Layered Half-space Research Articles

Broadband strong motion simulation in layered half-space using stochastic Green’s function technique

The stochastic Green’s function method, which simulates one component of the far-field S-waves from an extended fault plane at high frequencies (Kamae et al., J Struct Constr Eng Trans AIJ, 430:1–9, 1991), is extended to simulate the three components of the full waveform in layered half-spaces for broadband frequency range. The method firstly computes ground motions from small earthquakes, which correspond to the ruptures of sub-faults on a fault plane of a large earthquake, and secondly constructs the strong motions of the large earthquake by superposing the small ground motions using the empirical Green’s function technique (e.g., Irikura, Proc 7th Japan Earthq Eng Symp, 151–156, 1986). The broadband stochastic omega-square model is proposed as the moment rate functions of the small earthquakes, in which random and zero phases are used at higher and lower frequencies, respectively. The zero phases are introduced to simulate a smooth ramp function of the moment function with the duration of 1/fc s (fc: the corner frequency) and to reproduce coherent strong motions at low frequencies (i.e., the directivity pulse). As for the radiation coefficients, the theoretical values of double couple sources for lower frequencies and the theoretical isotropic values for the P-, SV-, and SH-waves (Onishi and Horike, J Struct Constr Eng Trans AIJ, 586:37–44, 2004) for high frequencies are used. The proposed method uses the theoretical Green’s functions of layered half-spaces instead of the far-field S-waves, which reproduce the complete waves including the direct and reflected P- and S-waves and surface waves at broadband frequencies. Finally, the proposed method is applied to the 1994 Northridge earthquake, and results show excellent agreement with the observation records at broadband frequencies. At the same time, the method still needs improvements especially because it underestimates the high-frequency vertical components in the near fault range. Nonetheless, the method will be useful for modeling high frequency contributions in the hybrid methods, which use stochastic and deterministic methods for high and low frequencies, respectively (e.g., the stochastic Green’s function method + finite difference methods; Kamae et al., Bull Seism Soc Am, 88:357–367, 1998; Pitarka et al., Bull Seism Soc Am 90:566–586, 2000), because it reproduces the full waveforms in layered media including not only random characteristics at higher frequencies but also theoretical and deterministic coherencies at lower frequencies.

Dynamics of a prestressed incompressible layered half-space under moving load

The paper addresses a plane problem: a concentrated force acts on a plate resting on an elastic half-space with homogeneous prestrain. The equations of motion of the plate incorporate shear and rotary inertia. The half-space is assumed to be incompressible and isotropic in the natural state. The elastic potential is given in general form and is only specified for numerical purposes. The dependence of the critical velocity of the load and the stress-strain state on the prestresses is analyzed for different ratios between the stiffnesses of the layer and half-space and different contact conditions. The calculations are carried out for a half-space with Bartenev-Khazanovich potential

Elastic and visco-elastic stress triggering in the South Iceland Seismic Zone due to large earthquakes since 1706

Damaging earthquakes in the South Iceland Seismic Zone (SISZ) occur fairly regularly and often as a series of events with a few days only between individual events. Tolerably reliable information on epicentre locations and mechanisms are available for 13 M ≥ 6 events between 1706 and 2000. For these events, we computed the co- and post-seismic stress fields, hereby approximating the SISZ by a mixed elastic/visco-elastic layered half-space. The horizontal shear stress and the Coulomb stress changes were analysed to detect possible trigger mechanisms, which may aid future earthquake mitigation efforts. We tested several criteria but must conclude that the start of an earthquake series in the SISZ cannot be explained by triggering through previous events. Inside an individual series, however, one may infer triggering. Our results are in contradiction with the findings in other regions of the world. The reason might be related to the fact that the SISZ is not a mature fault zone, in which old faults are re-activated if a certain stress level threshold is passed. In addition, uncertainties in the model parameters as well as the neglect of horizontal variations in the model and of possible stress transfer due to volcanic activity further complicate the evaluation of our results and need to be taken into account in future studies.

A 1D time-domain method for in-plane wave motions in a layered half-space

A 1D finite element method in time domain is developed in this paper and applied to calculate in-plane wave motions of free field exited by SV or P wave oblique incidence in an elastic layered half-space. First, the layered half-space is discretized on the basis of the propagation characteristic of elastic wave according to the Snell law. Then, the finite element method with lumped mass and the central difference method are incorporated to establish 2D wave motion equations, which can be transformed into 1D equations by discretization principle and explicit finite element method. By solving the 1D equations, the displacements of nodes in any vertical line can be obtained, and the wave motions in layered half-space are finally determined based on the characteristic of traveling wave. Both the theoretical analysis and the numerical results demonstrate that the proposed method has high accuracy and good stability.

Propagation of partially coherent non-stationary random waves in a viscoelastic layered half-space

This paper presents a highly accurate method based on the precise integration method (PIM) and on the pseudo excitation method (PEM). The method computes the propagation behaviour of partially coherent non-stationary random waves in a viscoelastic, transversely isotropic solid, which consists of a multi-layered soil resting on a homogeneous semi-infinite space. The excitation source is a local rupture between two layers, which causes a partially coherent non-stationary random field. The analysis of non-stationary random wave propagation is transformed into that for deterministic waves by using PEM. The resulting governing equations in the frequency-wavenumber domain are linear ordinary differential equations, which are solved very precisely by using PIM. The evolutionary power spectral densities of the ground level responses are investigated and some typical earthquake phenomena are explained.

Dynamic response of an infinite beam overlying a layered poroelastic half-space to moving loads

Dynamic response of an infinite beam resting on a layered poroelastic half-space subjected to moving loads is investigated in this study. The equivalent stiffness of the layered poroelastic half-space is obtained via the transmission and reflection matrices (TRM) method in the frequency wavenumber domain. Based on the obtained equivalent stiffness, the frequency wavenumber domain solution of the beam–half-space system is obtained by the compatibility condition between the beam and the half-space. The time domain solution for the beam and the layered half-space is obtained by means of the inverse Fourier transform method. Also, the influences of the load speed and material parameters of the poroelastic half-space on the responses of the beam as well as the layered half-space are investigated. In order to demonstrate the proposed method, some time–space domain examples and corresponding analysis are presented in the paper.

Source mechanism model for ground motion simulation

A simple analytical model for computing ground motion in a layered half-space due to a buried seismic source is presented in this paper. The buried earthquake source is represented as a distribution of double couples varying in time as a ramp function on the fault plane. The analysis is simplified by first decoupling the governing equations into P-SV and SH problem by a coordinate transformation in the frequency-wave number domain. These two problems are solved separately and the final solution is obtained by the sum of solutions of these individual problems. Explicit expressions for ground motion in a layered half-space due to an impulsive double couple are derived. In the sequel, Green’s function for the displacement field in an infinite medium is also presented. The developed source mechanism model is also demonstrated by simulating ground motion for the Kucth earthquake ( M w = 7.7) of 26th January 2001.

Dynamic response of a layered water-saturated half space to a moving load

The transmission and reflection matrices (TRM) method for a layered poroelastic half space subjected to moving loads is developed in this study. Applying the triple Fourier transformation, the general solutions for the displacements, the stresses and the pore pressure are derived from the governing equations of Biot’s theory. Utilizing the continuity conditions between each layer and the boundary conditions at the half space surface, the transformed domain solutions for the displacements, the pore pressures and the stresses are established by the transmission and reflection matrices (TRM) method. Numerical results in the time–space domain are obtained by performing the inverse Fourier transform with respect to frequency and the horizontal wavenumbers. Moreover, some numerical examples and corresponding analysis are presented in the paper. Numerical results show that the occurrence of a softer middle layer in the layered half space will enhance the vertical displacement and the pore pressure of the layered half space. Besides, the presence of a softer middle layer tends to make the response of the layered half space exhibit more oscillatory nature.

Pressure of the lateral surface of a cylinder on a periodically layered half space

We consider a two-dimensional contact problem of indentation of an absolutely rigid cylinder into an inhomogeneous half space with periodic structure. The zone of periodicity consists of two homogeneous elastic layers. The solution of the classical problem of the theory of elasticity is compared with the solution of the problem in which the inhomogeneous half space is replaced by a homogenized space. We propose simple engineering formulas for the evaluation of contact characteristics (the width of the contact zone and the distribution of contact pressure) approximating the solution of the problem in the case where the homogenized medium inadequately models the inhomogeneous medium.

Three-dimensional frequency-dependent infinite elements for soil–structure interaction

This paper presents three-dimensional elasto-dynamic infinite elements for the three-dimensional soil–structure-interaction problem in a multi-layered half-space. Five kinds of infinite elements are developed in Cartesian coordinates using approximate expressions of multiple wave components for the wave functions in the exterior far-field soil region. They are horizontal, horizontal-corner, vertical, vertical-corner and vertical–horizontal-corner elements. Modified wave numbers are introduced to make the wave lengths in the original radial coordinates fit into the Cartesian coordinates. The elements can be used for the multi-wave propagating problem. Numerical example analyses are presented for a rigid disk, rectangular footings, and an embedded caisson on homogeneous and layered half-spaces. The numerical results obtained show the effectiveness of the proposed three-dimensional infinite elements.

Theoretical Seismogram Calculation for Laterally Varying Crustal Structures

Summary A modification of the reflectivity method for calculating theoretical seismograms is introduced. This technique enables allowances to be made for differing crustal structures beneath the source and the receiver. The method has been applied to the 1971 long range refraction profile across France, and it is shown that the observed effects can only be accounted for by structure in the upper mantle. In this paper we develop a computational method which enables us to make such an allowance for the propagation effects of the different crustal structures beneath the source and the receiver using a modification of the reflectivity method. We then apply this technique to the results of the long seismic refraction profile in France in 1971 (Hirn et al. 1973) and use the experimentally determined crustal structures (Sapin & Prodehl 1973) to show that the observed effects can only be accounted for by structure in the upper mantle. Varying crustal structure and the reflectivity method The reflectivity method in its usual form (Fuchs & Miiller 1971) provides an effective computational technique for approximating the response of a horizontally layered elastic half space to excitation by a near-surface source. Since the com- putation of the complete response is exceedingly time consuming a number of simplifications are made. Only in the lower part of the layered half space, referred to as the ' reflection zone ' are all multiple reflections and interconversions of wave

A Procedure for Modelling Shallow Dislocation Sources*

Summary. Expressions for displacements on the surface of a layered half-space due to an arbitrary oriented shear dislocation are given in terms of generalized ray expansions. Useful approximations of these expressions for shallow events as recorded at teleseismic distances for realistic earth models are presented. The results of this procedure are used to generate synthetic P, SV and SH waveforms for various assumptions of stress-drop.

Transient response of SH waves in a layered half-space with sub-surface and interface cracks

The transient scattering of SH waves by sub-surface and interface cracks parallel to the free surface in a layered elastic solid is investigated. The problem in frequency domain is solved by using a hybrid method which combines the finite element method of the near field with the boundary integral representation of the far field. The transient responses are then obtained by inverting the spectra via fast Fourier transform with the incident pulse Ricker of wavelet. Numerical results are presented for the surface displacements, dynamic stress intensity factors and wave motion in the layered half-space. Furthermore, the propagations of reflected, diffracted, and direct impact waves at any instant are clearly identified by the present method. To understand the mechanism of elastic wave interaction is very important in the field of ultrasonic non-destructive evaluation (NDE) and fracture mechanics studies.

Inverse problem of reflection of electromagnetic waves from layered inhomogeneous plasma-like structures with refractive-index discontinuity

To reconstruct the distribution of free carriers in a layered inhomogeneous plasma-like half-space during the electromagnetic-wave reflection, we derive the Gelfrand-Levitan-Marchenko integral equation for structures with refractive-index discontinuity by the inverse problem method. Exact model solutions of the generalized equations are analyzed. The applicability limits and scope of these exact solutions for reconstruction problems are shown.

On axisymmetrical contact problem of pressure of a rigid sphere into a periodically two-layered semi-space

The axisymmetrical contact problem of pressure of a rigid sphere into a layered half-space with periodic structure is considered. The results obtained by using the classical elasticity and the homogenized model are compared. The range of problem parameters for which, the homogenized model stands for a good description of layered composites is presented. The contact parameters in which the homogenized model gives adequate large errors are also calculated.

Cone model for two surface foundations on layered soil

In this paper, the cone model is applied to the vibration analysis of two foundations on a layered soil half space. In the analysis, the total stress field in the subsoil is divided into the free-field and the scattering field. Seed’s simplified method is adopted for the free-field analysis, while the cone model is proposed for analyzing the dynamic scattering stress wave field. The shear stress field and the compressive stress field in the layered stratum with two scattering sources are calculated by shear cone and compressive cone, respectively. Furthermore, the stress fields in the subsoil with two foundations are divided into six zones, and the P wave and S wave are analyzed in each zone. Numerical results are provided to illustrate features of the added stress field for two surface foundations under vertical and horizontal sinusoidal force excitation. The proposed cone model may be useful in handling some of the complex problems associated with multi-scattering sources.

Critical Load Analysis for a Layered Half-Space under Sliding Indentation

Analyses are presented for cracking or delaminating problems of thin coatings on dissimilar thick substrate materials under a normal and a tangential force. Based on the energy principle associated with the consideration of substrate deformation, an analytical expression has been proposed to predict the critical load of an adherent stiff thin coating on a compliant substrate. The critical load has been shown to depend on the hardness, the coating thickness, the surface energy of adhesion, the coefficient of friction, contact areas of a scratch track, and elastic moduli of coatings and substrates. Experimental observations of microscratch testing on a CrN coating deposited on a soft coating, labeled as SC-I, over a Cu/Zn alloy substrate have shown that buckling and cracking in a semicircular arc ahead of an indenter are the predominant failure modes, thereby confirming the assumption of the theoretical calculation. Using Beuth's solution, a simple fracture model describing the cracking of thin coatings has been developed over a range of practical elastic mismatches and applied to solve cracking problems of compliant coatings in this work. Microscratch results from another soft coating, labeled as SC-II, on a Cu/Zn alloy substrate have revealed that cracks first occur along the edge of the circular contact area in the rear of the indenter due to the tensile stress. The stress formulae of Hamilton and Bower and Fleck are therefore introduced into this model to compute the critical load required to initiate a crack in the coating and the stress intensity factor of the coating. Review led by Dong Zhu

Response of Layered Half-Space Obtained Directly in the Time Domain, Part II: SV-P and Three-Dimensional Sources

In Part I of this set of two companion articles we presented a new, rigorous method to obtain the seismic response of horizontally layered, viscoelastic (or elastic) half-spaces caused by antiplane ( SH ) sources anywhere. This method is formulated directly in the time domain, and can be applied even in the absence of attenuation. This article (Part II) generalizes the concept to SV-P line sources and to three-dimensional point sources, including seismic moments.

Response of Layered Half-Space Obtained Directly in the Time Domain, Part I: SH Sources

A rigorous method is presented to obtain—directly in the time domain and without the need for an integral transform over frequencies—the seismic response of horizontally layered, viscoelastic half-spaces due to arbitrarily distributed (or concentrated) sources. The method is based on exact expressions for the response of the underlying half-space cast in the wavenumber-time domain, and a complete modal solution in that domain for the layers. The proposed method is highly effective and avoids the problems of the usual propagator matrix method that arise when numerical integrations over both frequencies and wavenumbers are carried out, especially the waviness of the kernels due to resonances and reverberations in the layers when damping is light or even absent. Part I presents the basic methodology for the relatively simple case of antiplane SH sources in two-dimensional space, while Part II generalizes the concept to two-dimensional SV - P line sources and to three- dimensional point sources, including seismic couples.

Inverse problem for wave propagation in a perturbed layered half-space

This paper is concerned with the inverse medium scattering problem in a perturbed, layered, half-space, which is a problem related to the seismologial investigation of inclusions inside the earth’s crust. A wave penetrable object is located in a layer where the refraction index is different from the other part of the half-space. Wave propagation in such a layered half-space is different from that in a homogeneous half-space. In a layered half-space, a scattered wave consists of a free wave and a guided wave. In many cases, only the free-wave far-field or only the guided-wave far-field can be measured. We establish mathematical formulas for relations between the object, the incident wave and the scattered wave. In the ideal condition where exact data are given, we prove the uniqueness of the inverse problem. A numerical example is presented for the reconstruction of a penetrable object from simulated noise data.