Multilayered Half-space Research Articles

Static Green's functions in multilayered half spaces

This paper presents an efficient and accurate method for the calculation of static Green's functions in a multilayered transversely isotropic or isotropic half space. The cylindrical system of vector functions and the propagator matrix method are used to derive the Green's functions in the transformed domain. The well-known exponentially growing elements in the propagator matrix are fractionated out by propagating the matrix either upwards or downwards, depending upon the relative vertical location of the source and field points. The Green's functions in the physical domain are evaluated numerically by an adaptive Gauss quadrature with continued fraction expansions. Numerical examples are presented to show that very accurate Green's functions with relatively less Gauss quadrature points can be obtained. These examples also show clearly the effect of material layering and anisotropy on the displacement and stress fields.

Elastic analysis of an annular slab-soil interaction problem using a hybrid method

An elastic interaction analysis of annular slab with soil has been made in which the soil has been modelled as either (i) homogeneous and isotropic half-space or (ii) multilayered half-space. The interaction problem has been solved by using a combined approach (hybrid method) in which the finite element method (FEA4) has been used to obtain the stiffness matrix for the annular slab while for the elastic half-space soil models, the stiffness matrix has been obtained using an analytical solution. In the analysis, the interface between the annular slab and the elastic half-space has been assumed to be frictionless. Later the combined stiffness matrix of the annular slab-elastic half-space has been obtained using the displacement continuity condition at the interface. Results of the contact pressure distribution, displacement (settlement) profile and bending moment variation in an annular slab have been presented for different half-space soil models. It is shown that the hybrid method presented here is easy to apply and computationally efficient.

The singular solutions of axisymmetric elastodynamic problem for multilayered halfspace by means of transfer matrix (in Chinese): Meng Fanshun, Guo Haiyan & Yie Tianqi, Earthquake Engineering & Engineering Vibration, 15(2), 1995, pp 44–51

The singular solutions of axisymmetric elastodynamic problem for multilayered halfspace by means of transfer matrix (in Chinese): Meng Fanshun, Guo Haiyan & Yie Tianqi, Earthquake Engineering & Engineering Vibration, 15(2), 1995, pp 44–51

Fluid pressure responses of a water-saturated porous elastic multi-layered half-space to water pumping and atmospheric loading : T. Maruyama, Journal of Physics of the Earth, 42(5), 1994, pp 331–375

Fluid pressure responses of a water-saturated porous elastic multi-layered half-space to water pumping and atmospheric loading : T. Maruyama, Journal of Physics of the Earth, 42(5), 1994, pp 331–375

Improper integrals for infinite elements in vibration of half spaces

In vibration of elastic and viscoelastic multilayered half spaces, the far field of each half space is discretized into infinite elements. Far-field displacement functions of three fundamental problems are used as the shape functions of appropriate infinite element nodal lines. Each fundamental problem solution is obtained as a linear combination of discrete waves. As a result, the mass and stiffness matrices of infinite elements involve improper integrals, which are evaluated by three schemes proposed herein. Different schemes are good for different ranges of the complex argument of the integrals.

Temperature field in a heat-sensitive multilayer half-space

A method is proposed for solving a nonlinear axisymmetrical boundary-value heat-conduction problem for a multilayered half-space with heat transfer, heated by internal heat sources.

EHL of Coated Surfaces: Part I—Newtonian Results

A complete solution for the Newtonian elastohydrodynamic lubrication of coated surfaces in line contact is introduced in this paper. The formulation is based mainly on solving the Reynolds equation coupled with the elasticity equations of the multilayered elastic half space. The effects of pressure inside the lubricated conjunction on the lubricant viscosity and density are considered. The mathematical model for this problem is a nonlinear one, and an iterative Newton-Raphson scheme is used for numerical solutions. The effects of coating material and coating thickness on the pressure profiles and the film shapes are presented. Two different coating materials (molybdenum disulfide as a soft coating and titanium nitride as a hard coating) are considered. The results show that the pressure spike is higher for hard coatings than for uncoated surfaces but that the pressure spike disappears for soft coatings. Furthermore, the coating thickness has a significant influence on the minimum film thickness and the maximum contact pressure for both soft and hard coatings.

Fluid Pressure Responses of a Water-Saturated Porous Elastic Multi-Layered Half-Space to Water Pumping and Atmospheric Loading.

This study deals with fluid pressure variations caused by pumping of water and by changes in atmospheric pressure for a semi-infinite body consisting of many horizontal layers and a substratum each of which is homogeneous, porous elastic, and water-saturated. The Thomson-Haskell matrix method is employed for axisymmetric geometry with cylindrical coordinates (r, z). Continuation is considered at the boundary of each layer with respect to 6 quantities, i.e., displacement components ur, uz; stress components σzr, σzz; pore pressure p, and fluid flow rate qz in the z-direction. A variant of the Sommerfeld integral in the wave theory is devised for pumping at a point with time dependence of exp(-iωt). On the campus of the University of Tokyo, an interesting phenomenon was found by Yamaguchi and has since been observed, that is, the groundwater level in a well about 380 m deep rises when pumping begins in another well. The fluid pressure responses to this pumping as well as to changes in atmospheric pressure are simulated numerically by the present model, with good agreement between theory and observation. By way of the same procedure, displacements, tilts or stresses can also be calculated at any point in the medium.

A systematic and efficient method of computing normal modes for multilayered half-space

We present a systematic and efficient approach for computing the dispersion curves (i.e. phase velocities for a given frequency) as well as the eigenfunctions of normal modes in a multilayered half-space medium. Our approach is superior to previous approaches in the following aspects. First, it is a simple and self-contained algorithm for simultaneously determining both the phase velocities and the corresponding eigenfunctions. From the basic principle that the normal modes are non-trivial solutions of the free elastodynamic equation under appropriate boundary conditions, we naturally derive the phase velocities and eigenfunctions. Second, because we use the reflection/transmission coefficients of Luco & Apsel's version (Luco & Apsel 1983), which intrinsically exclude the growth terms, our algorithm not only exhibits the physical mechanism of the formation of normal modes, i.e. constructive interference, as Kennett's did (1979, 1983), but also is numerically more stable for high-frequency cases. Furthermore, we derive a high-frequency asymptotic solution of the fundamental Rayleigh mode. Therefore, our approach can provide efficient and accurate solutions in any frequency range, and it is expected to be a powerful and useful tool for simulating the Lg and Rg phases and complete seismograms at regional distances.

Dynamic-stiffness matrix for SH-waves in a layered depth dependent randomly inhomogeneous half-space

This paper is concerned with the derivation of the dynamic-stiffness matrix for a multilayered viscoelastic half-space, with material parameters which are depth dependent random functions. The mean displacements and stresses are obtained for the out-of-plane motion resulting from inclined SH-waves. An explicit solution for the case of exponential correlation function is obtained.

Generation of synthetic seismograms with layer reduction

SUMMARY The surface motion in a plane multilayered half-space has been considered. Layer matrices of SH and P-SV motions are written as sums of matrices. Using these it is seen that when shear wave velocity of a layer, say the lth, is greater than the phase velocity, and layer thickness is large compared to wavelength, the surface motion can be written with the lth layer as the terminating half-space. Thus in computation of surface motion, the depth of the terminating half-space in a given structure depends on phase velocity and frequency. Further, if the source is below the terminating half-space, it has no contribution to surface motion at that phase velocity and frequency for which the terminating half-space has been obtained. This layer reduction formulation is applied in the generation of synthetic seismograms. As this generation requires computation over large ranges of phase velocity and frequency, the present method is important in avoiding excess computation time for unnecessary layers and in avoiding overflow due to the above mentioned Ith layer. The method is most useful in low phase velocity and high frequency ranges.

THREE DIMENSIONAL SOIL‐STRUCTURE INTERACTION ANALYSIS: DEFORMABLE STRUCTURES IN MULTILAYERED SOIL MASS

A new hybrid method based on three‐dimensional finite element idealization in the near field and a semi‐analytic scheme using the principles of wave propagation in multilayered half space in the far field is proposed for the dynamic soil‐structure interaction analysis. The distinguishing feature of this technique from direct or indirect boundary integral techniques is that in boundary integral techniques a distribution of sources are considered at the near field boundary. Strengths of these sources are then adjusted to satisfy the continuity conditions across the near‐field/far‐field interface. In the proposed method unknown sources are placed not at the near field boundary but at the location of the structure. Then the Saint‐Venant's principle is utilized to justify that at a distant point the effect of the structure's vibration can be effectively modelled by an equivalent vibrating point force and vibrating moment at the structure's position. Thus the number of unknowns can be greatly reduced here. For soil‐structure interaction analysis by this method one needs to consider only three unknowns (two force components and one in‐plane moment) for a general two‐dimensional problem and six unknowns (three force components and three moment components) for a general three‐dimensional problem. When a vertically propagating elastic wave strikes a structure which is symmetric about two mutually perpendicular vertical planes the structure can only vibrate vertically for dilatational waves and horizontally for shear waves. Under this situation the number of unknowns is reduced to only one whereas in boundary integral and boundary element techniques the number of unknowns is dependent on the number of nodes at the near field boundary, which is generally much greater than six. Several example problems are solved in this paper using this technique for both flexible and rigid structures in multilayered soil media.

Soil-structure interaction in a layered medium

A hybrid method based on the Green's function formalism is proposed for two-dimensional dynamic soil-structure interaction analysis in a multilayered half-space. The method models the near field (structure and surrounding soil) by finite elements and solves the far field problem through a continuum approach based on an adaptation of the Thomson-Haskell matrix formulation. Displacement continuity between the two regions is enforced by least-squares error minimization. The whole system is excited by steady plane waves (P and SV) propagating with oblique incidence and the structure can be assumed either on the surface or embedded at shallow depth. Special consideration of the scattered field permits the discretized region to be kept reasonably small.

Static deformation of a transversely isotropic multilayered half-space by surface loads

The method of layer matrices is used to solve the problem of the axially-symmetric deformation of a transversely isotropic, multilayered, clastic half-space by surface loads. It is shown that the general problem splits into two independent problems-the torsional problem and the plane problem. The solution of the plane problem is obtained by expressing the stresses in terms of the generalized Love's strain potential. Explicit expressions for the displacements and stresses at any point of the medium due to a vertical force and a torque are obtained. The present formulation avoids the cumbersome nature of the problem and is adaptable to numerical evaluation.

APPLICATION OF A TRANSFER MATRIX METHOD TO VIBRATION PROBLEM OF AN ELASTIC MULTI-LAYERED HALF-SPACE DUE TO BURIED SOURCES

This paper is concerned with the computational procedure of displacements and stresses of an elastic multi-layered half-space due to harmonic loads acting in the interior of the half-space. In calculating the fundamental solutions of displacements and stresses, the authors propose a transfer matrix method taking advantage of a vectormatrix form which expresses the general solutions of displacements and stresses being used in the derivation of dispersion function of an elastic multi-layered half-space. The results shown in this paper can be used as the fundamental solutions in the Boundary Element Method. As numerical examples, the stresses of an elastic half-space due to vertical and horizontal concentrated loads are calculated for the various non-dimensional frequency ωH1/Vs1.

On two-dimensional elastic dislocations in a multilayered half-space

The two-dimensional problem of a long displacement dislocation in a multilayered half-space is studied. Both plane strain and antiplane strain cases are considered. The Thomson-Haskell matrix method is used to obtain the static field caused by the dislocation. The source is represented in terms of the jumps at the source level in the depth-dependent factors in the Fourier integral representation of the displacements and stresses due to the source in an infinite medium. Explicit expressions for the surface displacements due to dip-slip and strike-slip faults of arbitrary dip are obtained. The formulation developed is quite convenient for numerical computation.

Elastostatic Infinite Elements for Layered Half Spaces

The far field behavior of a homogeneous half space and a layered half space is investigated under torsional, vertical, horizontal and moment loadings. Based on the derived far field behavior, three different finite‐element based algorithms, i.e., Ordinary Infinite Elements (OIE), Finite Elements by Singular Contraction (FESC) and Exactly Integrable Infinite Elements (EIIE) are developed to model the far field of multilayered half spaces. The coordinate mapping functions and displacement interpolation functions are selected in accordance with the derived far field model. All three schemes satisfy compatibility and completeness. Excellent results are obtained at a very low computational cost by modeling the near field using a small finite‐element mesh, together with any of the present schemes modeling the domain exterior to the finite elements. Considering the computational efficiency and ease in incorporating FESC into an existing finite element package, FESC is superior to the other two algorithms.

A note on calculation of Rayleigh wave solutions in the multilayered half space.

A calculation scheme for Rayleigh wave solutions at higher frequency range and/or higher modes was investigated. Process of long significant figures was made clear. By combining the computational algorithms developed by HASKELL (1953) and DUNKIN (1965), the loss of significant figures in evaluating the internal displacement was found to be improved. A suggestion was proposed for controlling the minimum layer number required to be put into calculation to maintain sufficient accuracy.

Consolidation of an elastic porous multilayer medium under the pressure of a circular press: PMM vol. 40, n≗ 5, 1976, pp. 909–918

A solution is given for the problem of impressing a circular cylindrical press with a flat base in a packet of an arbitrary number of plane-parallel layers lying on a half-space. The layer and half-space are considered elastic porous bodies saturated with a fluid and possessing different elastic characteristics. Two extreme cases are considered, when the outer surface is completely permeable or impermeable to the interstitial fluid. The condition of total adhesion of the layers is assumed on the boundary planes between the layers. The general solution, based on [1, 2], of the contact problem for an elastic porous multilayered medium being consolidated is presented in Sect. 1, as expressed in terms of multiple Bromwich and Hankel integrals. The solution of the contact problem is constructed in Sect. 2. Calculation methods used to obtain the numerical solution of the problem are described in Sect. 3 and results of calculating the magnitude of the press settlement and the intensity of its pressure in time are presented in the case of a two-layered half-space with different elastic and filtration characteristics. It is found that if the layer is sufficiently thin, but stiffer than the half-space, then inadmissible tensile stresses originate on the contact area as in the case of the analogous elasticity theory problem [3]. Therefore, the pressure distribution on the contact area between the press and the multilayered half-space being consolidated as a function of its elastc and geometric characteristics can differ qualitatively from the pressure distributions on the contact area between a press and a homogeneous halfspace being consolidated. Theoretical principles for the consolidation of an elastic porous medium have been developed by Biot [4, 5]. The first boundary value problem of the theory of consolidation for a half-space and layer is considered in [6–8] in particular loading cases. The plane contact problem of the theory of consolidation for the impression of a press with a flat base is a homogeneous half-space has been examined in [9, 10].

The acoustic response in a liquid layer overlying a multi-layered viscoelastic half-space

A Green function is constructed that represents the steady-state acoustic response in a liquid layer overlying a Voigt viscoelastic multi-layered half-space. The effect of the solid half-space is characterized by a ratio related to the impedance of the sub-bottom. This ratio is evaluated by first developing a matrix recurrence relation for the potentials in the solid which, when repeatedly applied, directly relates the first viscoelastic layer's response to the last layer's. After boundary conditions are applied at the liquid-solid interface, the impedance ratio is obtained from another matrix relation. The result is in a form convenient for digital computer calculations. The response for the homogeneous single layer or half-space is shown to be consistent with existing results.