Layered Viscoelastic Half-space Research Articles

In-plane oscillations of elastic rectangle on a layered viscoelastic half-space

In the present paper in the context of plane problem we study harmonic oscillations of the elastic rectangle above a viscoelastic layered half-space. The latter consists of an elastic half-space in which a viscoelastic layer of thickness h 1 is embedded at the depth h . By using the Fourier integral transform and Betti's integral identity, the problem is reduced to a set of integral equations. In view of practical applications, we study the possibility of protection from vibrations attained with the use of the viscoelastic layer.

A study of ground‐structure interaction in dynamic plate load testing

AbstractA mathematical treatment is presented for the forced vertical vibration of a padded annular footing on a layered viscoelastic half‐space. On assuming a depth‐independent stress distribution for the interfacial buffer, the set of triple integral equations stemming from the problem is reduced to a Fredholm integral equation of the second kind. The solution method, which is tailored to capture the stress concentrations beneath footing edges, is highlighted. To cater to small‐scale geophysical applications, the model is used to investigate the near‐field effects of ground‐loading system interaction in dynamic geotechnical and pavement testing. Numerical results indicate that the uniform‐pressure assumption for the contact load between the composite disc and the ground which is customary in dynamic plate load testing may lead to significant errors in the diagnosis of subsurface soil and pavement conditions. Beyond its direct application to non‐intrusive site characterization, the proposed solution can be used in the seismic analysis of a variety of structures involving annular foundation geometries. Copyright © 2002 John Wiley & Sons, Ltd.

Ground Vibration from High-Speed Trains: Prediction and Countermeasure

This paper outlines a test program in southern Sweden for measurement of the vibration induced in the ground and railway embankment by high-speed trains, together with a rigorous numerical model developed for the prediction of embankment/ground response. In this formulation the ground is modeled as a layered viscoelastic half-space, and the railway embankment is modeled as a viscoelastic beam excited by the moving loads of the train. The model uses the Kausel-Roeesset Green's functions to calculate the soil stiffness matrix at the ground-embankment interface and assembles it with the dynamic stiffness matrix of the embankment. The solution is carried out in the frequency domain, and the time histories of the motions are derived through a Fourier synthesis of the frequency components. Numerous simulations of train-induced vibration are presented for the ground conditions and embankment parameters at the test site and compared with measured records. The simulations agree well with the measurements, both in qualitative and quantitative terms. In particular, the large ground deformations registered for train speeds exceeding 140 km/h are reproduced by the simulations. With the help of the prediction model, the effectiveness of a remediation measure for the mitigation of ground vibration is explored.

Near field stress evolution after a strike-slip earthquake in a layered viscoelastic half-space

The stress recovery following an earthquake, on a fault where the initial strength threshold is preserved, is here considered in a “half-space” model. In this model a cracked horizontal elastic layer (the seismogenic crust) is welded to a viscoelastic, half-space substratum with Maxwell rheology (the ductile crust). The earthquake is modeled as a stress drop event ( crack) while in previous similar papers it was usually modeled as a “constant slip event”. Unlike crack models, constant slip dislocations are not suitable for describing the stress field in the proximity of the fault, particularly on the fault plane, which is the main concern of the present work. The fault becomes welded or “locked” immediately after the earthquake and can temporarily sustain the stress reloaded on it following the stress relaxation of the underlying ductile layer. Subsequent events are expected to be generated near the fault plane, at the location where the seismic stress drop is completely recovered by the postseismic reloading. Solutions show that, if the fault cuts most of the elastic upper layer, major aftershocks, or secondary earthquakes related to the first event, can occur in the deep section of the fault, within times comparable with the relaxation time of the ductile crust beneath the continents.

Dynamic response of a two-dimensional semi-circular foundation embedded in a layered viscoelastic half-space : F. C. P. De Barros & J. E. Luco, Soil Dynamics & Earthquake Engineering, 14(1), 1995, pp 45–57

Dynamic response of a two-dimensional semi-circular foundation embedded in a layered viscoelastic half-space : F. C. P. De Barros & J. E. Luco, Soil Dynamics & Earthquake Engineering, 14(1), 1995, pp 45–57

Dynamic response of a two-dimensional semi-circular foundation embedded in a layered viscoelastic half-space

The dynamic response of a massless rigid strip foundation of semi-circular cross-section embedded in a uniform or layered viscoelastic half-space is considered. The foundation is subjected to external harmonic forces and moments and to plane homogeneous P-, SV- and SH-waves. Numerical results are obtained by two indirect boundary integral equation methods: a reciprocity-based approach and a version of the least squares approach. The results for a uniform half-space are tested against the exact solution for longitudinal vibrations and against earlier results for the in-plane impedance functions and for the in-plane response to plane waves. The deffects of Poisson's ratio, material attenuation and layering on the impedance functions and on the scattering coefficients (response to waves) are described in some detail. The appearance of fictitious eigen-frequencies at which the solution may fail is also discussed.

Seismic response of a cylindrical shell embedded in a layered viscoelastic half-space. I: formulation : J. E. Luco & F. C. P. De Barros, Earthquake Engineering & Structural Dynamics, 23(5), 1994, pp 553–567

Seismic response of a cylindrical shell embedded in a layered viscoelastic half-space. I: formulation : J. E. Luco & F. C. P. De Barros, Earthquake Engineering & Structural Dynamics, 23(5), 1994, pp 553–567

Response of a layered viscoelastic half-space to a moving point load

A procedure to obtain the steady-state displacements and stresses within a multi-layered viscoelastic half-space generated by a buried or surface point load moving with constant speed parallel to the surface of the half-space is presented. The approach is based on an integral representation of the complete response in terms of wavenumbers. The effects of layering are included by use of an exact factorization of the displacement and stress fields in terms of generalized transmission and reflection coefficients. The results in the time domain are obtained by Fourier synthesis of the frequency response which in turn is obtained by numerical integration over one horizontal wavenumber. Numerical results for the displacement and stress fields on the surface and within the half-space are presented for surface and buried loads moving with various subsonic and supersonic speeds.

Dynamic response of coupled foundations on layered random medium for out-of-plane motion

The case of two basemats excited by incident SH-waves is considered. The dynamic-stiffness coefficients of surface foundations on a randomly inhomogeneous layered medium, based on the stiffness matrix approach, are calculated. The solution for the harmonic response of the coupled system of basemats on a layered viscoelastic half-space is obtained.

Diffraction of obliquely incident waves by a cylindrical cavity embedded in a layered viscoelastic half-space

The three-dimensional harmonic response in the vicinity of an infinitely long, cylindrical cavity of circular cross-section buried in a layered, viscoelastic half-space is obtained when the half-space is subjected to homogeneous plane waves and surface waves impinging at an oblique angle with respect to the axis of the cavity. The solution is obtained by an indirect boundary integral method based on the use of moving Green's functions for the viscoelastic half-space. Numerical results describing the motion on the ground surface and the motion and stresses on the wall of the cavity are presented for obliquely incident P-, SV-, SH- and Rayleigh waves with different horizontal angles of incidence.

Quasi-static response of a layered viscoelastic half-space to general surface loading : Phys Earth Planet Inter V66, N3/4, April 1991, P278–289

Abstract Analytical solutions to a prestressed, multilayered, elastic half-space subjected to general surface loads are derived using the Cartesian system of vector functions. The solutions are then applied to the Maxwell viscoelastic medium with time-dependent surface loading by means of the correspondence principle and the normal mode method. The number of layers in the model and the pattern of surface load, at least in principle, are unrestrained. This theoretical result is convenient for numerical calculation, and is applicable to the study of postglacial rebound, the inversion of physical parameters of the Earth's interior, the analysis of the evolution of a sedimentary basin, and so forth. As a numerical example, the evolution of the Williston Basin is investigated. The present study differs from an earlier analysis of that basin in two aspects: a four-layered model is taken instead of a two-layered one, and the surface sediment loading is treated as rectangular parallelopipeds with time-dependent heights rather than as a point load. The computed subsidence curves agree quite well with the data from drilling wells.

Quasi-static response of a layered viscoelastic half-space to general surface loading

Abstract Analytical solutions to a prestressed, multilayered, elastic half-space subjected to general surface loads are derived using the Cartesian system of vector functions. The solutions are then applied to the Maxwell viscoelastic medium with time-dependent surface loading by means of the correspondence principle and the normal mode method. The number of layers in the model and the pattern of surface load, at least in principle, are unrestrained. This theoretical result is convenient for numerical calculation, and is applicable to the study of postglacial rebound, the inversion of physical parameters of the Earth's interior, the analysis of the evolution of a sedimentary basin, and so forth. As a numerical example, the evolution of the Williston Basin is investigated. The present study differs from an earlier analysis of that basin in two aspects: a four-layered model is taken instead of a two-layered one, and the surface sediment loading is treated as rectangular parallelopipeds with time-dependent heights rather than as a point load. The computed subsidence curves agree quite well with the data from drilling wells.

Dynamic response of embedded foundations: A hybrid approach

A hybrid approach is presented to obtain the dynamic response of foundations of arbitrary shape embedded in a layered viscoelastic half-space when subjected to external forces and elastic waves. The approach is based on the use of Green's functions for the half-space continuum combined with a finite element discretization of the finite portion of soil excavated for the foundation. Compared with direct or indirect boundary integral equation methods this particular combination permits a reduction of the number of calculations of Green's functions at the expense of additional finite element calculations. The validity and accuracy of the hybrid approach is investigated by comparison with solutions obtained by other methods for axisymmetric foundations embedded in a uniform viscoelastic half-space and in a layered medium. It is found that the proposed hybrid approach with an appropriate discretization of the soil excavated for the foundation can achieve excellent accuracy.

Dynamic interaction between rigid foundations in a layered half-space

A computationally efficient boundary integral equation technique to calculate the dynamic response of a group of rigid surface foundations bonded to a layered viscoelastic half-space and subjected to external forces and seismic waves is presented. The technique relies on an iterative scheme which minimizes in-core memory requirements and takes advantage of any geometrical symmetry of the foundations. Extensive results for the case of two rigid square foundations placed at different separations and bonded to a viscoelastic half-space are presented. It was found that the choice of discretization of the foundations has a marked effect on the calculated impedance functions for extremely small separations. Illustrative results for a case of several closely-spaced foundations bonded to a layered half-space are also presented.

Dynamic-stiffness matrix of embedded and pile foundations by indirect boundary-element method

The boundary-integral equation method is well suited for the calculation of the dynamic-stiffness matrix of foundations embedded in a layered visco-elastic halfspace (or a transmitting boundary of arbitrary shape), which represents an unbounded domain. It also allows pile groups to be analyzed, taking pile-soil-pile interaction into account. The discretization of this boundary-element method is restricted to the structure-soil interface. All trial functions satisfy exactly the field equations and the radiation condition at infinity. In the indirect boundary-element method distributed source loads of initially unknown intensities act on a source line located in the excavated part of the soil and are determined such that the prescribed boundary conditions on the structure-soil interface are satisfied in an average sense. In the two-dimensional case the variables are expanded in a Fourier integral in the wave number domain, while in three dimensions, Fourier series in the circumferential direction and Bessel functions of the wave number in the radial direction are selected. Accurate results arise with a small number of parameters of the loads acting on a source line which should coincide with the structure-soil interface. In a parametric study the dynamic-stiffness matrices of rectangular foundations of various aspect ratios embedded in a halfplane and in a layer built-in at its base are calculated. For the halfplane, the spring coefficients for the translational directions hardly depend on the embedment, while the corresponding damping coefficients increase for larger embedments, this tendency being more pronounced in the horizontal direction. For the rocking direction, both coefficients increase. The frequency dependence is more marked for increasing embedment. For the layer large oscillations of the dynamic-stiffness coefficients arise. For an undamped layer the damping coefficients vanish for frequencies below the fundamental frequency of the layer. It is demonstrated that for the foundation embedded either in a halfplane or in a layer, a two-dimensional model cannot be used to represent a truly three-dimensional situation.

Vibration of Flexible Plate on Viscoelastic Medium

The dynamic response of a massless flexible circular plate with a rigid core supported on a layered viscoelastic half-space and subjected to harmonic vertical and rocking excitation is studied. The mixed boundary-value problem for the case of relaxed contact conditions between the plate and the half-space is reduced to Fredholm integral equations of the second kind which are solved numerically. The effects of flexibility of the plate on the force-displacement relationship, on the motion of different points on the plate, and on the distribution of contact stresses beneath the plate are studied numerically. In general, it has been found that all of these aspects of the response of the plate are highly dependent on the flexibility of the plate relative to that of the supporting half-space.

Vibrations of a rigid disc on a layered viscoelastic medium

A method of obtaining the dynamic impedance functions for a rigid circular foundation placed on a layered viscoelastic half-space is presented. Both hysteretic and Voigt models of internal damping are considered. The results obtained indicate that the presence of internal damping introduces important changes in the dynamic response of the foundation for vertical, rocking and horizontal steady-state excitation.

On The Three-dimensional Seismic ResponseOf A Class Of Cylindrical Inclusions EmbeddedIn Layered Media

In this paper we examine the three-dimensional response of a class of cylindrical inclusions of infinite length embedded in a layered viscoelastic half-space and subjected to plane waves impinging at an oblique angle with respect to the axis of the inclusion. The cases considered include topographical canyons of arbitrary cross section, buried cavities or unlined tunnels, buried pipelines or lined tunnels, and layered sedimentary basins or valleys. All of these cases are formulated and solved by means of a unified indirect boundary integral method based on the use of moving Green's functions for a layered viscoelastic half-space. Results for the three-dimensional response of canyons, cavities, pipelines and valleys as well as comparisons with previous solutions for simple cases are presented for excitations corresponding to P, SV and SH-waves.