Homogeneous Elastic Half-space Research Articles

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.

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.

Beyond elasticity: Are Coulomb properties of the Earth's crust important for volcano geodesy?

Geodetic modelling has become an established procedure to interpret the dynamics of active volcanic plumbing systems and magma transfer within the crust. Most established geodetic models implemented for inverting geodetic data share similar physical assumptions: (1) the Earth's crust is modelled as an infinite, homogeneous elastic half-space with a flat surface, (2) there is no anisotropic horizontal stress to simulate tectonic stresses, (3) the source boundary conditions are kinematic, i.e. they account for an instantaneous inflation or deflation of the source. Field and geophysical observations, however, provide evidence that significant inelastic shear deformation of the host rock can accommodate the propagation of dykes and sills. We show that inelastic processes accommodating the emplacement of dykes in the brittle crust have large implications for dyke-induced surface deformation patterns. We present two quantitative laboratory experiments of dyke emplacement, during which the syn-emplacement surface deformation is monitored. In one experiment, the host material is elastic gelatine, whereas in the other experiment the host material is cohesive Coulomb, plastic silica flour. Even if both experiments produce sub-vertical dykes of similar shapes, their emplacement mechanisms and their associated surface deformation strongly differ. In the gelatine experiment, the dyke propagates as a tensile fracture in a dominantly elastic host, and the surface deformation exhibits two uplifting bulges separated by a trough parallel to, and above the apex of, the underlying dyke. Conversely, in the silica flour experiment, the dyke propagates as viscous indenter through a dominantly plastic host, and the surface deformation exhibits a single uplifting area that narrows through time. The comparison of our experiments shows that (1) plastic deformation (e.g., shear failure, compaction) of the host has large effects on dyke-induced surface deformation patterns and needs to be considered in geodetic models, and (2) dyke emplacement mechanisms matter in geodetic modelling, strongly suggesting that commonly used kinematic geodetic models such as the opening rectangular dislocation model (Okada 1985) are limited for revealing the physics and dynamics of volcano plumbing systems. Finally, our silica flour experiment shows that pure uplift geodetic signals can result from the emplacement of a dyke emplaced as viscous indenter, whereas such signals are commonly modelled using geodetic models of inflating spherical/elliptical or horizontal planar source. Our experiments call for the design of new geodetic models that account, even partly, for the plastic deformation component of the Earth's brittle crust.

Cracked Equivalent Beam Models for Assessing Tunneling-Induced Damage in Masonry Buildings

This paper proposes simple numerical models that simulate the influence of tunneling-induced ground movements on masonry buildings founded on shallow strip footings. The focus is on developing an improved understanding of the impact of structural discontinuities due to pre-existing cracks and joints between adjacent buildings. The soil–structure system is idealized with equivalent Timoshenko beams connected to a homogenous elastic half-space with rigid links. The flexibility of the equivalent beam at the location of the crack or building interface is determined using linear-elastic fracture mechanics. The proposed models are first applied to a generic numerical example. Then, they are used to interpret the detailed displacement and strain measurements gathered from a masonry church during nearby tunneling. The results show that pre-existing cracks can have variable influence on the soil–structure interaction depending on their extent and position relative to the building and settlement trough. For the investigated case study, the proposed models illustrate how pre-existing large cracks in the hogging zone localized deformations and how the interaction between adjacent buildings drastically altered response mechanisms.

Effect of material damping on the impedance functions of an embedded circular foundation under vertical and horizontal excitation

Machine foundations are generally affected by the vibratory shocks from different machines. The behavior of these foundations is influenced by the properties of underlying soil and the excitation frequency of the applied dynamic load. The influence of material damping on the dynamic impedance functions of a circular disk embedded in homogeneous elastic half space is analyzed using one-dimensional wave propagation in cones (cone model) and the results are presented in the form of dimensionless plots to observe the more realistic response of machine foundations. Three different types of material damping models viz., Hysteretic, Voigt and Kelvin model are introduced in the above elastic solutions using correspondence principle. The spring and damping coefficients of the embedded foundation are then computed in a wide range of frequency of excitation under vertical and horizontal mode of vibration varying the influencing parameters namely dimensionless frequency (a0), Poisson’s ratio (ν), embedment ratio (e/r0) and damping ratio (ξ). The outcomes from the present analysis suggest that the spring coefficient is nonlinearly affected by the dimensionless frequency and embedment ratio, for both the modes of vibration. The effect of material damping on spring coefficient is only significant for a0> 2, irrespective of the damping model used.

On the Lamb problem: forced vibrations in a homogeneous and isotropic elastic half-space

The problem of vibrations generated in a homogeneous and isotropic elastic half-space by spatially concentrated forces, known in Seismology as (part of) the Lamb problem, is formulated here in terms of Helmholtz potentials of the elastic displacement. The method is based on time Fourier transforms, spatial Fourier transforms with respect to the coordinates parallel to the surface (in-plane Fourier transforms) and generalized wave equations, which include the surface values of the functions and their derivatives. This formulation provides a formal general solution to the problem of forced elastic vibrations in the homogeneous and isotropic half-space. Explicit results are given for forces derived from a gradient, localized at an inner point in the half-space, which correspond to a scalar seismic moment of the seismic sources. Similarly, explicit results are given for a surface force perpendicular to the surface and localized at a point on the surface. Both harmonic time dependence and time $$\delta $$ -pulses are considered (where $$\delta $$ stands for the Dirac delta function). It is shown that a $$\delta $$ -like time dependence of the forces generates transient perturbations which are vanishing in time, such that they cannot be viewed properly as vibrations. The particularities of the generation and the propagation of the seismic waves and the effects of the inclusion of the boundary conditions are discussed, as well as the role played by the eigenmodes of the homogeneous and isotropic elastic half-space. Similarly, the distinction is highlighted between the transient regime of wave propagation prior to the establishment of the elastic vibrations and the stationary-wave regime.

The critical forces in the weak ground under the influence of technogenic relief

The initial critical force that the ground is exposed to from the weight of a structure is one of the main characteristics of the “structure-ground environment” system. A significant part of grounds are classified as weak, and the strain modulus value does not exceed 5 MPa. These soils have a weak load-carrying capacity and they are significantly deformed by the weight of the structures and the static load of technogenic relief. The theory and methodology of determining the initial critical force and its maximal action depth for various two-dimensional technogenic land forms, the force diagrams of which are described by trapezes and triangles of various types, are considered. The models grounds considered herein are a homogeneous elastic half-space or horizontally layered strata on the surface of a homogeneous elastic half-space. The nomograms of the relations of the initial critical force and its maximum depth of action to the ground’s physical and mechanical properties are attained. The identified relations are quasi-linear and the value of the initial critical force increases with an increase in the ground’s physical and mechanical parameters. The presented nomograms significantly simplify the calculation of the initial critical force in the ground.

Rayleigh wave motions in an orthotropic half-space under time-harmonic loadings: A theoretical study

Rayleigh surface wave motion in a homogeneous orthotropic elastic solid half-space under time-harmonic sources is theoretically investigated. In this article, closed-form expressions of the Rayleigh wave field as a result of a time-harmonic concentrated line load are simply determined by using reciprocity theorems. For the verification purpose, analytical predictions are also obtained through Fourier transform technique with the aid of the residue theorem. The solutions found by the two approaches are shown to be mathematically the same. As an example of calculation, surface waves due to a set of buried loadings in the orthotropic half-plane are then examined in detail. A discussion of these results is presented to explore the influence of the loading types on the displacement amplitudes of the generated wave fields.

Exact Rayleigh-Wave Solution from a Point Source in Homogeneous Elastic Half-Space

ABSTRACTA Rayleigh wave, often the most visible component in the far-field seismograms, is an important type of seismic-wave motion associated with the Earth’s surface. In this study, we explore some of the general properties of the Rayleigh wave in a homogeneous elastic half-space. Starting from the displacement expressed in the form of a wavenumber integral in the frequency domain, we extract the contribution from the pole in the complex wavenumber plane to obtain the excitation formulae of the Rayleigh wave by the residue theorem for complex integrals. Numerical results are compared with the full wavefield solutions to validate our solutions. By examining the analytical expressions obtained, we explore some basic properties of Rayleigh waves such as the particle motion and geometrical spreading. We also demonstrate that these properties of the Rayleigh wave excited by a point source are slightly different from but mostly consistent with the well-known classical properties of plane Rayleigh waves.

A numerical study on the transverse seismic response of lined circular tunnels under obliquely incident asynchronous P and SV waves

The numerical analysis of the transverse seismic response of tunnels is an important issue in the design and safety evaluation of tunnel structures, where the spatial variation of earthquake ground motion induced by wave passage effect has been recognized as an important influence factor that cannot be ignored. In this paper, a 2-D transient dynamic finite element modelling technique is applied to investigate the seismic response of two soil-structure interaction systems subjected to obliquely incident plane P and SV waves. Based on the theory of wave input, the incident earthquake motion is modeled as equivalent nodal forces applied on the truncated boundary, and the free field ground response is calculated from the 1-D time-domain finite element method by considering the wave passage effect. The numerical scheme is implemented into the commercial software ABAQUS, and its accuracy is verified by comparing with the analytical solutions of soil-structure interaction problems. Firstly, the verified method is applied to study the transverse seismic response of a circular tunnel embedded in a homogenous elastic half-space under obliquely incident P and SV waves, where the influence of the incident wave type, incident angle, burial depth and tunnel diameter is discussed. Secondly, the response of circular tunnels embedded in a multi-layered non-homogenous elastic foundation is studied, where the influence of tunnel position in the foundation, flexibility ratio between soil and tunnel and stiffness difference of adjacent soil layers in the foundation are extensively investigated. Moreover, comparison is drawn between the tunnel responses in the homogenous and non-homogenous foundation systems, and the importance of conducting asynchronous seismic response analysis of circular tunnels is highlighted.

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.

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.

Eigenvalue approach for generalized thermoelastic porous medium under the effect of thermal loading due to a laser pulse in DPL model

The aim of the present study is concerned with the thermal loading due to laser pulse on thermoelastic porous medium under dual-phase-lag model DPL. The material is a homogeneous isotropic elastic half-space and heated by a non-Gaussian laser beam with the pulse duration of 8 ps. Eigenvalue approach is proposed to analyze the problem and obtain the analytical solutions of the displacement components, stresses, temperature distribution and change in the volume fraction field. The results of the physical quantities have been illustrated graphically by the comparison between the coupled theory, DPL model and Lord–Shulman theory of the certain value of time and distance. The results presented in this article should prove useful for researchers in material science (porous media), designers of new materials, physicists as well as for those working on the development of heat laser pulse practical situations as in geophysics, optics, acoustics and geomagnetic.

BEM modeling of a 3D homogeneous anisotropic elastic half space under dynamic load

In this paper, we consider a problem of a homogeneous anisotropic and linearly elastic half space subjected to dynamic loading. Zero body forces and vanishing initial conditions are assumed. The problem is solved using the Boundary Element Method in the Laplace transformed domain. Integral expressions of the three-dimensional dynamic fundamental solutions for displacements and tractions are utilized. We employ the displacement boundary integral equations which are regularized using the static part of the dynamic anisotropic traction fundamental solution. For the spatial discretization of the boundary integral equations mixed boundary elements are adopted. The geometry of the considered domain is approximated with quadrilateral quadratic eight-noded elements. On the boundary elements displacements and tractions are interpolated using linear and constant shape functions, respectively. Time-domain solutions are obtained using suitable scheme for numerical inverse Laplace transform. The boundary-element solutions for the illustrative problem of an anisotropic elastic half space subjected to a Heaviside-type load are provided.

Closed-form stiffnesses of multi-bucket foundations for OWT including group effect correction factors

Offshore Wind Turbine (OWT) support structures need to satisfy different Limit States (LS) such as Ultimate LS (ULS), Serviceability LS, Fatigue LS and Accidental LS. Furthermore, depending on the turbine rated power and the chosen design (all current designs are soft-stiff), target natural frequency requirements must also be met. Most of these calculations require the knowledge of the stiffnesses of the foundation which, especially in the case of large turbines in intermediate waters (30–60 m), might need to be configured using multiple foundation elements. For this reason, this paper studies, for a homogeneous elastic halfspace, the static stiffnesses of groups of polygonally arranged non-slender suction bucket foundations in soft soils modeled as rigid solid embedded foundations. A set of formulas for correcting the stiffnesses obtained from isolated foundation formulation are proposed. It is shown through the study of several multi-megawatt OWTs that, as expected, group effects becomes more relevant as spacing decreases. Also, group effects are sensitive mainly to shear modulus of soil, foundation shape ratio and diameter, and the number of foundations. The results obtained from the soil-structure system show that ignoring group effects may add significant errors to the estimation of OWT fundamental frequencies and leads to either overestimating or underestimating it by 5%. This highlights the importance of adequately modeling the interaction between elements of closely-separated multi-bucket foundations in soft soils, when current guidelines specify the target fundamental frequency to be at least 10% away from operational 1P and blade passing frequencies (2P/3P frequencies).

УПРУГОЕ МОМЕНТНОЕ ПОЛУПРОСТРАНСТВО ПОД ДЕЙСТВИЕМ ОСЕСИММЕТРИЧНЫХ НЕСТАЦИОНАРНЫХ ПОВЕРХНОСТНЫХ КИНЕМАТИЧЕСКИХ ВОЗМУЩЕНИЙ

The article deals with elastic homogeneous isotropic half-space filled with the Cosserat medium. A cylindrical coordinate system is used. A closed system of equations includes equations for the non-trivial components of the displacement and rotation potentials, as well as relations relating the displacements to the potentials and components of the stress and moment stress tensors with displacements and the angle of rotation. On the boundary plane, normal displacements are specified, and the angle of rotation and tangential displacements are absent. Initial conditions are zero. All components of the stress-strain state are assumed to be limited. A system of dimensionless quantities is used. The solution is represented as a convolution with respect to time and a generalized convolution with respect to the radius of surface perturbations with influence functions. To construct them, Hankel transforms are applied along the radius and Laplace in time, as well as expansion in power series in a small parameter in the linear approximation. Found image of all surface influence functions. For example, the following function is considered corresponding to the normal voltage. Its original is on the border of a half-space using the connection of axisymmetric and plane problems, namely, taking into account the proportionality of the Hankel and Fourier images respectively. It uses the previously constructed solution of the planar problem. As a result, the desired function is represented as integrals, which are understood in the sense of regularized values. Analytic expressions for these integrals and the influence function itself are found. Examples of calculations of the influence functions of a granular composite of aluminum shot in an epoxy matrix are given. Two variants of action on the half-space of surface normal displacements are also considered: the disturbance concentrated at the origin and distributed in a circle. Analysis of the results shows that the effect of the moment properties of the medium depends significantly on the value of the parameter characterizing the relationship of elastic displacements and rotational motions. For the material in question in quantitative terms it is not great. However, consideration of moment stress leads to qualitative changes. Namely, there is an additional wave front.

Nonstationary Axisymmetric Motion of an Elastic Momentum Half-Space Under Nonstationary Normal Surface Displacements

An elastic homogeneous isotropic half-space filled with the Cosserat medium has been considered. The deformed state is characterized by independent displacement and rotation vectors. At the initial time and at infinity, there are no perturbations.On the boundary of a half-space, normal displacements are given. All components of the stress–strain state are supposed to be limited. A cylindrical coordinate system with an axis directed inward to the half-space has been used. With axial symmetry, the resolving system of equations includes three hyperbolic equations with respect to the scalar potential and the nonzero components of the vector potential and the rotation vector. The components of displacement vectors, rotation angle, stress tensors, and stress moments are related to potentials by known relationships. The solution of the problem has been sought in the form of generalized convolutions of a given displacement with corresponding superficial influence functions. These functions have been constructed using a Hankel transform with respect to the radius and a Laplace transform with respect to time. All images have three terms. The first of these terms corresponds to the tension–compression wave, and the remaining two are determined by the associated shear and rotation waves. The originals of the first components have been found accurately through successive inversion of transforms. For the remaining terms, we have used expansion in power series in a small parameter characterizing the relation between shear and rotation waves. The images of the first two coefficients of these series have been found. The corresponding originals have been determined by successive inversion of transforms. Examples of calculations of the regular components of the influence function of a granular composite of aluminum shot in an epoxy matrix have been given.

On the complexity of seismic waves trapped in irregular topographies

Documented observations from strong seismic events have repeatedly shown that the ground surface topography significantly affects the characteristics of seismic waves (amplitude, frequency and duration) travelling from the deeper layers of the crust compared to what the ground motion would have been on the surface of a flat homogeneous linear elastic half-space. Although numerous theoretical studies have qualitatively corroborated these observations, they systematically underestimate the absolute level of topographic amplification up to an order of magnitude or more in some cases. In this paper, we try to bridge the quantitative gap between previous theoretical studies and observations by systematically studying the role of geometry, stratigraphy, and ground motion characteristics through a series of elaborate numerical analyses. We show a collection of examples that highlight the effects of topography on seismic ground shaking, and we point out what these results suggest in the context of the current state of earthquake engineering practice. Examples range from semi-analytical solutions of wave propagation in infinite wedges to three-dimensional numerical simulations of topography effects using digital elevation map-generated models and layered geologic features. We conclude by demonstrating that topography effects vary strongly with the stratigraphy and material properties of the underlying geologic materials, and thus it cannot be accurately predicted by studying the effects of ground surface geometry alone.

Soil Structure Interaction Effects on Hysteretic Energy Demand for Stiffness Degrading Systems Built on Flexible Soil Sites

This paper aims to investigate the effect of soil-structure interaction on plastic energy demand spectra directly derived from the energy-balance equations of soil-shallow-foundation structure with respect to an ensemble of far-field strong ground motions recorded on alluvium soil. The superstructure is modeled as a single-degree-of-freedom (SDOF) oscillator with Modified Clough stiffness degrading modelresting on flexible soil. The soil below the superstructure is modeled as a homogeneous elastic half space and is considered through the concept of Cone shallow foundation Models. A parametric study is carried out for 2400 soil-structure systems with various aspect ratios of the building as well as dimensionless frequency with wide range of fundamental fixed-base period and target ductility demand values subject to an ensemble of 19 earthquakes. Results show that generally for the structure located on softer soils severe dissipated energy drop will be observed with respect to the corresponding fixed-base system. The only exception is for the case of short period slender buildings in which the hysteretic energy demand of soil-structure systems could be up to 70% larger than that of their fixed-base counterparts. Moreover, dissipated energy spectra are much more sensitive to the variation of target ductility especially for the case of drastic SSI effect.

Can a discrete dynamic model ever perfectly simulate a continuum?

This article elaborates on the seemingly impossible notion that a continuous elastic body subjected to dynamic sources on its outer surface could ever be substituted in all of its essential qualities by a discrete model accomplished with finite elements. This topic is taken up herein and discussed in the context of a very simple two-dimensional model involving the propagation of SH shear waves (or acoustic waves) in a homogeneous elastic half-space. It is shown that there exists at least one discrete solid, referred to here as the Guddati Solid, which from its external surface behaves exactly like the continuum and is able to transmit waves of any frequency and any wavelength. This is a rather surprising finding in that it seems to contradict some well-known elastodynamic representation theorems, not to mention falsify the widespread belief that a discrete system can never behave like the continuum it purports to model. The purpose of this article is thus to present one example which disproves this widely believed postulate.