Elastic Half-space Research Articles

The slow slip event cycle along the Izmit segment of the North Anatolian Fault

The occurrence of aseismic creep along seismogenic faults significantly impacts seismic hazard assessment by releasing accumulated stress and reducing the slip deficit. Since the 1999 Mw7.6 Izmit earthquake on the North Anatolian Fault in Türkiye, while aseismic creep has been observed as a postseismic response to the Izmit rupture, additional slow slip events were detected in 2015 and 2016, accommodating several millimeters of relative displacement over periods of approximately one month. By automating Interferometry Synthetic Aperture Radar time series processing from 2016 to 2021 (FLATSIM project) and applying specific post-processing, we extract the tectonic signal to estimate the slip dynamics of the Izmit segment, including the detection and characterization of slow slip events. Modeling the slip distribution at depth on a 2D fault interface within a layered elastic half-space, we estimate a locking depth of 11km and steady creep between 2 and 5km. Above the steady creep zone, we identify two new shallow slow slip events in March 2018 and November 2019, with moment magnitudes of 4.3 and 4.4, respectively. Based on creepmeter measurements, we estimate a lateral propagation velocity of 6.4km/day for the 2019 event. The location of these shallow slow slip events above the sedimentary-bedrock interface suggests a critical role of variations in frictional properties in the occurrence of transient slip events.

Love wave along the corrugated interface between an elastic half-space and the corrugated porous layer containing two fluids

This article presents a comprehensive analysis of Love wave propagation along the interface between a porous layer saturated with two immiscible fluids and an underlying elastic half-space. The study places particular emphasis on the corrugated nature of both the interface and the upper boundary of the porous layer. A complex dispersion relation, governed by appropriate boundary conditions, is derived and subsequently simplified into four distinct subcases. The effects of corrugation parameters at the upper boundary of the porous layer and the interface between the layer and the elastic half-space are investigated. The influence of the undulation parameter, which represents the combined effects of wave number and porous layer thickness is also analyzed on the phase speed of the Love wave. In the numerical discussion, a geological structure analogous to oil-contaminated sandstone aquifers in contact with sedimentary bedrock, a common formation in coastal regions, is considered as a realistic example. The key findings reveal that the phase speed of Love waves is highly sensitive to both the corrugation and undulation parameters. Increased corrugation at the upper boundary of the layer enhances phase speed, while corrugation at the interface decreases it. An increase in the undulation parameter decreases the phase speed of the Love wave. Additionally, the presence of oil and water within the porous sandstone significantly reduces phase speed, particularly at lower frequencies, due to higher viscosity and internal friction. This study provides valuable insights into wave propagation dynamics in complex geological settings, with implications for hydrogeology, environmental monitoring, and seismic risk assessment.

Long-Range Electrostatic Adhesive Contact Between an Elastic Half-Space and a Rigid Indenter With Power-Law Profile

Abstract In this study, the electrostatic adhesive contact between a smooth indenter with a power-law geometry and an elastic half-space is studied using both a theoretical and numerical approach. Both the indenter and substrate are coated with an electrically insulating layer. The Maxwell stress and hard-wall constraint are applied to describe the interaction between the indenter and elastic counter face. By assuming electrostatic adhesion as a long-range interaction, we derived a theoretical relation between external load and contact radius. We show that the theoretical and numerical results are plausible when the Tabor parameter is small. However, when the Tabor parameter is large, the numerical results get closer to the Johnson–Kendall–Roberts (JKR) limit. The generalized Tabor parameter, which depends on the applied voltage and indenter shape, has been derived by following the technique of dimensional analysis.

Theoretical and numerical modeling of a shallow foundation stiffness based on the theory of elastic half-space

Abstract The paper presents the derivation of equations for the calculation of the spring constants ky, kz, and kϕ and the spring coefficients βy, βz , and βϕ used in the modeling of a foundation with a rectangular base on elastic soil. All the equations were derived based on information provided in literature, for example, Barkan, Gorbunov-Possadov, and Whitman et al. To demonstrate the application of these equations in practice, two numerical models of a reinforced concrete frame structure that was built on soil were created using Abaqus FEA software. Model A represents a complex three-dimensional numerical model that consists of a reinforced concrete frame structure, which has a soil layer beneath it. Model B represents a simple three-dimensional numerical model consisting of a reinforced concrete frame structure, where the stiffness of the soil layer beneath the structure was modeled with vertical, horizontal, and rocking spring constants applied to the bottom of each foundation. Due to the nonlinear boundary condition used in the supports of the concrete frame model, such as contact and friction, all the involved loads were incorporated into a single load case, and a large displacement formulation was used in the analysis. The authors focused on the method of a simplified modeling of frame structures founded on soil. To conduct comparative analyses, two columns and two beams from each model were selected, from which the internal forces and displacements were compared. The findings of the comparative analysis are presented in tables and then discussed.

An analytical approach to incomplete spherical conformal contact: application on self-lubricating spherical plain bearings

Purpose Contact pressure is a critical factor that significantly influences the wear of self-lubricating spherical plain bearings. The purpose of this paper is to address the issue of conformal contact in spherical plain bearings with a self-lubricating fabric liner, and then a universal theoretical analytical model for conformal contact between frictionless spheres is proposed. Design/methodology/approach This study establishes an analytical model to calculate the conformal contact in spherical plain bearings and verifies the new model by finite element analysis. Findings The new model proposed in this paper overcomes the limitations of elastic half-space and small-deformation assumptions. After conducting accuracy validation, it was observed that the computational error of the new model has significantly decreased in comparison to the Johnson model. For a conformal contact with a clearance of 0, the error is nearly 0. Practical implications The analytical model can calculate the contact pressure distribution of self-lubricating spherical plain bearings bonded with a self-lubricating layer and can be extended to the conformal contact problem of spherical contact surfaces in biomechanics and other fields. Originality/value The model presented here overcomes the limitations of the elastic half-space and small deformation assumptions. It accurately calculates the contact pressure distribution of self-lubricating spherical bearings. Moreover, the complex nonlinear relationship between variables such as normal force, clearance, maximum contact pressure and contact radius was investigated using this model. The model can also be extended and applied to the conformal contact problem of spherical contact surfaces in various fields, including biomechanics. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-08-2024-0296/

Rayleigh wave propagation in an initially stressed orthotropic elastic solid half-space lying under inviscid liquid and viscous liquid layers

PurposeThe purpose of this article is the propagation of Rayleigh waves in a homogeneous, isotropic, initially stressed orthotropic elastic solid half-space lying under a homogeneous, viscous, inviscid liquid layer of finite thickness.Design/methodology/approachIn the presence of both viscous and inviscid liquids, it derives the phase velocity, initial stress, and wave number-dependent frequency equation for an orthotropic elastic solid. With the help of the MATLAB program, the thickness effects of liquid layers, initial stress, and viscosity on the phase velocity and attenuation coefficient of the Rayleigh wave are explained for a particular model.FindingsThe phase velocity-dependent dispersion relation of Rayleigh waves at the interface of viscous liquid and solid half-space is a function of initial stress and wave number. Rayleigh waves along the free surface of an orthotropic elastic half-space are also derived as a particular case. The classical results of an inviscid liquid are achieved when the thickness of a viscous liquid approaches zero. Well-known classical results for initially stressed orthotropic elastic solids were also derived.Originality/valueSo far, many researchers have looked into the propagation of surface waves at the interfaces of solid–inviscid liquid, solid–solid, and multilayer interfaces. But in this article, the dispersion behavior of Rayleigh wave propagation in an initially stressed homogeneous orthotropic elastic solid half-space under a double layer of viscous liquid and inviscid liquid is studied.

Modeling of reflected ultrasonic fields in composed samples

Ultrasonic nondestructive testing involves the study of propagation, reflection and refraction patterns of elastic waves excited by contact or non-contact piezoelectric transducers in the inspected object. The finite element modeling usually requires high computational costs and additional postprocessing to select individual waves from the total solution. When probing joints of homogeneous materials, such as turbine blades made of heat-resistant monocrystalline alloys, the joint boundary is low-contrast, and the reflected signals are relatively weak. This causes additional difficulties for their separation from the total wave field and correct interpretation of the information they bring. To solve this problem, explicit asymptotic representations for reflected and transmitted waves in a two-layer elastic half-space with a surface source are proposed in the present work, which allow fast parametric analysis. They can be used to analyze ultrasonic probing data, for example, to estimate the state of the junction zone or to determine the mutual orientation of the crystals’ principal axes.

Research of Dynamic Processes in a Layer During Collision With an Impactor

Abstract The article concerns the modeling of the transverse impact of an impactor (test sample) on the surface of an infinite elastic layer. The Laplace transform with respect to time and the Hankel transform with respect to the radius for the axisymmetric case were applied. The propagation of elastic waves in the layer and local deformations in the contact zone are taken into account. Impact force, impact time and the coefficient of restitution were examined. The results are compared with the elastic half-space. The calculations carried out showed that for layer thicknesses of more than five impactor diameters, the layer can be considered as a half-space.

Axisymmetric Indentation of Circular Rigid Plate on Layered Elastic Halfspace with Transverse Isotropy

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate.

SELF-CONSISTENCY CONDITIONS IN STATIC THREE-BODY ELASTIC TANGENTIAL CONTACT

The contact problem for an elastic third-body particle between two elastic half-spaces is considered. The contact is assumed to consist of three Hertzian contact spots. The normal and tangential contact problems are analyzed analytically considering partial slip in the contacts and the influence of third-body weight. Self-consistency conditions between global equilibrium and the contact solution are formulated to give criteria, under which circumstances static slip and stationary sliding are possible states for the third-body particle. The sliding case is solved in detail.

THE IMPACT OF TUNNEL LINING ON THE REACTION OF THE GROUND SURFACE UNDER THE INFLUENCE OF TRANSPORT LOADS

When a tunnel is subjected to transport loads (loads from the moving transportation within it), vibrations occur in the tunnel lining and the surrounding massif. Traditional quasi-static methods do not account for the dynamic behavior of tunnel structures. Therefore, this paper aims to develop a dynamic calculation method using modern mechanics. The purpose of this paper is to develop such a method. The relevance of the research in this article is due to the trend of increasing the speed of vehicles in recent years. This paper considers an unsupported and lined circular cylindrical shallow tunnel. The tunnel is modeled as an extended circular cylindrical cavity or reinforcing shell located in an elastic half-space. The surface of the cavity or the inner surface of the shell is subjected to a normal load (the effect of the pressure of a moving object on the tunnel) and a tangential load parallel to this axis (the effect of the friction forces of a moving object on the tunnel) moving uniformly along its axis. The motion of the half-space and the shell are described by the dynamic equations of elasticity theory and the equations of classical shell theory, respectively, in moving coordinate systems. The integral Fourier transform method is used to solve the problem. In the case of moving axisymmetric normal and axial tangential loads acting on the tunnel, a numerical study of the influence of the tunnel lining on the stress-strain state of the ground surface is carried out.

Analysis of Rayleigh wave dynamic response and propagation characteristics in layered site

Rayleigh waves are crucial in earthquake engineering due to their significant contribution to structural damage. This study aims to accurately synthesize Rayleigh wave fields in both uniform elastic half-spaces and horizontally layered elastic half-spaces. To achieve this, we developed a self-programmed FORTRAN program utilizing the thin layer stiffness matrix method. The accuracy of the synthesized wave fields was validated through numerical examples, demonstrating the program’s reliability for both homogeneous and layered half-space scenarios. A comprehensive analysis of Rayleigh wave propagation characteristics was conducted, including elliptical particle motion, depth-dependent decay, and energy concentration near the surface. The computational efficiency of the self-programmed FORTRAN program was also verified. This research contributes to a deeper understanding of Rayleigh wave behavior and lays the foundation for further studies on soil-structure interaction under Rayleigh wave excitation, ultimately improving the safety and resilience of structures in seismic-prone regions.

Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity

In this paper, we investigate the axisymmetric Hertzian contact problem at micro-/nanoscale. The deformation of material bulk is described by a simplified theory of strain gradient elasticity, and the influence of surface tension is integrated based on the surface elasticity theory. Using the Mindlin's potential function method and double Fourier integral transform, the normal surface displacement induced by a concentrated force is derived in a closed form. Following this, the contact between a rigid sphere and an elastic half-space is formulated in terms of singular integral equation, which is numerically solved by applying the Gauss-Chebyshev method. The results indicate that the distribution of contact pressure is distinctly different from that in classical elasticity theory. The indented substrate tends to perform stiffer due to the effects of surface tension and strain gradient elasticity. When the contact radius is comparable with the material length parameter, the indentation force (depth) can be ten (three) times of that given by classical Hertz theory.

Surface wave mitigation by periodic wave barriers under a moving load: theoretical analysis, numerical simulation and experimental validation.

Periodic wave barriers (PWB) open a new window for vibration mitigation. However, the Doppler effect is rarely considered in most of the previous investigations on the control of ambient vibration induced by moving loads. This article reveals the significance of the speed and frequency of moving loads on surface waves, and improves the design method of PWB for ambient vibration reduction and isolation. First, the theoretical expression of the main frequency band of surface waves propagating in an elastic half-space caused by a moving load was obtained. Comparisons with the numerical results under three different types of traffic loads were also conducted and good agreement was found. Second, the theoretical expression and numerical results were verified by experimental studies. Some inherent properties of wave propagation caused by a moving load in an elastic half-space were also revealed. Third, two kinds of PWBs, i.e. periodic empty trench barrier and periodic pile barrier, were introduced to mitigate wave propagation. It has been confirmed that if the attenuation zones of PWB match the target frequency bands given by the theoretical expression, good vibration mitigation can be achieved. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

THEPORE: A software package for modeling THErmo-PORo-elastic displacements

THEPORE (THErmo-POro-Elastic solutions) is an open source software to perform forward and inverse modeling of ground displacements induced by thermo-poro-elastic sources. The software, implemented in MATLAB, offers a library of analytical and semi-analytical solutions to compute ground displacements induced by thermo-poro-elastic deformation sources of different geometries, embedded in an elastic, homogeneous and isotropic half-space. The solutions have been verified against finite-element simulations. THEPORE includes also an inversion procedure of the deformation data to constrain the source parameters that better fit the observed signals.The software's functionality is showcased by inverting the GPS deformation data recorded on Vulcano Island at the onset of the 2021 unrest, in order to estimate the position and volume change of the source responsible for the observed deformations. The results encourage to consider THEPORE as a practical tool suitable for a fast preliminary estimation of the deformation source during a volcanic crisis.

High-Speed Train-Induced Vibration of Bridge–Soft Soil Systems: Observation and MTF-Based ANSYS Simulation

In this paper, a multi-transmitting formula (MTF) was integrated into ANSYS software through secondary development, enabling dynamic finite element simulation of wave propagation in infinite domains. The numerical reliability and accuracy of the MTF were verified through a plane wave problem involving a homogeneous elastic half-space, as well as 3D scattering and source problems in a three-layered soil site. Additionally, a comparative analysis of various artificial boundaries was conducted to highlight the advantages of the MTF. Field observations of environmental vibrations caused by high-speed railway operations revealed localized amplification of vibrations along the depth direction at the Kunshan segment of the Beijing–Shanghai high-speed railway. Based on these observations, a series of numerical analyses were conducted using the customized ANSYS integrated with the MTF to investigate the underlying causes and mechanisms of this phenomenon, as well as the spatial variation characteristics of foundation vibrations induced by bridge vibrations during high-speed train operations. This study reveals the mechanism by which the combined effect of bridge piles and soft soil layers influences the depth variation in peak ground accelerations during site vibrations. It also demonstrates that the presence of bridge piers and pile foundations effectively reduces vibration intensity in the vicinity of the railway, playing a crucial role in mitigating vibrations induced by high-speed train operations.

On the existence of Rayleigh waves with full impedance boundary condition

The existence of Rayleigh waves (propagating in isotropic elastic half-spaces) with the tangential and normal impedance boundary conditions was investigated. It has been shown that for the tangential impedance boundary condition (TIBC), there always exists a unique Rayleigh wave, while for the normal impedance boundary condition (NIBC), there exists a domain (of impedance and material parameters) in which exactly one Rayleigh wave is possible and outside this domain a Rayleigh wave is impossible. In this paper, we consider the existence of Rayleigh waves with the full impedance boundary condition (FIBC) that contains both TIBC and NIBC. It is shown that the existence picture of Rayleigh waves for this general case is more complicated. It contains domain for which exactly one Rayleigh wave exists, domain where a Rayleigh wave is impossible, and domain for which all three possibilities may occur: two Rayleigh waves exist, one Rayleigh wave exists, and no Rayleigh wave exists at all. The obtained existence results recover the existence results established previously for the cases of TIBC and NIBC. The formulas for the Rayleigh wave velocity are derived. As these formulas are totally explicit, they are very useful in various practical applications, especially in the non-destructive evaluation of the mechanical properties of structures. In order to establish the existence results and derive formulas for the Rayleigh wave velocity, the complex function method, which is based on the Cauchy-type integrals, is employed.

Semi-analytical modelling of realistic tire-pavement contact: an analysis of pavement surface cracking damage under critical driving conditions

This paper aims to evaluate the surface cracking phenomenon on a flexible pavement subjected to diverse vehicle running conditions. Top-down cracking initiation is addressed through a continuum damage framework coupled with a semi-analytical model of tire-pavement rolling contact. The acceleration, braking and cornering scenarios were selected to represent the effect of combining normal and tangential loads. The differences between these loading cases are accurately computed by simulating the realistic contact stresses on the pavement surface through semi-analytical modelling. The computational cost of this modelling technique is reduced, allowing it to be used in industrial applications. It is assumed that tires and pavement are linear elastic multilayered half-spaces. The critical locations of the cracks are identified for each loading scenario and the cracking directions are discussed. This study emphasises the influence of driving conditions on premature pavement cracking which is not typically considered in the current pavement design guides. It is shown that a standard axle load may cause about 9 times more surface damage on a circular intersection compared to a straight-line pavement section.

Analysis of axisymmetric thermoelastic diffusive half-space with variable material properties using hyperbolic two-temperature model

The aim of the current study is to investigate the variable thermal conductivity and diffusivity impact on the transient response of an axisymmetric thermodiffusive elastic half-space under axisymmetric thermal, mechanical, and concentration loads in the light of hyperbolic two-temperature generalized fractional thermoelastic diffusion model. With the help of Kirchhoff’s transform, the governing equations are converted into linear form. The resulting equations are solved by imposing the combined Laplace–Hankel transform. In converted space, the solutions for different field quantities are presented in compact form. The copper material is considered for numerical computation. By employing a mathematical inversion procedure, the solutions are converted into the original space. Numerically computed solutions for various physical quantities are illustrated in the form of graphs to show the impact of variable thermal conductivity and diffusivity, hyperbolic two temperature, and fractional parameters. Validation of the obtained results is also presented.

Partial Slipping of Flat Punch in Thermomechanical Contact with Elastic Half-Space

Partial Slipping of Flat Punch in Thermomechanical Contact with Elastic Half-Space