Layered Half-space Research Articles

Axisymmetric BEM analysis of layered elastic halfspace with volcano-shaped mantle and cavity under internal gas pressure

This paper presents an axisymmetric BEM analysis of layered elastic halfspace with volcano-shaped mantle and cavity under internal gas pressure. The problem is of interest to understand the behavior of volcanoes, tectonic earthquakes and other oil and gas reservoirs. The volcano-shaped mantle ground topography and the internal spherical or ellipsoidal cavity are added to the conventional model of layered halfspace. The analysis uses the axisymmetric boundary element method (BEM) associated with the fundamental solution of a multilayered elastic fullspace subject to the concentrated ring-body force. The BEM is further verified for its high efficiency and accuracy by comparing its result with the analytical and FEM solutions for the problem of a homogeneous elastic halfspace whose spherical cavity is under internal pressure. The BEM is used to examine the effect of the volcano-shaped mantle, the cavity shape, the layered material properties and the internal pressure on the elastic behavior of the halfspace under the internal pressure loading within the cavity. The displacements and stresses at the external and internal boundaries and within the layered halfspace are presented and analyzed in detail. The presence of the volcano-shaped mantle can reduce significantly the swelling ground movement induced by the expansive pressure in the cavity in the layered halfspace. The cavity shape can have significant effect to the location and magnitude of the maximum tensile principal hoop stress on the cavity surface induced by its internal pressure.

Modelling of Love Waves in a Heterogeneous Medium Demarcated by Functionally Graded Piezoelectric Layer and Size-Dependent Micropolar Half-Space

Modelling of Love Waves in a Heterogeneous Medium Demarcated by Functionally Graded Piezoelectric Layer and Size-Dependent Micropolar Half-Space

Genetically nonlinear combined model of pile field under dynamic impacts

The authors propose a combined approach for definition of the shear rigidity of the multilayered soil which is cut through by a pile. The solution for the vertical direction is presented in the view of an axisymmetric problem. As to the horizontal direction, the solution is presented in view of a beam on elastic subsoil with genetically non-linear transition to equivalent horizontal rigidity of the wide pile field in condition of dynamic forces action. The axisymmetric solution provides visual clarity in the analysis of the stress-strain state of the pile and near-pile soil in comparison with the approved analytical methods. To speed up calculations at the stage of the main combination of constant and long-term impacts, the vertical rigidity of the base under the foot of the pile can be calculated analytically as for a stamp on an elastic-plastic base. The horizontal rigidity is considered as for a discrete single bent pile in the medium of an elastic layered half-space at the stage of formation of the stress-strain state of the system under the main combination of static loads. At the final stage of short-term or special load, a transition to the integral shear rigidity of a pile field is proposed.

A novel Rayleigh-type viscoelastic Perfectly-Matched-Layer for wave propagation analysis: Formulation, implementation and application

The Perfectly-Matched-Layer (PML) has been recognized as a robust absorbing boundary condition and adopted in many research areas, such as electromagnetic, acoustic, and elastodynamic problems. In geotechnical engineering, the PML is widely utilized to mimic the soil domain’s semi-infinite extent, via efficiently absorbing the outgoing scattered waves, while regardless of their frequencies and directions. Up to date, most of the existing PMLs are designed for elastic soil media. In this research, we develop both two- and three-dimensional viscoelastic PMLs that incorporate the effects of Rayleigh damping. Detailed formulations of the viscoelastic PML are presented. The PML is then successfully implemented in ABAQUS through a user-defined element (UEL) subroutine. Comprehensive validation studies are conducted, including all 1D, 2D and 3D problems, homogeneous and layered half-space. Finally, as illustrations of applications of the proposed PMLs, we extract the impedance functions of two different foundation structures that embedded in/ rest on viscoelastic soil media with Rayleigh damping, by attaching the PML-truncated domain.

Scattering of elastic spherical P, SV, and SH waves by three-dimensional hill in a layered half-space

In this paper, the method of combining indirect boundary element method (IBEM) with the region-matching technique is used to solve the seismic responses of a three-dimensional (3D) hill in a layered half-space subjected to spherical P, SV, and SH waves. The direct stiffness method and Hankel transform method are used to solve the free field of spherical wave source field, and the scattering field is constructed by Green's function in half-space. The correctness and accuracy of this method are verified by comparing with the results of several literatures. This method can be used to deal with arbitrary-shaped hill topography, and has the advantages of wide applicability, high accuracy and fast convergence. Taking Gaussian hill topography as an example, the scattering phenomenon of hill under the incidence of spherical P, SV and SH waves is studied, and the influences of hill topography, source location and frequency on site seismic responses are explored. The results show that: Existence of the hill topography has evident focusing effect, which significantly enlarges the seismic response on the hill surface, and the hill has evident isolation effect in the process of energy propagation; With higher frequency of the incident wave, the fluctuation amplitude of displacement on the hill surface increases; The fluctuation amplitude of horizontal displacement on the hill surface is SV wave > SH wave > P wave, and the vertical displacement is P wave > SV wave.

Thermoelastic wave propagation in a piezoelectric layered half-space within the dual-phase-lag model

Thermoelastic wave propagation in a piezoelectric layered half-space within the dual-phase-lag model

REALIZATION OF GROUND MOTION INPUT IN ABAQUS FOR LAYERED FOUNDATION UNDER SV WAVE OF OBLIQUE INCIDENCE OVER CRITICAL ANGLE

In the finite element numerical simulation, it is an urgent challenge to determine the ground motion input for the layered foundation under SV wave of oblique incidence over critical angle. Exact dynamic stiffness matrix in the frequency domain (i.e., frequency domain stiffness matrix method) of layered foundation is used to derive the formula for calculating the equivalent node force of ground motion input under arbitrary angular oblique incidence of SV wave, and the validity and accuracy of the ground motion input applied to ABAQUS numerical simulation are demonstrated by simulating seismic wave field of homogeneous half-space and layered foundation under oblique incidence of SV wave through ABAQUS software. It is shown that the inclined SV wave incidence with an arbitrary angle can be realized through ABAQUS simulation by means of stiffness matrix method, and the method is of high accuracy, especially in the case of SV wave incidence with an inclined angle greater than the critical one. The oval like motion trails of surface particles in uniform half space and the wave motion characteristics of layered half space are simulated commendably by finite element numerical modelling. On the above basis, the frequency domain stiffness matrix method is further combined with equivalent linearization method, which solves the ground motion input problem of two-dimensional layered half space considering soil nonlinearity for the inclined incidence with arbitrary angle.

Coseismic slickenlines record the emergence of multiple rupture fronts during a surface-breaking earthquake

Coseismic changes in slip direction recorded by curved slickenlines on fault surfaces are commonly observed following surface-breaking earthquakes. Such observations represent a dynamic record of seismic slip and may provide a new set of constraints on the evolution of propagating rupture and hence earthquake dynamics. We test this hypothesis by conducting dynamic rupture simulations of the 2011 Mw 6.6 Fukushima-Hamadori Earthquake (Japan). These simulations aim to reproduce the well-documented field observations of curved slickenlines that formed during coseismic fault displacement at the ground surface. We consider relatively simple dynamic rupture models with a dipping fault embedded into a homogeneous or layered elastic halfspace. Among a wide range of model parameters tested, we find a model with shallow (<1.5km) low-velocity layers derived from a regional 3D velocity model combined with a depth-dependent prestress result in curved slip trajectories that closely match slickenline observations. This same model also generates a slip distribution and rupture propagation direction consistent with published inversions constrained by seismological and geodetic data. Furthermore, the characteristics of slickenline curvature are consistent with theoretically predicted earthquake rupture direction. Unlike previous theoretical studies, coseismic changes in rake angle occur due to the emergence of multiple slip fronts within the shallow low-velocity medium. Our results indicate that on-fault geological observations can supplement seismological studies of earthquake rupture evolution beyond traditional datasets.

HVSR analysis of a layered saturated half-space using diffuse-field theory

SUMMARYThe recently constructed diffuse field theory from isotropic energy equipartition has been well developed in elasticity for full-wave interpretation of horizontal-to-vertical ratio (HVSR), which links the signal autocorrelation with the imaginary part of Green's function. Here, the theory is extended to the saturated layered medium within the framework of Biot's theory to account for the offshore environment. The imaginary parts of Green's functions are obtained using direct stiffness method accompanied with Fourier–Hankel transform. In particular, the upgoing wave amplitudes are modified to tackle the overflow during wavenumber integral and allow for fast calculations. After validating the method from the perspectives of Green's function calculation, emphasis is laid on evaluating the inaccuracies of HVSR calculation induced by model misuses in the lack of prior geological and geotechnical information. The numerical results considering the effects of layer sequence, impedance ratio, porosity and drainage condition show that the predominant frequency of the one-phase medium is slightly less than the two-phase medium with the maximum shift no more than 0.1 Hz, while their amplitude differences can be prominent as impedance ratio and porosity increase, with the maximum difference up to 29 per cent. The shallowest soft layer has the dominant effects on HVSR amplitudes, whereas the buried low-velocity layer at depth over one-wavelength contributes little to the peak amplitude. Finally, the method is applied to a realistic case at Mirandola, Northorn Italy, which suffered extensive liquefaction-induced damages in 2012 Emilia earthquake. The well identified pattern of the experimental HVSR using the two-phase medium model illustrates the application potential of our method to further assist the subsurface geology retrieval.

Modelling of torsional wave propagation in a two-layer anisotropic inhomogeneous media

The current article reflects the behavior of torsional waves in a two-layer inho-mogeneous model, where rigidities and density vary along with the depth parameter z. The variation for the upper layer and lower half-space has been taken as the product of two exponential functions and the product of one linear and exponential function, respectively. A mathematical model is developed by following the concept of elastic waves. The separation of the variable method has been imposed on both layers in order to calculate the displacement equations. Intrinsic boundary conditions are imposed to derive the frequency equation in closed form. It has been observed a substantial effect of rigidity and density on the phase velocity of the torsional wave. Graphical presentations have been performed in order to support the different findings of the dispersion equation.

A substructure method for underground structure-dry soil-saturated soil–bedrock interaction under obliquely incident earthquake and its application to groundwater effect on tunnel

This study presents a substructure method with good accuracy and excellent efficiency for seismic underground structure-dry soil-saturated soil–bedrock interaction problem under obliquely incident earthquake. A combined zigzag-paraxial on-surface condition attached on structural surface is proposed to provide the dynamic stiffness of structural surrounding media including dry soil, saturated soil and underlying half space bedrock. The on-surface boundary condition can be implemented to any desired degree of accuracy and is compatible with numerical method such as the finite element method and the finite difference method. The condition is then coupled with structural finite element scheme and seismic wave input method to establish a substructure method. The reliability and accuracy of the proposed substructure method are verified by applying it in two degenerated interaction problems and comparing with other solutions. Finally, the proposed substructure method is employed to establish a lined tunnel-dry soil-saturated soil–bedrock interaction model for evaluating the seismic response of a lined tunnel embedded in the layered half space rich in groundwater, with an emphasis on the effects of changing groundwater levels on seismic response of tunnel lining.

Obliquely incident P-SV wave scattering by multiple structures in layered half space using combined zigzag-paraxial boundary condition

Seismic wave scattering by multiple structures in layered half space is a classically multiple scattering problem in infinite domain. The computationally efficient substructure method is commonly adopted to solve this problem, because the structures can be modelled by the finite element scheme with the combination of the high-accuracy boundary condition to provide the dynamic stiffness of the truncated infinite rock or soil media. However, it is difficult to establish a high-accuracy boundary condition for multiple scattering problem in layered half space. In this paper, a combined zigzag-paraxial boundary condition is presented, and it is coupled with the seismic wave input method, as well as with the finite element scheme, to form the substructure method for solving the multiple scattering problem in layered half space. To verify the proposed substructure method, it is firstly applied to obliquely incident P-SV wave scatterings by a canyon, a cavity and an alluvial basin in homogeneous half space. The results are validated by comparing with several existing analytical and numerical methods. To demonstrate the capability of the proposed substructure method, it is finally applied to obliquely incident P-SV wave scattering by multiple tunnels in layered half space, and the effects of twin tunnels, soft soil interlayer and a V-shaped canyon on the structural response are discussed.

Hypothesis tests on Rayleigh wave radiation pattern shapes: a theoretical assessment of idealized source screening

SUMMARYShallow seismic sources excite Rayleigh wave ground motion with azimuthally dependent radiation patterns. We place binary hypothesis tests on theoretical models of such radiation patterns to screen cylindrically symmetric sources (like explosions) from non-symmetric sources (like non-vertical dip-slip or non-VDS faults). These models for data include sources with several unknown parameters, contaminated by Gaussian noise and embedded in a layered half-space. The generalized maximum likelihood ratio tests that we derive from these data models produce screening statistics and decision rules that depend on measured, noisy ground motion at discrete sensor locations. We explicitly quantify how the screening power of these statistics increase with the size of any dip-slip and strike-slip components of the source, relative to noise (faulting signal strength) and how they vary with network geometry. As applications of our theory, we apply these tests to (1) find optimal sensor locations that maximize the probability of screening non-circular radiation patterns and (2) invert for the largest non-VDS faulting signal that could be mistakenly attributed to an explosion with damage, at a particular attribution probability. Finally, we quantify how certain errors that are sourced by opening cracks increase screening rate errors. While such theoretical solutions are ideal and require future validation, they remain important in underground explosion monitoring scenarios because they provide fundamental physical limits on the discrimination power of tests that screen explosive from non-VDS faulting sources.

Multiple zero-group velocity resonances in elastic layered structures

To date, the phenomenon of zero group velocity (ZGV) Lamb wave in a stress-free elastic plate has been well studied and used in non-destructive material characterization, electroacoustic devices and some other applications. This phenomenon is associated with the backward mode appearing in the range limited by the ZGV and cutoff frequencies. The modal frequency response of the structure, primarily the spectrum of the corresponding guided wave (GW) excited in the plate by a force source, is featured by a sharp peak at the ZGV frequency, allowing its experimental detection. Other peaks appearing at the cutoff frequencies indicate thickness resonances. In a layered half-space of infinite thickness, the backward-mode bend of the dispersion curve turns into a double bend that gives rise to three GWs associated with the same branch (one backward and two forward modes). Moreover, several adjacent branches with double bends, and, thus, several peaks, can appear in a narrow range as the number of layers increases, indicating multiple ZGV resonances. Such dispersion curves with double ZGV points were also found in waveguides of finite thickness with anisotropic or contrast material properties; however, their resonance manifestation is poorly studied. In this article, we consider and discuss several kinds of multi-peak ZGV responses using examples of layered samples of various materials. Along with the expected increase in the number of peaks in multilayer structures with contrast layers, it was noted that some ZGV points may or may not appear in peaks, depending on the type of load.

Generation of 6-C synthetic seismograms in stratified vertically transversely isotropic media using a generalized reflection and transmission coefficient method

SUMMARYWe develop a generalized reflection and transmission coefficient method (GRTM) for generating six-component (6-C) synthetic seismograms in horizontally layered vertically transversely isotropic (VTI) media. Compared with the traditional seismic modelling approaches that only consider translational motion, our method can simultaneously produce three-component translational and three-component rotational data excited by a point vector force or a moment tensor source in a layered half-space. Horizontally layered models are widely used in near surface applications as the properties of near surface formations generally show small lateral variations and change mainly along the depth direction. The use of the VTI constitutive relation can make our method applicable to more general situations because it takes into account the characteristics of sedimentary formations. We compare our method with a finite-difference method (FDM) for a variety of velocity models and acquisition geometries. The numerical results demonstrate that accurate 6-C synthetic seismograms can be calculated using our method. The computational efficiency of our method for 6-C seismic modelling is much higher than the finite-difference method, because it can reduce a 3-D modelling problem to 2.5-D by eliminating the azimuthal dimension. Also, our method does not require to perform additional spatial interpolations to obtain the rotational components. These advantages make our method suitable to serve as a forward modelling tool for rotational seismology.

SH surface wave propagating in a strain-gradient layered half-space

By employing Mindlin’s Form II gradient elasticity theory (strain gradient elasticity theory), we investigate the SH surface wave propagating in a strain-gradient layered half-space. The dispersion equation of the SH surface wave is derived analytically. For the general case where both the surface layer and the half-space are strain-gradient elastic materials, the dispersion equation involves 10 material constants. The dispersion equation of the general case can degenerate to that of several special cases by dropping some material parameters. The existence regions of the dispersion curves are discussed in detail. The effects of strain-gradient elastic constants on the dispersion curves are examined. It is seen that the features of SH surface waves in strain-gradient elastic materials are much richer than those in classical elastic materials.

Analysis of curved thin wires radiating over a layered medium and some engineering applications

The paper deals with modelling of radiation from arbitrarily shaped thin wires above a multilayer and related engineering applications. The formulation is based on the set of corresponding Pocklington integro-differential equations (IDEs) for curved wires. The influence of lower layered half-space is taken into account by means of simplified reflection coefficient stemming from modified image theory (MIT). The set of coupled IDEs is handled via Galerkin Bubnov variant of indirect boundary element method (GB-IBEM) for curvilinear elements. The related field radiated from a wire configuration is obtained from corresponding integral expressions including integration of the currents along a given wire configuration. Some illustrative computational examples for certain wire configurations pertaining to dipole antennas, power communication systems (PLC) and instrumental landing systems (ILS) are given in the paper.

Propagation of Rayleigh Waves in Layered Elastic Half-Space with Finite Sliding Contact

Propagation of Rayleigh Waves in Layered Elastic Half-Space with Finite Sliding Contact

Model errors caused by rigid-foundation assumption in soil-structure interaction: a comparison of responses of a soil-structure-flexible foundation system and a rigid foundation system

This study investigates model errors caused by the rigid-foundation assumption in dynamic Soil-Structure Interaction (SSI), which has been widely accepted in the past decades to reduce computational effort. A linear two-dimensional model is used for a qualitative analysis that compares the dynamic responses of a rigid system, comprising a rigid foundation embedded in a layered half-space with a superstructure mounted on top, and a corresponding flexible system with the same parameters but a flexible foundation with a variable stiffness. The Indirect Boundary Element Method combined with non-singular Green’s functions of distributed line loads is employed to calculate the system responses accurately. Transfer functions computed for a range of parameters show that the rigid-foundation assumption leads to overestimating the system natural frequency and changes the peak deformations to a different extent. It is also shown through a case study of 42 earthquakes that the rigid-foundation assumption may either overestimate or underestimate the system responses by up to approximately 50%, and in some cases even by approximately 100%, depending on the frequency content of excitation and SSI dynamic characteristics.

Accuracy of Near-Fault Fling-Step Displacements Estimated Using the Discrete Wavenumber Method

ABSTRACTNear-fault fling steps might cause severe damage to near-fault structures such as bridges or base-isolated buildings. Therefore, the accurate simulation of ground displacements including fling steps based on fault models is an important issue not only for seismological but also for engineering purposes. The discrete wavenumber (DWN) method (e.g., Bouchon, 2003) has been established as a method to calculate complete elastic wavefield, including permanent displacement for a homogeneous or a layered half-space. However, the accuracy of the permanent displacements calculated by the DWN method is influenced by the selection of parameters, such as the imaginary part of the complex frequency and the subfault size in the case of extended sources. The objective of this study is to clarify the requirement for these parameters for the accurate simulation of fling-step displacements to further enhance the use of the DWN method. Honda and Yomogida (2003) also addressed the issue of calculating fling-step displacements using the DWN method; however, their study was focused on cases in which a large amount of seismic moment is released at depth. This study was focused on fling-step displacements due to rather shallow slip, in which the fault distance was as small as several meters in an extreme case, motivated by recent damaging earthquakes such as the 2016 Kumamoto, Japan, earthquakes. The ground displacements including fling steps were calculated by the DWN method and compared with the analytical solutions for the static displacements (Okada, 1985, 1992), both for point sources and extended sources in a homogeneous half-space. According to the results, following recommendations were made. For the imaginary part of the complex frequency, ωc=ω−λi, λ=ξπ/Tw with ξ≥2.0 can be recommended, with the understanding that the waveforms are effective only within the range of [0,Tw/ξ]. For extended sources, the subfault size should be as small as 0.5 times the fault distance to accurately simulate fling steps.