Linear Elastic Half-space Research Articles

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.

The mechanics of the Gentle Driving of Piles

Gentle Driving of Piles (GDP) is a new vibratory installation technology for tubular (mono)piles. It is characterized by the simultaneous application of low-frequency axial and high-frequency torsional vibrations, envisaged to achieve both high installation performance and reduced underwater noise emissions. The concept of GDP has been demonstrated experimentally in a medium-scale onshore field campaign, showcasing the potential of the method in terms of installation and post-installation performances. To further comprehend the mechanics of the GDP method, the driving process is studied by means of a novel pile–soil model; this framework has been recently developed and successfully applied to the problem of axial vibratory driving. In particular, the pile is treated as a thin cylindrical shell via a Semi-analytical Finite Element (SAFE) approach and a linear elastic layered soil half-space is considered via the Thin-Layer Method (TLM) coupled with Perfectly Matched Layers (PMLs). The pile–soil coupling is realized through a hereditary frictional interface and an elasto-plastic tip formulation, both characterized by standard geotechnical in-situ measurements. The comparison of numerical results with field data is favourable for drivability purposes, showcasing the potential of the numerical framework for the analysis of GDP. Conclusively, the mechanics of the installation process are deciphered and the redirection of the friction force vector – induced by high-frequency torsion – is identified as the main driving mechanism of GDP.

Rigid circular footing on the surface of a transversely isotropic linear elastic half-space

It is sometimes convenient to treat a relatively rigid footing as a single element subjected to up to six force resultants: vertical load, horizontal load in two directions, overturning moment about two axes and torsion. Applications include calculations of initial, elastic settlement; foundation responses under seismic loading or for machine vibrations; and fixity of an offshore jack-up foundation under cyclic wave loading. Present codes of practice provide formulae that assume fully isotropic and linear elastic soil and in some case assume no friction between the footing and soil. This paper uses recent theoretical advances to develop solutions for stiffnesses of rigid circular footings on transversely isotropic linear elastic soil, including with interface friction. Closed-form solutions are developed for vertical load and overturning moment on a frictionless interface and for vertical load and torsion on a frictional interface. A boundary-element analysis is presented for the cases of lateral load and overturning moment on a frictional interface. By fitting expected forms to the results, new formulae are proposed for stiffnesses for these cases. The effects of interface friction on limiting loads are calculated. Practical applications are outlined, and some limitations are briefly discussed.

Point loads on the surface of a transversely isotropic linear elastic half-space

General solutions for the displacements, strains and changes in stress in a transversely isotropic linear elastic (TILE) half-space subject to a point load were published in 1998. They represent a massive advance on the nineteenth-century solutions by Boussinesq, Cerruti and others, which were for fully isotropic linear elastic half-spaces. The 1998 solutions were for loads at a general depth below the surface and were criticised for being difficult. The present paper deduces more accessible expressions for the special case of loads that are on the surface. This allows one clarification and one potential error in the original equations to be identified and avoided. With support from existing data of geological materials, agreement is found with a previous claim that a limit proposed in 1970 on one of the shear moduli for a TILE material is invalid. Corrected equations derived herein can be the basis of calculating ground stresses and movements due to linear responses to distributed surface loads, behaviours of flexible foundations and stiffnesses for rigid surface footings.

A numerical study of 3D topographic site effects considering wavefield incident direction and geomorphometric parameters

The topographic site effect plays a vital role in controlling the characteristics of earthquake ground motions. Due to its complexity, the factors affecting topographic amplification have not been fully identified. In this study, 100 ground motion simulations generated by double-couple point sources in the homogeneous linear elastic half-space are performed based on the 3D (three-dimensional) Spectral Element Method, taking the Menyuan area of Qinghai Province, China as a local testbed site. A relationship between incident direction and the strength of topographic amplification has been observed. The horizontal ground motion is affected by the back-azimuth, which is typically chosen to be the direction from seismic station to seismic source measured clockwise from north. Specifically, the east-west PGA (Peak Ground-motion Acceleration) is significantly amplified when back-azimuth is about 90° or 270°, and the north-south PGA is significantly amplified when back-azimuth is around 0° or 180°. The vertical ground motion is affected by the dipping angle, which is the angle from vertical at which an incoming seismic wave arrives. The vertical PGA is strongly amplified when the seismic wave is almost horizontally incident (e.g., dipping angle = 78°). A correlation study between geomorphometric parameters and frequency-dependent topographic amplification indicates that relative elevation and smoothed curvature contain similar information, both of which are closely related to the topographic amplification of horizontal components, but not the vertical component. Our study reveals the influence of source and propagation path on topographic amplification and provides a reference for considering the topographic site effect in real engineering sites.

Novel Open Trench Techniques in Mitigating Ground-Borne Vibrations due to Traffic under a Wide Range of Ground Conditions

This study proposes the novel geometrical design of open trenches in contrast to the traditional rectangular type of open trench, in order to improve the efficiency in mitigating the traffic-induced ground vibrations. The novel geometric design involves the variation in cross-sectional shape, cross-sectional area, sidewall inclination, depth of trenches, and the number of trenches. In addition, the efficiency of the multirow trenches with shallower depths, in contrast to the single-row deep trench, is also assessed for further mitigating the low-frequency ground vibrations. It is argued that these novel open trenches would change the wave propagation direction while also weakening the wave propagation capability in the original direction. A two-dimensional (2D) finite-element study is performed to investigate the isolation efficiency of novel open trench techniques with varying cross sections in a linear elastic, isotropic, and homogeneous half-space. A steady-state vertical harmonic excitation of the ground surface was simulated. The numerical results show that compared with the rectangular open trench, the part of the area behind the new cross section trench has better isolation efficiency, Cross section 6 seems to be the most efficient of the six cross sections, and the isolation efficiency is improved by nearly 10.2% compared. Increasing the cross-sectional area could not effectively improve the isolation efficiency of the open trenches. Selection of inclination of the hypotenuse to an appropriate angle, increasing the number and depth of open trenches could improve the isolation efficiency. The vibration isolation performance of the trenches in the soft soil was better than the stiff soil. Besides, an excessively high groundwater table is usually detrimental to the isolation efficiency of the trenches.

Scattering attenuation of transient SH-wave by an orthotropic gaussian-shaped sedimentary basin

• Scattering attenuation of transient SH -wave. • Orthotropic Gaussian-shaped sedimentary basin. • Time-domain boundary element method. • Illustrating the responses in time and frequency domains. • Isotropy-factor, wave angle, and geometrical effect of sedimentary basin. The surface motion of an orthotropic Gaussian-shaped sedimentary basin embedded in a linear elastic half-space was successfully obtained under transient SH -wave propagation. In the use of the time-domain boundary element approach, a simple model was developed only by discretizing the interface. To achieve better convergence of the response, an attenuation rule was implemented with the assistance of the reduction exponential function applied in the boundary integral equations. After a brief overview of the method, the heterogonous sub-structured basin model was created and decomposed into two parts to satisfy the modified continuity conditions based on the normal direction at the interface node position. Then, the coupled matrices were temporarily solved at each time-step to obtain the boundary values. The developed computer code was tested by solving a validation example. Finally, to illustrate the valley surface response in the time/frequency-domain, a general sensitivity analysis was performed by considering the parameters of frequency, shape-ratio, isotropy-factor, and wave angle in the form of snapshots, seismograms, and amplification patterns. The results showed that the effect of orthotropic anisotropy was very efficient on the ground response and different seismic patterns on the surface of sedimentary basins were extracted by changes in the quantity of this parameter.

Phase Field Modeling of Hertzian Cone Cracks Under Spherical Indentation

A phase field model of brittle fracture has been developed to simulate the Hertzian crack induced by penetration of a rigid sphere to an isotropic linear-elastic half-space. The fracture formation is regarded as a diffusive field variable, which is zero for the intact material and unity if there is a crack. Crack growth is assumed to be driven by a strain invariant. The numerical implementation is performed with the finite element method and an implicit time integration scheme. The mechanical equilibrium and the phase field equations are solved in a staggered manner, sequentially updating the displacement field and the phase field variable. Numerical examples demonstrate the capability of the model to reproduce the nucleation and growth of the Hertzian cone crack.

Indentation of a Periodically Layered, Planar, Elastic Half-Space

We investigate indentation by a smooth, rigid indenter of a two-dimensional half-space comprised of periodically arranged linear-elastic layers with different constitutive responses. Identifying the half-space’s material parameters as periodic functions in space, we utilize the theory of periodic homogenization to approximate the layered heterogeneous material by a linear-elastic, homogeneous, but anisotropic medium. This approximation becomes exact as the layer thickness becomes infinitesimal. In this way, we reduce the original problem to the indentation of an anisotropic, homogeneous, linear-elastic half-space by a smooth, rigid indenter. The latter is solved analytically by formulating and resolving the corresponding matrix Riemann–Hilbert boundary-value problem in complex analysis. Thus, we obtain an approximate, but analytical solution for the indentation of a layered heterogeneous medium. We then compare this solution with finite element computations of the indentation on the original layered, heterogenous half-space. We conclude that (a) the contact pressure on the layered, heterogenous half-space is well approximated by that obtained through homogenization, and the approximation improves as the layer thickness is decreased, or if the indentation force is increased; (b) the upper bound of the difference between the two contact pressures depends only upon the ratio of the Young’s moduli of the two materials constituting the heterogenous medium and their Poisson’s ratio; and (c) the average variation of the discontinuous von Mises stress in the layered half-space is well approximated by the one found in the homogenized half-space. The approach presented here can be utilized for a diverse array of indentation and contact problems of finely mixed heterogeneous media, and is also amenable to systematic improvements.

The Community Code Verification Exercise for Simulating Sequences of Earthquakes and Aseismic Slip (SEAS)

Abstract Numerical simulations of sequences of earthquakes and aseismic slip (SEAS) have made great progress over past decades to address important questions in earthquake physics. However, significant challenges in SEAS modeling remain in resolving multiscale interactions between earthquake nucleation, dynamic rupture, and aseismic slip, and understanding physical factors controlling observables such as seismicity and ground deformation. The increasing complexity of SEAS modeling calls for extensive efforts to verify codes and advance these simulations with rigor, reproducibility, and broadened impact. In 2018, we initiated a community code-verification exercise for SEAS simulations, supported by the Southern California Earthquake Center. Here, we report the findings from our first two benchmark problems (BP1 and BP2), designed to verify different computational methods in solving a mathematically well-defined, basic faulting problem. We consider a 2D antiplane problem, with a 1D planar vertical strike-slip fault obeying rate-and-state friction, embedded in a 2D homogeneous, linear elastic half-space. Sequences of quasi-dynamic earthquakes with periodic occurrences (BP1) or bimodal sizes (BP2) and their interactions with aseismic slip are simulated. The comparison of results from 11 groups using different numerical methods show excellent agreements in long-term and coseismic fault behavior. In BP1, we found that truncated domain boundaries influence interseismic stressing, earthquake recurrence, and coseismic rupture, and that model agreement is only achieved with sufficiently large domain sizes. In BP2, we found that complexity of fault behavior depends on how well physical length scales related to spontaneous nucleation and rupture propagation are resolved. Poor numerical resolution can result in artificial complexity, impacting simulation results that are of potential interest for characterizing seismic hazard such as earthquake size distributions, moment release, and recurrence times. These results inform the development of more advanced SEAS models, contributing to our further understanding of earthquake system dynamics.

Anti-plane seismic ground motion above twin horseshoe-shaped lined tunnels

The ground surface response was obtained in the presence of twin horseshoe-shaped lined tunnels embedded in a linear elastic half-space, subjected to obliquely propagating incident plane SH-waves. A numerical approach known as direct half-plane time-domain boundary element method was used to prepare the model. First, the heterogeneous problem was decomposed into three parts, two closed linings and a double-hole half-plane. Then, the method was applied to each part to compute the required matrices. Finally, the problem was solved by satisfying the continuity conditions at the interfaces. After solving a verification example, an advanced parametric study was carried out for obtaining the surface response with shallow twin horseshoe-shaped lined tunnels as synthetic seismograms and 3-D amplification patterns. Some intended parameters were also considered to illustrate the ground response including the horizontal/vertical overlapping of the tunnels, depth and distance between them and incident wave angle. To summarize the results, some graphs were presented to obtain the maximum amplification ratio of the surface. Numerical results showed that the existence of twin horseshoe-shaped tunnels could amplify the ground surface response up to three and half times the free-field movement at dimensionless frequencies greater than one. Also, achievements indicated that the greater role of vertically overlapped tunnels in creating safe areas on the surface compared to the horizontally placed tunnels. The results can be practically used in clarifying seismic codes in the field of urban transportation geotechnics.

Calculated diagrams of a deformed soil mass at transfer of external and internal power influences to array

The analysis of the existing laws of stress distribution in the soil from external and internal force effects. It is shown that the existing classical solution of the stress-strain state of an elastic half-space from exposure to external and internal influences to use in calculations without additional assumptions. In this regard, we propose a method of determining the stress-strain state (elastic half-space) of the soil from external and internal influences transmitted via limited in terms of the area that gives the opportunity to determine the state of stress (elastic half-space) of the soil with regard to real values, going from ∞ no regular assumptions. Determination of deformations of foundation was based on the solution of one-dimensional problem of compacting of the soil massif limited to the thickness proposed by prof. M. Gersevanov. In the normative documents, approved by the USSR state Committee for construction (SNiP) there are two calculation schemes, which were determined by precipitation of the bases and foundations: 1. Linear deformable elastic half-space with a conditional limitation of the compressible strata (elementary method of summation) and the calculated scheme in the form of linear-deformed layer (for major in terms of the bases b ≥ 10 m). To solve this problem, we proposed a method for determining the capacity of the compressible strata with the assumption that soil mass will start to show deformation properties (deformation) when the ground pressure (σ) will exceed the calculated resistance of the soil mass, that is, performs the equilibrium condition σ ≥ R. On this basis it is shown that underground structures are carried out by way of trap hole and wall in the ground, do not show the residue grounds.

Effects of soil-foundation-structure interaction on fundamental frequency and radiation damping ratio of historical masonry building sub-structures

Large-scale simulations and forensic analyses of the seismic behaviour of real case studies are often based on simplified analytical approaches to estimate the reduction in fundamental frequency and the amount of radiation damping induced by dynamic soil-foundation-structure (SFS) interaction. The accuracy of existing closed-form solutions may be limited because they were derived through single degree-of-freedom structural models with shallow rigid foundations placed on a homogeneous, linear elastic half-space. This paper investigates the effectiveness of those formulations in capturing the dynamic out-of-plane response of single load-bearing walls within unreinforced masonry buildings having either a shallow foundation or an underground storey embedded in layered soil. To that aim, analytical predictions based on the replacement oscillator approach are compared to results of two-dimensional dynamic analyses of coupled SFS elastic models under varying geotechnical and structural properties such as the soil stratigraphy, foundation depth and number of building storeys. Regression models and a relative soil-structure stiffness parameter are proposed to quickly predict the frequency reduction induced by SFS interaction, accounting for the presence of an embedded foundation, an underground storey and a layered soil. The effects of SFS interaction are also evaluated in terms of equivalent damping ratio, showing the limitations of simplified approaches. Since the geometric layouts considered in this study are rather recurrent in the Italian and European built heritage, the proposed procedure can be extended to similar structural configurations.

Boundary-polarized subsonic Rayleigh waves under conditions of semi-simple Stroh degeneracy

We undertake a study of subsonic free surface (Rayleigh) waves in linear elastic half-spaces of general anisotropy when the wave polarization vector lies in the half-space boundary , if and when the formalism due to A. N. Stroh exhibits semi-simple degeneracy at the Rayleigh speed v R . It is shown that the class of subsonic steady wave motions at any subsonic velocity exhibiting semi-simple degeneracy includes a free surface wave whenever the first transonic state is not exceptional, in accordance with general surface wave theory. Furthermore, such a free surface wave is always necessarily boundary-polarized ! In general, the restrictions on the half-space elastic constants permitting the existence of semi-simplicity in steady motion at a subsonic phase speed are not satisfied in any physically realized medium which is elastically stable, but we outline an algorithm which allows one to construct the elastic stiffnesses of media which exist mathematically and allow for the existence of subsonic free surface waves (which are necessarily boundary-polarized) under conditions of semi-simple degeneracy in the sense of Stroh's formalism.

Numerical investigation of plane wave propagation in three soil models and a simple viscoelastic treatment of the boundary considering gravity using MPM

The importance of plane wave propagation in geotechnical engineering problems considering both small and large deformation is well understood but the theory of wave propagation in the latter is maybe not mature. In this paper, the mechanism of plane wave propagation in constitutive models such as linear elastic model, Mohr Coulomb model (MC) and elastoplastic density-dependent model (DDSM) considering finite deformations is extensively studied using the material point method. The classical laws of wave propagation and reflection in a continuum are found to provide reasonable explanations to the considered problems. The stress propagation mechanism in the three models considered is mainly related to the variation of the material wave impedance. An accurate silent boundary condition is found difficult even impossible to be determined for complicated models, which is verified by a simple deduction. Moreover, a simple viscoelastic silent boundary is proposed to consider the realistic effects of gravity in geotechnical engineering problems and its performance is compared with that of the classical viscous boundary for truncating a 1D linear elastic half-space.

Seismic interaction between a semi-cylindrical hill and a nearby underground cavity under plane SH waves

Seismic waves might be unavoidably triggered during underground energy and resource exploitation. Surface irregularities and subsurface cavities have significant effects on seismic wave propagation and cause amplification or reduction of ground motion. Hence, it is significant to address the scattering problem of SH waves induced by topography features and subsurface cavities in earthquake engineering. In this paper, based on the wave function expansion method coupling with the conjunction concept and the Graf’s addition formula, a series solution to this scattering problem of SH waves induced by a semi-cylindrical hill and a nearby cylindrical cavity in a homogeneous, isotropic, linear elastic half-space is derived. The displacement amplitudes on the hill surface and its environs are calculated, and compared with the results in the existing works to verify the validity of the deriving process. Then, parametric studies are performed to analyze the effects of the cavity (its location and size), and the characteristics of incident waves (the frequency and incident angle) on the seismic response of the semi-cylindrical hill. Finally, the seismic behavior of the cylindrical cavity near or away from the hill for different frequencies and incident angles is examined.

Satisfying Boundary Conditions in Homogeneous, Linear-Elastic and Isotropic Half-Spaces Subjected to Loads Perpendicular to the Surface: Distributed Loads on Adjacent Contact Areas

Abstract This work originates from an experimental program on strain distribution near the loaded surface of an airfield concrete pavement,which provided us with results that contrast with the rheological predictions of Boussinesq for a homogeneous, linear-elastic and isotropic half-space. We already reviewed and extended the original work carried out by Boussinesq in previous papers, to provide a closed form second order solution that enabled us to establish a good match between analytical and experimental findings for point-loads. In this paper, we have explained why Boussinesq’s closed form solution for a homogeneous linear-elastic and isotropic half-space subjected to a point-load is not exact, as believed until now, but approximated. Then, we have shown that our second order solution is the actual solution of Boussinesq’s problem. We have also presented the numerical analysis of second order for rectangular and elliptical contact areas, both loaded by uniform and parabolic laws of external pressure. Moreover, we have evaluated the interaction effect provided on the surface of a concrete half-space by the twin wheels of an aircraft landing gear. Extension of the solution to layered systems is also possible, for improving the knowledge of stress propagation into airfield pavements and promoting more effective design standards.

Analytical investigation of a two-layered elastic half-space containing a cylindrical elastic inhomogeneity under forced torsional rotation

This paper presents a rigorous investigation for a two-layered linear elastic half-space containing a cylindrical elastic inhomogeneity with length equal to the thickness of the top layer undergoing forced torsional rotation applied on a rigid circular disc with the same radius as the cylinder, which is welded on the top of the cylinder. To investigate this complicated system analytically, a combination of Fourier sine and cosine integral transforms for depth, respectively in the domain of inhomogeneity and its matrix at the top layer, and Hankel integral transforms for radial distance in the lower half-space as the remaining part of the matrix are used, which translate the related boundary value problem to a system of coupled singular integral equations for the non-dimensional shear stresses resulted from the continuity and boundary conditions. The system of coupled singular integral equations is solved for some collocation points with a smoothed variable of distance, which is adapted with the use of a free parameter. The validities of both the analytical procedure and numerical evaluations are verified with its degenerated case for the homogeneous case, which is the Reissner–Sagoci problem. In addition, the degree of singularities of the shear stresses in between different regions are estimated in terms of material properties of the layer, inhomogeneity, and the half-space. Moreover, some new graphical results are presented for more understanding of related engineering behavior.

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.

Longitudinal impact into viscoelastic media

We consider several one-dimensional impact problems involving finite or semi-infinite, linear elastic flyers that collide with and adhere to a finite stationary linear viscoelastic target backed by a semi-infinite linear elastic half-space. The impact generates a shock wave in the target which undergoes multiple reflections from the target boundaries. Laplace transforms with respect to time, together with impact boundary conditions derived in our previous work, are used to derive explicit closed-form solutions for the stress and particle velocity in the Laplace transform domain at any point in the target. For several stress relaxation functions of the Wiechert (Prony series) type, a modified Dubner–Abate–Crump algorithm is used to numerically invert those solutions to the time domain. These solutions are compared with numerical solutions obtained using both a finite-difference method and the commercial finite element code, COMSOL Multiphysics. The final value theorem for Laplace transforms is used to derive new explicit analytical expressions for the long-time asymptotes of the stress and velocity in viscoelastic targets; these results are useful for the verification of viscoelastic impact simulations taken to long observation times.