Homogeneous Half-space Research Articles

Coupling the first and second attenuation zones in seismic metasurface

Semi-infinite elastic metamaterials and metasurfaces on homogeneous elastic half-spaces have attracted significant attention in the past two decades as efficient artificial structures to control and mitigate surface waves. In this research, the first two attenuation zones of an elastic metasurface composed of different arrangements of pillars on a substrate are investigated. First, the lowest frequency attenuation zone (LFAZ) is numerically investigated. Then, the rainbow trapping effect of the second attenuation zone (SAZ) is demonstrated by numerical simulations and experiments. The different characteristics of these two attenuation zones are brought together to connect the lower bound of the SAZ and the upper bound of the LFAZ, allowing to design a seismic metasurface that only uses one kind of artificial structure on a half-space to attenuate Rayleigh waves in the frequency range extending from 5 to 22 Hz.

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.

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.

2-D Analysis of Generalized Thermoelastic Porous Medium under the Effect of Laser Pulse and Microtemperature

The paper presents the analytical solutions for a generalized thermoelastic medium consisting of microtemperatures and voids subjected to a laser pulse loading the medium thermally. The 0.02 ps pulse duration of the non-Gaussian laser beam is apt for heating a homogenous isotropic elastic half-space. A method called the normal mode analysis is employed to evaluate numerically the effects of various variables such as the micro-temperature vector, variation in the fraction field of the volume, first heat flux moment tensor, temperature distribution on the stresses and displacement components of the medium. In addition, the graphical illustration of the physical response of the medium has been presented in the presence and absence of void parameters, as well as in the presence of laser pulse with two different acting periods.

Effect of longitudinal joint on train-induced vibrations from subway shield-driven tunnels in a homogeneous half-space

With the rapid increase of urban rail transit networks, the environmental problems induced by vibrations from subway traffic become more and more prominent. Most subway tunnels are constructed by the shield machine, which are assembled by tunnel linings connected by joints. This study investigates the effect of the tunnel longitudinal joints on the train-induced vibrations from a subway shield-driven tunnel in a homogeneous half-space. The shield-driven tunnel is considered as an elastic lining ring connected by longitudinal joints, while the ground is simplified as a homogeneous elastic half-space. The general solution for the tunnel and the ground are derived by using the thin-shell theory and the elastic continuum theory, respectively. By satisfying the boundary conditions on the tunnel-ground interface, the tunnel is coupled to the ground in a half-space domain. The responses during the running of a metro train are calculated by coupling the tunnel-ground model with the vehicle-track model. The longitudinal joint has a significant impact on ground vibrations, which increases the running RMS values of the vertical velocity by 20% ∼ 70% and of the horizontal verticity by 10% ∼ 20%. As the stiffness of the longitudinal joint decreases, the ground vibrations increase.

Stoneley waves in mild auxetics

The current research concerns with the analysis of Stoneley waves propagation along an interface between two dissimilar homogeneous and isotropic halfspace having negative Poisson’s ratios (auxetics). The main attention is paid to the case when both contacting media satisfy the Wiechert condition ensuring the existence of Stoneley wave. Closed form expressions are derived for derivatives of the Stoneley secular equation. The velocity of Stoneley waves is analyzed both theoretically and numerically in a broad range of the Wiechert parameter variation.

Solving a one mixed problem in elasticity theory for half-space with a cylindrical cavity by the generalized Fourier method

Designing and constructing underground structures for various purposes, such as tunnels, mines, mine workings, necessitate the development of procedures for calculating their strength and reliability. The physical model of such objects worth considering is a homogeneous isotropic half-space that contains an infinitely long hollow cylinder, located parallel to its border. One can explore problems related to the mechanics of deformable solids for such a multiply connected body. This paper reports the proofs of addition theorems for the basic solutions to the Lamé equation regarding the half-space and cylinder recorded, respectively, in the Cartesian and cylindrical coordinate systems. This result is important from a theoretical point of view in order to substantiate a numerical-analytical method ‒ the generalized Fourier method. This method makes it possible to solve spatial boundary problems from the theory of elasticity and thermo-elasticity for isotropic and transversal-isotropic multiply connected bodies. Similar to the classical Fourier method, the general solutions to equilibrium equations have been used here but in several coordinate systems rather than one. From a practical point of view, this method has made it possible to investigate the combined problem of elasticity theory regarding the multiply-connected body described above. The analysis of the stressed-strained state of this elastic body has made it possible to draw conclusions on determining those regions that are most vulnerable to destruction. The highest values are accepted by normal stresses in the region between the boundaries of the half-space and the cylinder. Changing the σy component along the Ox axis corresponds to the displacements assigned on the half-space. The τxy component contributes less to the distribution of stresses than σx and σy. The applied aspect of using the reported results is the possibility to apply them when designing underground structures

Stress and deformation induced in layered media by cylindrical thermo-poro-elastic sources: An application to Campi Flegrei (Italy)

In recent literature thermo-poro-elastic (TPE) inclusions are proposed to contribute to deformation and stress in volcanic and hydrothermal areas. With respect to previous works, the present study implements the modeling improvements requested for a more realistic description of a TPE source and the embedding medium. Firstly we propose a numerical method to represent inclusions with an arbitrary geometry. In the case of cylindrical-shaped sources, the proposed approach allows us to study the effects of the source thickness on both deformation and stress fields. Moreover, we are allowed to consider depth dependent pore pressure and temperature changes within the inclusion, as expected during the transient stage of hot fluid propagation. Secondly, we take into account elastic stratification of the crust, in order to perform unbiased inversion of ground deformation data. The inversion of geodetic data collected during the 1982–84 unrest episode in the Campi Flegrei caldera (Italy), performed in a layered elastic medium, leads to a lower misfit and a deeper deformation source with respect to the one obtained inverting data in a homogeneous half-space. Furthermore, the geometry and location of the thermo-poro-elastic inclusion retrieved from the inversion in the layered medium leads to a shear stress distribution in good agreement with the relocated seismicity observed at Campi Flegrei. Most hypocenters are found to be located near and inside the TPE inclusion boundaries, where the greatest shear stress is predicted. With respect to previous works we also found that vertical gradients of pore pressure and temperature and elastic layering can promote both normal and thrust earthquakes within the inclusion.

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.

Seismic evaluation of steel plate shear wall systems considering soil-structure interaction

This study investigates the various effects of Soil-Structure Interaction (SSI) on the seismic behavior of steel frames with Steel Plate Shear Wall (SPSW) lateral resisting systems. Nine steel frames with various aspect ratios are studied under multiple seismic hazard levels. The SPSWs are modeled based on the strip model concept, and the Soil-Structure Systems are simulated using the substructure method. The soil beneath the structure is considered as a homogeneous elastic half-space. The Endurance Time method is exploited for nonlinear dynamic analysis of the fixed-base structures and soil-structure systems. Results indicate that use of fixed-base models leads to the significant overestimation of the damage of SPSWs in the lower stories of the structure. Neglecting SSI in squat structures is undoubtedly a conservative choice, while in slender structures, it may lead to the underestimation of structural responses, especially in the upper stories. In the design procedure of the structures with SPSW lateral load resisting system located on soft soils, ignoring SSI effects results in very heavy and cost-intensive code-based designs.

A Mode-I crack for a rotational fibre-reinforced thermoelastic medium with thermal relaxation time

A general model of the equations with coupled theory (CD) and Lord-Shulman (L-S) theory with one relaxation time is applied to study the influence of reinforcement on the total deformation for an infinite space weakened by a finite linear opening Mode-I Crack is solving. We study the influence of reinforcement on the total deformation of rotating thermoelastic half-space and the interaction with each other. The material is homogeneous isotropic elastic half space. The crack is subjected to prescribed temperature and stress distribution. The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses and temperature. The variations of the considered variables with the horizontal distance are illustrated graphically. Comparisons are made with the results in two theories in the presence and absence of rotation. A comparison also is made between the two theories for different depths.

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.

Field calculation of electric dipole excitation of a metafilm printed on a finite‐thickness dielectric slab using the susceptibility homogenization method

Here, the authors calculate the fields due to the electric dipole that excites a homogenized metasurface which is printed on a finite thickness dielectric slab. The problem is solved by replacing the total parts below the periodic array which contain the dielectric slab and the half-space under it, with an effective homogenous half-space. Furthermore, the static interaction constants between dipoles which are located at two different materials’ interface are calculated. Then, with the help of the Lorentz method, the analytical formulae are presented to calculate the surface susceptibility densities of the homogenized metasurface located at two materials’ interface. These analytical formulae can be applied to arrays with both square and rectangular periodicity. An array of circular patches printed on dielectric slabs with different thicknesses is considered to verify the results. The results show good agreement compared to the fields calculated by commercial numerical software FEKO.

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.

Solution of three-dimensional generalized thermodiffusive elastic half-space with phase lag in the presence of chemical potential shock and thermal pulse

An isotropic homogeneous thermoelastic half-space has been considered with permeating gas, thermal pulse, and chemical potential on the bounding plane. A three-dimensional generalized thermo-diffusion model with three-phase lag has been constructed in the context of Memory-dependent derivative (MDD). Laplace transform (time domain) and Double Fourier transform (space domain) have been used in the formulation of the problem and thus the solutions for temperature increment, invariant stress, chemical potential and diffusion concentration are obtained using eigenvalue approach. Finally the effect of kernel function has been discussed and also the results are depicted in several dimensions.

Three-dimensional finite-difference modeling of time-domain electromagnetic responses for a large-loop source

The time-domain/transient electromagnetic (TEM) method employs an alternating magnetic field that is established by passing a bidirectional pulsed current through a transmitter loop. A large wire loop on or above the earth is one of the most widely used sources in TEM survey systems. To simulate TEM responses of the large loop system, the effect of the loop size must be considered in a numerical modeling scheme. We have developed an efficient three-dimensional (3D) TEM modeling algorithm using a staggered-grid finite-difference time-domain (FDTD) method. In this paper, we first derive FDTD TEM equations using fictitious displacement currents based on an explicit mid-point method. Then, we employed a conversion method to efficiently obtain stepoff TEM responses from step-on responses. Further, following the concept of the conversion method, the charging and discharging processes of current waveform in loop does not need to be identical, implying we can apply the simulation scheme for more general source waveforms. The developed algorithm has been verified through comparison of numerical results for a homogeneous halfspace medium. Finally, we analyze TEM responses generated from a large loop source for a 3D conductive body buried in a homogeneous half space.

A 2.5D coupled FEM-BEM-MFS methodology for longitudinally invariant soil-structure interaction problems

This paper is concerned with a new prediction methodology for longitudinally invariant soil-structure interaction problems in elastodynamics. The method uses the finite-element method to model the structure, the boundary-element method to model the local soil surrounding the structure and the method of fundamental solutions to model the wave propagation through the soil, all of them formulated in the two-and-a-half dimensional domain. The proposed method firstly obtains the displacement field on the soil-structure interaction boundary making use of a two-and-a-half dimensional coupled finite element–boundary element method. The method of fundamental solutions is used then as a post-processing tool to compute the response of the soil, increasing the computational efficiency of the overall methodology with respect to a methodology that considers the boundary element method as a model of the wave propagation through the soil. The accuracy of the methodology is verified for four calculation examples: a solid cylinder and a circular thin shell embedded in a homogeneous full-space and also in a homogeneous half-space. This verification is performed comparing the results with available analytical or semi-analytical solutions and a conventional two-and-a-half dimensional coupled finite element–boundary element method. Furthermore, a control methodology to increase the robustness of the method is presented.

Mathematical Modeling on Rotational Magneto-Thermoelastic Phenomenon under Gravity and Laser Pulse considering Four Theories

The aim of this investigation is making mathematical model for the variation in laser pulse, rotational gravity, and magnetic fields on the generalized thermoelastic homogeneous isotropic half-space. The governing dynamical system equations have been formulated considering the four thermoelastic models: coupled (CT) model, Lord and Shulman (LS) model, Green and Lindsay (GL) theory, and Green and Naghdi (GN III) model. Normal mode analysis technique is used to obtain the analytical expressions for the displacement components, temperature, and mechanical and Maxwell’s stresses distribution. The effect of laser pulse, gravity, and magnetic field is studied by numerical examples and displayed graphically. A comparison has been made between the theories as well as the present results and agreement with it as a special case from this study. The results predict the strong effect of magnetic field, laser pulse, and gravity field on the wave propagation phenomenon.

Simple approach for including foundation–soil–foundation interaction in the static stiffnesses of multi-element shallow foundations

In common engineering practice, foundation–soil–foundation interaction of shallow foundations is frequently ignored. This is presumably for cost/benefit reasons, since computationally demanding finite-element and/or boundary element models are required for that purpose, and its effects are usually assumed to be negligible. In this sense, the present paper provides a simple and inexpensive way of incorporating foundation–soil–foundation interaction through a numerically explicit stiffness matrix formulation. The necessary ingredients for homogeneous and non-homogeneous (shear modulus power-law variation with depth) half-spaces are given. The proposed approach is then applied to offshore wind turbines’ multiple suction caisson foundations (tripod and tetrapod), where it is observed that the foundation–soil–foundation interaction is significant. Its range of validity is also established, and valuable ready-to-use closed-form formulas for the correction factors of the stiffnesses of tripod and tetrapod groups are also derived. The methodology is applicable as long as the spacing between foundations is somewhat greater than the foundation depth.

Mathematical Modeling of Vibration Dampers of Vibration-Insulated Structures

Vibration dampers are installed on the machine foundations in order to reduce the vibration level. Such technological solutions are most expedient in the case of a harmonic load with a low instability of the vibration frequency. Unfortunately, dampers do not provide such a large reduction in the dynamic effect on the base, as vibration isolation, but in some cases their efficiency turns out to be quite sufficient with a relatively simple implementation and low manufacturing cost. The use of dynamic vibration dampers gives a great effect when an increased vibration of foundations occurs during the operation of equipment in metallurgical production, for example, when processing materials by pressure, reconstructing enterprises and replacing heavy equipment. During the operation of heavy forging equipment and manipulators for various purposes, the foundations of these devices can be considered as a rigid body. The model soil on which this foundation is installed can be considered a homogeneous elastic isotropic half-space. When calculating with such mathematical models, one can use solutions of the corresponding dynamic contact problems. A comparative analysis of the effectiveness of damping foundation vibrations using different foundation models, including the model of an elastic, homogeneous half-space and a system of semi-infinite rods, the modulus of elasticity of which increases with depth according to the quadratic law, shows a fairly close agreement.