Homogeneous Half-space Research Articles

Simplified frequency-independent model for vertical vibrations of surface circular foundations

PurposeThis paper aims to simplify a new frequency-independent model to calculate vertical vibration of rigid circular foundation resting on homogenous half-space and subjected to vertical harmonic excitation is presented in this paper.Design/methodology/approachThe proposed model is an oscillator of single degree of freedom, which comprises a mass, a spring and a dashpot. In addition, a fictitious mass is added to the foundation. All coefficients are frequency-independent. The spring is equal to the static stiffness. Damping coefficient and fictitious mass are first calculated at resonance frequency where the response is maximal. Then, using a curve fitting technique the general formulas of damping and fictitious mass frequency-independent are established.FindingsThe validity of the proposed method is checked by comparing the predicted response with those obtained by the half-space theory. The dynamic responses of the new simplified model are also compared with those obtained by some existing lumped-parameter models.Originality/valueUsing this new method, to calculate the dynamic response of foundations, the engineer only needs the geometrical and mechanical characteristics of the foundation (mass and radius) and the soil (density, shear modulus and the Poisson’s ratio) using just a simple calculator. Impedance functions will no longer be needed in this new simplified method. The methodology used for the development of the new simplified model can be applied for the resolution of other problems in dynamics of soil and foundation (superficial and embedded foundations of arbitrary shape, other modes of vibration and foundations resting on non-homogeneous soil).

Seismic wave input method for three-dimensional soil-structure dynamic interaction analysis based on the substructure of artificial boundaries

The method of inputting the seismic wave determines the accuracy of the simulation of soil-structure dynamic interaction. The wave method is a commonly used approach for seismic wave input, which converts the incident wave into equivalent loads on the cutoff boundaries. The wave method has high precision, but the implementation is complicated, especially for three-dimensional models. By deducing another form of equivalent input seismic loads in the finite element model, a new seismic wave input method is proposed. In the new method, by imposing the displacements of the free wave field on the nodes of the substructure composed of elements that contain artificial boundaries, the equivalent input seismic loads are obtained through dynamic analysis of the substructure. Subsequently, the equivalent input seismic loads are imposed on the artificial boundary nodes to complete the seismic wave input and perform seismic analysis of the soil-structure dynamic interaction model. Compared with the wave method, the new method is simplified by avoiding the complex processes of calculating the equivalent input seismic loads. The validity of the new method is verified by the dynamic analysis numerical examples of the homogeneous and layered half space under vertical and oblique incident seismic waves.

Two Solution Techniques for the Problem of Diffraction of a Plane Wave by a Revolution Body Located in a Dielectric Half-Space

Two techniques for calculation of the problem of diffraction of a plane wave by a dielectric revolution body located in a homogeneous dielectric half-space are suggested. The case, in which the permittivity of the medium inside the body is the coordinate function independent of the angular coordinate in the cylindrical coordinate system, is considered. When the first technique is used (in the case of a homogeneous medium of the body), the discrepancy of the boundary condition on the contour of the axial section of the body is constructed. Two methods are compared using the example of the problem of the plane wave scattering by a homogeneous sphere and circular cylinder of finite dimensions that are located in a half-space. The results of calculation of the field scattering pattern are presented for the case of diffraction by a finite circular cylinder having variable material characteristics.

The effects of asteroid layering on ejecta mass-velocity distribution and implications for impact momentum transfer

Most bodies in the Solar System do not have a homogeneous structure. Understanding the outcome of an impact into regolith layers of different properties is especially important for NASA’s Double Asteroid Redirection Test (DART) and ESA’s Hera missions. Here we used the iSALE shock physics code to simulate the DART impact into three different target scenarios in the strength regime: a homogeneous porous half-space; layered targets with a porous weak layer overlying a stronger bedrock; and targets with exponentially decreasing porosity with depth. For each scenario we determined the sensitivity of crater morphology, ejecta mass-velocity distribution and momentum transferred from the impact for deflection, β−1, to target properties and structure. We found that for a homogeneous porous half-space, cohesion and porosity play a significant role and the DART impact is expected to produce a β−1 between 1 and 3. In a two-layer target scenario, the presence of a less porous, stronger lower layer close to the surface can cause both amplification and reduction of ejected mass and momentum relative to the homogeneous upper-layer case. For the case of DART, the momentum enhancement can change by up to 90%. Impacts into targets with an exponentially decreasing porosity with depth only produced an enhancement in the ejected mass and momentum for sharp decreases in porosity that occur within 6 m of the asteroid surface. Together with measurements of the DART crater by the Hera mission, these results can be used to test the predictive capabilities of numerical models of asteroid deflection.

Wave Propagation of Generalized Magneto-Thermoelastic Interactions in an Elastic Medium Under Influence of Initial Stress

Magneto-thermoelastic interactions in an isotropic homogeneous elastic half-space with two temperatures are studied using mathematical methods under the purview of the Lord–Şhulman (LS) and Green–Lindsay (GL) theories, as well as the classical dynamical coupled theory (CD). The medium is considered to be permeated by a uniform magnetic field. The general solution obtained is applied to a specific problem of a half-space and the interaction between them under the influence of magnetic field subjected to one type of heating the thermal shock type. The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses, temperature and couple stress distribution. The variations in the considered variables through the horizontal distance are illustrated graphically. Comparisons are made with the results between the three theories. Numerical work is also performed for a suitable material with the aim of illustrating the results.

Rayleigh-type waves on a coated elastic half-space with a clamped surface.

Elastodynamics of a half-space coated by a thin soft layer with a clamped upper face is considered. The focus is on the analysis of localized waves that do not exist on a clamped homogeneous half-space. Non-traditional effective boundary conditions along the substrate surface incorporating the effect of the coating are derived using a long-wave high-frequency procedure. The derived conditions are implemented within the framework of the earlier developed specialized formulation for surface waves, resulting in a perturbation of the shortened equation of surface motion in the form of an integral or pseudo-differential operator. Non-uniform asymptotic formula for the speeds of the sought for Rayleigh-type waves, failing near zero frequency and the thickness resonances of a layer with both clamped faces, follow from the aforementioned perturbed equation. Asymptotic results are compared with the numerical solutions of the full dispersion relation for a clamped coated half-space. A similarity with Love-type waves proves to be useful for interpreting numerical data. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.

Ground Deformation Revealed by Sentinel-1 MSBAS-InSAR Time-Series over Karamay Oilfield, China

Fluid extraction or injection into underground reservoirs may cause ground deformation, manifested as subsidence or uplift. Excessive deformation may threaten the infrastructure of the oilfield and its surroundings and may even induce earthquakes. Therefore, the monitoring of surface deformation caused by oil production activities is important for the safe production of oilfields and safety assessments of surrounding infrastructure. Karamay oilfield is one of the major oil and gas fields in China. In this study, we take the Karamay oilfield in Xinjiang as a case study to detect surface deformation caused by subsurface fluid injection. Sentinel-1A images of 32 ascending (Path 114) and 34 descending (Path 165) tracks spanning March 2017 to August 2018, were used to derive vertical and horizontal deformation over Karamay oilfield using the MSBAS-InSAR method. The observed two-dimensional deformation exhibited significant vertical and east-west deformation in this region. The maximum uplift and horizontal velocity was approximately 160 mm/year and 65 mm/year, respectively. We also modeled one of the typical deformation zones using a dislocation model in a homogenous elastic half-space.

1D finite element artificial boundary method for transient response of ocean site under obliquely incident earthquake waves

Ocean site consisting of seawater-seabed-bedrock system is modeled as horizontally layered fluid-poroelastic-solid media. Earthquake wave is assumed as a plane P or SV wave of oblique incidence from the solid bedrock into the site. A time-domain numerical method called one-dimensional finite element artificial boundary (1D-FE-AB) method is proposed to effectively and accurately calculate the transient response of the ocean site under the obliquely incident earthquake waves. In this method, a horizontally line-shaped AB is firstly introduced in the solid bedrock to divide the whole site into the fluid-poroelastic-solid layer and the remaining homogenous solid half space. The former is semi-discretized by FE method along depth, and the latter with the incident earthquake wave is simulated by an exact AB condition. The resulting semi-discrete system with AB is then transformed into a spatially 1D dynamic equation along depth by using the Snell's law of plane body wave propagation in multilayer media. The 1D equation can be solved by the time integration algorithm such as Newmark method. Numerical examples are finally given to demonstrate the effectiveness of the proposed 1D-FE-AB method.

Reflection of plane waves in a thermoelastic diffusive medium under the effect of microtemperatures

The present analysis is aimed to model and study the characteristics of various reflected waves in a homogeneous and isotropic thermoelastic diffusive half-space with microtemperatures. It is shown that there exist four kinds of coupled longitudinal waves in addition to transverse and microtemperature waves in such type of medium. The reflection coefficients and energy ratios of these reflected waves have been computed numerically with the help of MATLAB programing when (i) a set of coupled longitudinal waves is made incident and (ii) a transverse wave is made incident. The numerical values of modulus of reflection coefficients are presented graphically to depict the effect of thermodiffusive parameter. It has been verified that there is no dissipation of energy at the free plane boundary during reflection phenomena.

Transmission and reflection of waves at an interface between ordinary material and metamaterial

The propagation of waves though bulk composite material (or metamaterial) is to a large extent understood, but the problem of how waves are transmitted and reflected at any interface between two different composites has hardly been considered, particularly if the composites have random microgeometry. The present work makes a contribution towards the solution of this problem by studying a configuration of particular simplicity. A randomly inhomogeneous composite which occupies the half-space x2 > 0 is bonded at x2=0 to a half-space x2 < 0 of a homogeneous medium, and an acoustic wave is incident on the interface from below. The composite has two constituents which have the same elastic modulus but different densities. A variational approximation of Hashin–Shtrikman type yields an explicit formula for the Fourier/Laplace transform of its effective density and a corresponding dispersion relation for the plane mean waves that the composite can support. When its two-point correlation is chosen as h(r)=e−r/a, the dispersion relation admits two distinct mean acoustic waves, and the only directly available conditions at the interface x2=0 are continuity of displacement and traction, which are insufficient in number to fix the two transmission coefficients and one reflection coefficient. It is possible, however, to apply the variational approach directly to an infinite medium comprising the composite in x2 > 0 and the homogeneous medium in x2 < 0, leading to a problem of Wiener–Hopf type. When the homogeneous half-space has the same properties as one of the constituents of the composite, the Wiener–Hopf problem involves only a single unknown “polarization” supported in x2 > 0 and is solved explicitly.

Surface-to-Underground Transient Electromagnetic Detection of Water-Bearing Goaves

Water-bearing goaves are one of the main causes for water inrush accidents in Chinese coal mines. At present, both the surface and subsurface geophysical technologies cannot fully meet the technical requirements of high precision detection for coal mines safety development and production. Based on the surface-to-borehole electromagnetic method, we propose a surface-to-underground transient electromagnetic prospecting (SUTEM), in which the large rectangular loop is laid on the surface as the transmitting source and the receivers are laid in the subsurface roadways of the coal mines. First, we describe and discuss the SUTEM signal response using numerical simulation for a homogeneous half space. The numerical results show that the curve of the induced electromotive force (EMF) first increases to a peak, and then decreases gradually till the end of the time window, the shape of the curve closely reassembles a parabolic. So the conductive bodies can be identified with the characters of the EMF curve. Second, we conducted a numerical simulation using the 3-D, unstructured tetrahedral grids vector finite-element total-field electromagnetic forward modeling algorithm for the response of typical 1-D layered models and 3-D models. The results show that the relative sensitivity is no relation with the depth of the abnormal layer when receiving points are under the layer, the deeper the abnormal layer is, the more obvious the advantage of SUTEM is. Finally, the field experiment was carried out at a coal mine, where the positions of water-bearing goaves are known. The SUTEM detection results do not exhibit a clear trend in the variation of electrical properties in the vertical direction of the formation. However, the variation in the lateral electrical properties is clear, the expression of the low-resistivity goaves is much clearer than the results from the surface TEM. The field experiment result is in agreement with the suggestions of the computer simulation and confirms the effectiveness of SUTEM.

Mathematical study of Rayleigh waves in piezoelectric microstretch thermoelastic medium

Abstract This paper is concerned with the study of propagation of Rayleigh waves in a homogeneous isotropic piezo-electric microstretch-thermoelastic solid half-space. The medium is subjected to stress-free, isothermal boundary. After developing a mathematical model, the dispersion curve in the form of polynomial equation is obtained. Phase velocity and attenuation coefficient of the Rayleigh wave are computed numerically. The numerically simulated results are depicted graphically. Some special cases have also been derived from the present investigation.

Effects of inhomogeneity on rolling contact fatigue life in elastohydrodynamically lubricated point contacts

PurposeThis paper aims to investigate the effects of inhomogeneities on the rolling contact fatigue (RCF) life in elastohydrodynamically lubricated (EHL) point contacts.Design/methodology/approachA numerical model for predicting the RCF life of inhomogeneous materials in EHL contacts was established by combining the EHL model and the inclusion model through the eigen-displacement and then connecting to the RCF life model through the subsurface stresses. Effects of the type, size, location and orientation of a single inhomogeneity and the distribution of multiple inhomogeneities on the RCF life were investigated.FindingsThe RCF life of a half-space containing manganese sulfide (MnS) inhomogeneity or the mixed inhomogeneity of aluminium oxide (Al2O3) and calcium oxide (CaO) was longer than that for the case of Al2O3inhomogeneity. For a single ellipsoidal MnS inhomogeneity, increases of its semi-axis length and decreases of its horizontal distance between the inhomogeneity and the contact center shortened the RCF life. Furthermore, the relationship between the depth of a single MnS inhomogeneity and the RCF life was found. For the half-space containing multiple inhomogeneitites, the RCF life decreased remarkably compared with the homogeneous half-space and showed discreteness.Originality/valueThis paper implements the prediction of the RCF life of inhomogeneous materials under EHL condition.

Effects of dust layers on thermal emission from airless bodies

We have investigated the influence of thin thermally opaque dust layers on the thermal emission of rocks and regolith and determined the thermal response of these dust-covered surfaces to diurnal insolation cycles. Results are computed for Hayabusa2’s target asteroid (162173) Ryugu, which was observed by thermal infrared instruments on the orbiter and in situ. We show that even a very thin (10..100 μm) fine-grained porous dust layer with thermal inertia of 25 J m−2 K−1 s−1/2 can have a significant influence on surface temperatures and alter the apparent thermal inertia of the underlying material derived under the simplified assumption of a homogenous half space by more than 20%. The masking of the underlying material is complete at about 1 diurnal skin depth, corresponding to ~ 10 mm on Ryugu. Between 0.1 and 1 diurnal skin depths, we find a thermal lag smaller than what would be predicted for a surface consisting of dust only.If a dust cover were present on Ryugu, this should be clearly visible in the data returned by the orbiter’s thermal infrared imager (TIR) and the MASCOT lander’s radiometer (MARA), which observed a single boulder at the landing site. However, this appears not to be the case, and dust seems to play a minor role in the thermal emission from the asteroid.

3D frequency-domain airborne EM forward modelling using spectral element method with Gauss–Lobatto–Chebyshev polynomials

ABSTRACTThe spectral element method (SEM) based on high-order complete orthogonal polynomials is an accurate and efficient numerical method for electromagnetic modelling due to its spectral accuracy and exponential convergence. The SEM combines the flexibility of the finite-element method and the high accuracy of the spectral method. In this paper, we introduce SEM into three-dimensional frequency-domain airborne electromagnetic forward modelling. Starting from Maxwell's equations, we obtain a vector Helmholtz equation for the electric field. We use the Galerkin method to discretise the Helmholtz equation, in which the curl-conforming Gauss–Lobatto–Chebyshev polynomials are used as basis functions. The GLC polynomials help to derive the analytical expressions of entries in the system matrix and thus guarantee the modelling accuracy. Finally, we use the direct solver MUMPS to solve for the electric field and calculate the magnetic field by interpolation. For numerical experiments, we first compare our results with the semi-analytical solutions of a homogeneous half-space to verify the accuracy of our algorithm. We then analyse the characteristics of SEM by assuming different orders of interpolation polynomials and meshes. We also compare our method with the finite element method and SEM based on Gauss–Lobatto–Legendre polynomials. The results show that SEM is an efficient and effective method for electromagnetic modelling, it can deliver very accurate results and is less sensitive to mesh quality than the finite element method.

Dynamic Soil–Structure–Equipment Interaction (I): Closed-Form Analytical Solution for Incident Plane SH wave Based on Rigid Foundation Model

ABSTRACT Soil–structure–equipment interaction is investigated using a two-dimensional (2D) whole model of soil–structure–equipment interaction with rigid foundation embedded in a homogenous half-space. A closed-form analytical solution is presented for the model response to incident plane SH wave based on wave function expansion method, and the accuracy of the solution is verified by comparison with published results. The analysis focuses on the effects of soil–structure interaction (SSI) on the response of structure–equipment interaction. The results show that SSI has significantly influence the response of structure–equipment interaction. The traditional fixed base model of structure–equipment interaction may not reflect the response of structure–equipment interaction properly. After considering the effect of SSI, the peak effective input motion of structure–equipment system is magnified 35.5% than that of fixed base model, and the structure relative response and the equipment relative response are all smaller. The peak relative response of structure and that of the equipment are highly dependent on the equipment–structure frequency ratio and mass ratio. Especially, the peak of equipment relative response will tend to linear increase, instead of rapid decreasing when the equipment–structure frequency ratio is in the range of 0.5 ~ 1.0. Although the model of this paper is simple, it may be helpful to obtain insight into the effects of SSI on structure–equipment interaction and provide a comparison for the further research about soil–structure–equipment interaction with flexible foundation.

Effects of full transmitting-current waveforms on transient electromagnetics: Insights from modeling the Albany graphite deposit

In transient electromagnetic (TEM) methods, the full transmitting-current waveform, not just the abrupt turn-off, can have effects on the measured responses. A 3D finite-element time-domain forward-modeling solver was used to investigate these effects. This was motivated by an attempt to match, via forward-modeling, real data from the Albany graphite deposit in northern Ontario, Canada. Initial modeling results for homogeneous half-spaces illustrate the effects that a full waveform can have on TEM responses, especially the durations of the steady stage and turn-off time. For the Albany data set, a geophysical conductivity model was developed from a geologic model that itself had been constructed predominantly from drillhole information. The conductivities of the various geologic units in the model were first estimated based on typical conductivity values for the respective rock types, then adjusted to fit the measured TEM data as closely as possible. We found that the TEM responses differed significantly from the pure step-off response and that incorporating the effects of the full waveform (particularly the linear ramp turn-off) greatly improved the match between observed and computed responses, especially for the early measurement times. In addition, this Albany example illustrates the presence of sign changes in TEM data caused primarily by localized conductivity targets.

Elastic field by a dislocation loop in an anisotropic elastic half-space with general boundary conditions and its application in nanoindentation of single crystals

The elastic displacement and stress fields induced by a dislocation loop of polygonal shape within an anisotropic homogeneous half-space with general boundary conditions on the flat surface z = 0 are obtained by employing the corresponding point-force Green's functions. We first study in detail the effects of the eight sets of boundary conditions on the elastic fields of dislocation loops. We then identify the elastic fields of dislocation loops corresponding to three of the eight sets of boundary conditions. These are the traction-free, completely, and partially sticking contact between the indenter and the indented single crystals, which are employed to establish the distribution dislocation loop model for simulating nanoindentation of single crystals. It is the first time that significant differences in the magnitude of sink-in and pile-up of the indented profile, and of the residual stress components, corresponding to different frictional boundary conditions, are observed, which could serve as benchmark in future nanoindentation study.

Analytical solutions for elastic SH-waves propagating through an isotropic inhomogeneous layer

Plane time-harmonic elastic SH-wave propagation through an isotropic inhomogeneous layer surrounded by two homogeneous half-spaces is studied in this article. The material properties of the inhomogeneous layer are assumed to be non-uniform along the thickness direction according to a distribution law described by the triconfluent Heun functions or their polynomial forms that contain a number of optional parameters. The general analytical solution of the governing equation for elastic SH-waves in the layer is presented. Employing optional parameters, the material-property profiles can be varied to a relatively large extent without the need to seek new solutions of the governing equation for a chosen material-property profile. If the wave speed is constant in the inhomogeneous layer, the derived analytical solution is exact; otherwise the analytical solution is approximate. As a part of this article, the method enabling to find an approximate analytical solution of the governing equation for predetermined material functions is also presented. The applicability of the analytical solutions are tested and discussed based on the representative examples, and at the same time, the analytical results are compared with numerical ones to demonstrate their validity.

Dual boundary element method for analyzing three-dimensional cracks in layered and graded halfspaces

This paper presents a dual boundary element analysis of three-dimensional cracks in layered and graded halfspaces. The fundamental solution of a multilayered solid is used to develop the dual boundary element method so that only the external boundary surface and the crack surface need to be discretized while the material interfaces do not need to be discretized. Infinite boundary elements and crack-tip discontinuous elements are introduced to consider the far-fields of a layered halfspace and capture the crack-tip behavior, respectively. Special attentions are given to strongly singular and hypersingular integrals in the discretized displacement and traction boundary integral equations. For square-shaped, penny-shaped and elliptical cracks located in a homogeneous halfspace, the stress intensity factors obtained with the present formulation are in very good agreement with existing numerical results and closed-form solutions. The square-shaped cracks in horizontally layered halfspaces and the penny-shaped and elliptical cracks in graded halfspaces are analyzed. Results show that the material heterogeneity in layered and graded halfspaces can have a profound effect on the stress intensity factors.