equations are then discretized and solved numerically to determine the unknown normal contact pressure and electric charge. The effects of the gradient index and punch geometry on the surface electro-mechanical contact behaviors are discussed in detail.

Wave scattering of complex local site in a layered half-space by using a multidomain IBEM: incident plane SH waves

Wave scattering of complex local site in a layered half-space by using a multidomain IBEM: incident plane SH waves

Responses of an anisotropic magnetoelectroelastic and layered half-space to internal forces and dislocations

Layered multiferroic heterostructures have applications in various mechanical and electronic devices where the coupling among the mechanical, electric and magnetic fields provides unique opportunity for tailoring the device properties. In this paper, we investigate the coupled magnetoelectroelastic (MEE) field in a layered anisotropic MEE half-space induced by the internal force (traction) and dislocation. Utilizing the Fourier transform and propagator matrix method, we derive the induced field quantities in terms of the Stroh formalism. The analytical solution in the corresponding homogeneous MEE half-space is also derived to remove the concentration/singularity in the layered system. Numerical examples are presented in terms of contours for the extended displacement and stress fields in a three-layered MEE half-space made of piezoelectric and piezomagnetic layers induced by an internal force and dislocation applied over a horizontal circle. These results not only show various interesting features, they can also serve as benchmarks for future numerical analysis and perhaps as guidance for design engineers.

The effect of initial stress on the propagation of surface waves in a layered half-space

In this paper the propagation of small amplitude surface waves guided by a layer with a finite thickness on an incompressible half-space is studied. The layer and half-space are both assumed to be initially stressed. The combined effect of initial stress and finite deformation on the speed of Rayleigh waves is analyzed and illustrated graphically. With a suitable simple choice of constitutive law that includes initial stress, it is shown that in many cases, as is to be expected, the effect of a finite deformation (with an associated pre-stress) is very similar to that of an initial stress (without an accompanying finite deformation). However, by contrast, when the finite deformation and initial stress are considered together independently with a judicious choice of material parameters different features are found that do not appear in the separate finite deformation or initial stress situations on their own.

The dispersion of Rayleigh waves in orthotropic layered half-space using matrix method

In this paper, the secular equation of Rayleigh surface waves propagating in an orthotropic layered half-space is derived by the matrix method. All the layers and the half-space are assumed to have identical principle axes. The explicit form of the matrizant for each layer is obtained by the Sylvester's theorem. The derived secular equation takes only real values and depends only on the dimensionless variables and dimensionless material parameters. Hence, it is convenient in numerical calculation.

Impact of coseismic deformation fields with different time scales on finite fault modelling in 2010 California Baja Earthquake

In this study, the static coseismic deformations with different time scales in the target event are extracted from the long term and short term solutions of continuous GPS and the epoch-by-epoch solution of high rate GPS respectively. The mean discrepancies among these deformations are less than 1 cm. The earthquake source parameters are inverted in the layered inhomogeneous half-space under the static dislocation theory from the static coseismic displacements with different time scales. It can be concluded that the time scale of the static coseismic displacement shows a relatively small impact on the source parameters in this moderate earthquake. The disturbances on the fault slips are less than 20 cm, and less than 0·1 on the moment magnitudes. As the time scale grows, the postseismic deformation in the static coseismic displacement from the short term solution increases, which indicates that apparent early postseismic displacements exist on the day after the termination of the earthquake.

Effect of loosely bonded undulated boundary surfaces of doubly layered half-space on the propagation of torsional wave

This paper investigates the propagation of torsional wave in an initially stressed poroelastic layer with corrugated as well as loosely bonded boundary surfaces, sandwiched between a corrugated fiber-reinforced layer and a viscoelastic half-space under initial stress. The velocity equation has been obtained in closed form analytically and the substantial effect of affecting parameters on the phase velocity of torsional surface wave has been demonstrated numerically and graphically. Comparative study has been made to observe the effect of flatness parameter, reinforcement, viscoelasticity and porosity on the phase velocity, meticulously. Some particular cases have also been discussed and it is found that velocity equation is in well-agreement to the classical Love wave equation. Moreover, some remarkable observation has been made through numerical computation and graphical demonstration for fiber-reinforced layer of carbon fiber-epoxy resin, poroelastic layer of sandstone and a viscoelastic half-space.