Case Of Half-space Research Articles

Subsonic Free Surface Waves in Linear Elasticity

For general anisotropic linear elastic solids with smooth boundaries, Rayleigh-type surface waves are studied. Using spectral factorizations of matrix polynomials, a self-contained exposition of the case of a homogeneous half-space is given first. The main result is about inhomogeneous anisotropic bodies with curved surfaces. The existence of subsonic free surface waves is shown by giving ray series asymptotic expansions, including formulas for the transport equation.

Calculation of impedances for axial symmetric foundation embedded in half-space medium using solutions for one layer stratum

The paper is devoted to develop a new scheme to calculate impedance matrices for axial-symmetric foundations embedded in half-space medium. The half-space medium can be approximated by increasing thickness of one layer stratum on rigid bedrock due to the effect of material damping. However, as thickness increases, the numerical problem will arise. This problem is caused by the numerical contamination by some negligible reflection waves from rigid bedrock. In reality, the effect of these reflection waves on impedance is getting small as the thickness of the one layer stratum increases. Therefore, the scheme will employ the solutions for one layer stratum with suppressing these reflection waves to generate the impedance for the case of half-space. The numerical results by the presented scheme are compared with the results by other scheme in order to show that the new numerical scheme is effective and the solutions in layered medium can be extended to obtain the results for the cases of layered half-space medium. Some numerical results of torsional, vertical, horizontal, coupling and rocking impedances with different embedded depths will be presented and comments on the numerical scheme will be given.

Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane

The normal indentation of a rigid circular disk into the surface of a transversely isotropic half-space reinforced by a buried inextensible thin film is addressed. By virtue of a displacement potential function and the Hankel transform, the governing equations of this axisymmetric mixed boundary value problem are represented as a dual integral equation, which is subsequently reduced to a Fredholm integral equation of the second kind. Two important results of the contact stress distribution beneath the disk region as well as the equivalent stiffness of the system are expressed in terms of the solution of the Fredholm integral equation. When the membrane is located on the surface or at the remote boundary, exact closed-form solutions are presented. For the limiting case of an isotropic half-space the results are verified with those available in the literature. As a special case, the elastic fields of a reinforced transversely isotropic half-space under the action of surface axisymmetric patch loads are also given. The effects of anisotropy, embedment depth of the membrane, and material incompressibility on both the contact stress and the normal stiffness factor are depicted in some plots.

Computations of Rayleigh waves in anisotropic elastic media and impedance

An algorithm of computing Rayleigh waves in layered anisotropic elastic media with an arbitrary position of the boundary plane and an arbitrary direction of propagation is considered. With the help of translation operators, the problem is reduced to the construction of an impedance operator for a homogeneous anisotropic half-space and to the construction with its help the Rayleigh waves in the half-space. Computations of the velocities of Rayleigh waves and their polarization for different anisotropic media are given for the case of a homogeneous half-space at the end of the paper. Bibliography: 3 titles.

Impedance Functions for Surface Rigid Rectangular Foundations on Transversely Isotropic Multilayer Half-Spaces

A horizontally multilayered Green elastic transversely isotropic half-space is considered as the domain of the boundary value problem involved in this paper, such that the axes of material symmetry of different layers are parallel to the axis of material symmetry of the lowest half-space, which is depthwise. The domain is assumed to be affected by an arbitrary time-harmonic forced vibration due to a rigid rectangular surface foundation. With the use of a potential function method and the Hankel integral transforms, the displacements and stresses Green's functions are determined in each layer. The unknown functions due to integrations in each layer are transformed to the unknown functions of the surface layer with the use of the concept of propagator matrix and the continuity conditions. The mixed boundary conditions at the surface of the whole domain are numerically satisfied with the assumption of piecewise constant distribution of tractions in the contact area. It is numerically shown that the surface displacement and stress boundary conditions are satisfied very well. The vertical and horizontal impedance functions of the rectangular foundation are determined, which may be used as lumped parameters in time-harmonic soil-structure interaction with transversely isotropic horizontally layered domain as the soil. It is shown that the impedance functions determined in this paper coincide with the same functions for the simpler case of isotropic homogeneous half-space as degenerations of this study.

Quasilinear Elliptic Equations on Half- and Quarter-spaces

Abstract We consider quasilinear elliptic problems of the form Δpu + f(u) = 0 over the half-space H = {x ∈ ℝN : x1 > 0} and over the quarter-space Q = {x ∈ ℝN : x1 > 0, xN > 0}. In the half-space case we assume u ≥ 0 on ∂H, and in the quarter-space case we assume that u ≥ 0 on {x1 = 0} and u = 0 on {xN = 0}. Let u ≢ 0 be a bounded nonnegative solution. For some general classes of nonlinearities f , we show that, in the half-space case, limx1→∞ u(x1, x2, ..., xN) always exists and is a positive zero of f ; and in the quarter-space case, where V is a solution of the one-dimensional problem ΔpV + f(V) = 0 in ℝ+, V(0) = 0, V(t) > 0 for t > 0, V(+∞) = z, with z a positive zero of f . Our results extend most of those in the recent paper of Efendiev and Hamel [6] for the special case p = 2 to the general case p > 1. Moreover, by making use of a sharper Liouville type theorem, some of the results in [6] are improved. To overcome the difficulty of the lack of a strong comparison principle for p-Laplacian problems, we employ a weak sweeping principle.

Lateral translation of an inextensible circular membrane embedded in a transversely isotropic half-space

Abstract The asymmetric problem of lateral translation of an inextensible circular membrane embedded in a transversely isotropic half-space is addressed. With the aid of appropriate Green's functions, the governing equations of the problem are written as a set of coupled integral equations. With further mathematical transformations, the system of dual integral equations is reduced to two coupled Fredholm integral equations of the second kind which are amenable to numerical treatments. The exact closed-form solutions corresponding to two limiting cases of a membrane resting on the surface of a half-space and embedded in a full-space are derived. The jump behavior of results at the edge of the membrane for the case of an infinitesimal embedment is highlighted analytically. For the special case of an isotropic half-space, the results are in exact agreement with those available in the literature. The detailed numerical scheme for solving the coupled Fredholm integral equations is presented. Selected numerical results are depicted and the effects of anisotropy on the lateral stiffness factor are discussed.

Ultrasonic wave’s interaction at fluid-porous piezoelectric layered interface

The complete description of acoustic propagation in a multilayered system is of great interest in a variety of applications such as non-destructive evaluation and acoustic design and there is need for a flexible model that can describe the reflection and transmission of ultrasonic waves in these media. The reflection and transmission of ultrasonic waves from a fluid loaded porous piezoelectric layered structure is studied analytically. The layered structure is considered to be consisting of n number of layers of porous piezoelectric materials. Transfer matrix technique is used to study the layered materials. The analytical expressions for the reflected, transmitted, interaction energy ratios and surface impedance are obtained. The effects of frequency, porosity, angle of incidence, layer thickness and number of layers on the energy ratios and surface impedance are studied for different configurations of the layered materials. The results obtained are deduced for the poro-elastic and fluid loaded porous piezoelectric half space case, which are in agreement with earlier established results. A comparison of the results, obtained by alternate numerical techniques, is made.

Amplification of in-plane seismic ground motion by group cavities in layered half-space (II): with saturated poroelastic soil layers

As the continuation study on amplification of in-plane seismic ground motion by underground group cavities in layered half-space, this study extends to the case of poroelastic half-space with dry poroelastic and saturated poroelastic soil layers. The influence of poroelastic layers on the amplification of seismic ground motion is studied both in frequency domain and time domain using indirect boundary element method (IBEM). It is shown that for the example of a saturated poroelastic site in Tianjin under the excitation of Taft wave and El Centro wave, the amplification of seismic ground motion in poroelastic case is slightly smaller than that in the elastic case, and the amplification of PGA (peak ground acceleration) and its PRS (peak response spectrum) can be increased up to 38.8% and 64.6%; the predominant period of response spectra in poroelastic case becomes shorter to some extent compared with that in the elastic case. It is suggested that the effect of underground group cavities in poroelastic half-space on design seismic ground motion should be considered.

A concave indenter on a piezoelectromagneto-elastic substrate or a layer elastically supported

Indentation of piezoelectromagneto-elastic half-space or a layer on a two-parameter elastic foundation by a cylindrical indenter with a slightly concave base is considered. Full-field magnetoelectro-elastic solutions in elementary functions are obtained for the case of full contact and half-space. If the axial load is small, the contact area will be an annulus the outer circumference of which coincides with the edge of the punch. The inner circumference will shrink with increasing load and there will be a critical load above which the stratum makes contact with the entire punch base. The contact problem for high loads can therefore be treated by classical methods. The more interested case in which the load is less the critical value and the contact region is annulus remains. By use the methods of triple integral equations and series solution technique the solution for an indentured substrate over an annular contact region is also given. For parabolic and conical concave punches the exact or approximate solutions are obtained for full contact or annular contact region, respectively. For the layer on two-parameter elastic foundation and concave punch approximate solution is established.

The complete convergence theorem holds for contact processes in a random environment on [formula omitted

In this article, we consider the basic contact process in a static random environment on the half space Zd×Z+ where the recovery rates are constants and the infection rates are independent and identically distributed random variables. We show that, for almost every environment, the complete convergence theorem holds. This is a generalization of the known result for the classical contact process in the half space case.

A missed persistence property for the Euler equations and its effect on inviscid limits

We consider the problem of strong convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier–Stokes equations, under the Navier slip-type boundary condition (1.4), to the solution of the Euler equations under the no-penetration condition. In two dimensions, the above strong convergence holds in any smooth domain. Furthermore, in three dimensions, arbitrarily strong convergence results hold in the half-space case. In spite of the above results, recently we presented an explicit family of smooth initial data in the 3D sphere, for which the result fails. The result was proved as a by-product of the lack of time persistency for the above boundary condition under the Euler flow. Our aim here is to show a more general, and simpler proof, displayed in arbitrary, smooth domains.

DETERMINATION OF THE FRESNEL REFLECTION COEFFICIENT OF A HALF-SPACE FOR MEDIUM ESTIMATION PURPOSES

A novel procedure for estimating the Fresnel re∞ection coe-cient of a transversally homogeneous half-space medium is introduced. GPR multistatic measurements are exploited as data whereas the Fresnel coe-cient (as a function of the incidence angle) is retrieved by inverting a linear integral equation. Having estimated the re∞ection coe-cient it can be exploited to determine the medium electromagnetic parameters. Numerical examples are used to show the procedure efiectiveness for difierent types of homogeneous half-space media within a two-dimensional scalar geometry. The case of a layered half-space is also discussed.

Equivalence of the Green's Function for a Full-Space to the Direct-Wave Contributions for a Half-Space and a Layered Half-Space

Abstract As a part of a series of studies on the dynamic rupture processes based on a modern boundary integral equation method, we prove that contribution of direct waves to the Green’s function for a half-space and that to the Green’s function in source layer for a layered half-space, which are both expressed in a cylindrical coordinate system, are strictly equivalent to the well-known analytic Green’s function for full-space. Because all of the hypersingularities in boundary integral equations (BIEs) come from the direct waves and contribution of direct waves can be evaluated analytically, the results obtained in this study are very useful in the simplification of the BIEs, especially for the case of layered half-space, in which the treatment on separation of singular parts is otherwise extremely complicated in mathematics and hence impracticable.

Electromagnetic fields induced by a point source in a uniaxial multiferroic full-space, half-space, and bimaterial space

Multiferroic magnetoelectrics are materials that are both ferroelectric and ferromagnetic in the same phase. In addition, electric and magnetic polarizations are strongly coupled in some magnetoelectric multiferroic materials. In this work, by virtue of the image method, exact closed-form Green’s functions are derived for a uniaxial multiferroic full-space, half-space, and bimaterial space. While for the bimaterial space case the interface is assumed to be perfect, for the half-space case four different sets of surface conditions are considered. The point source can be either an electric or a magnetic charge. Numerical results are presented to demonstrate the differences among the infinite-space, half-space, and bimaterial space Green’s functions.

On the Stokes and Navier-Stokes equations in a perturbed half-space

We discuss $L^p$-$L^q$ type estimates of the Stokes semigroup and their application to the Navier-Stokes equation in a perturbed half-space. Especially, we have the $L^p$-$L^q$ type estimate of the gradient of the Stokes semigroup for any $p$ and $q$ with $1 < p \leqq q < \infty$, while the same estimate holds only for the exponents $p$ and $q$ with $1 < p\leqq q \leqq n$ in the exterior domain case, where $n$ denotes the space dimension, and, therefore, we can get better results concerning the asymptotic behaviour of solutions to the Navier-Stokes equations compared with the exterior domain case. Our proof of the $L^p$-$L^q$ type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case, and, then, such better asymptotics near the boundary are also obtained in the perturbed half-space by a perturbation argument. This is one of the reasons why the result in the perturbed half-space case is essentially better compared with the exterior domain case.

Regularity of solutions to a non homogeneous boundary value problem for general Stokes systems in R n +

We give a simple and very complete proof of the existence of a strong (H2) solution to the non-homogeneous problem (1.1) under the non homogeneous boundary conditions (1.6). Here we consider the half-space case Ω = Rn+, n≥ 3, see theorem 1.2. This regularity result was previously obtained by Solonnikov and Scadilov in reference [33] for the classical Stokes system (μ=λ=0, g(x)=0) in the 3−D homogenous case (a=0, b=0) and Ω a suitable open subset of R3.

3-D Modelling of Campi Flegrei Ground Deformations: Role of Caldera Boundary Discontinuities

Campi Flegrei is a caldera complex located west of Naples, Italy. The last eruption occurred in 1538, although the volcano has produced unrest episodes since then, involving rapid and large ground movements (up to 2 m vertical in two years), accompanied by intense seismic activity. Surface ground displacements detected by various techniques (mainly InSAR and levelling) for the 1970 to 1996 period can be modelled by a shallow point source in an elastic half-space, however the source depth is not compatible with seismic and drill hole observations, which suggest a magma chamber just below 4 km depth. This apparent paradox has been explained by the presence of boundary fractures marking the caldera collapse. We present here the first full 3-D modelling for the unrest of 1982–1985 including the effect of caldera bordering fractures and the topography. To model the presence of topography and of the complex caldera rim discontinuities, we used a mixed boundary elements method. The a priori caldera geometry is determined initially from gravimetric modelling results and refined by inversion. The presence of the caldera discontinuities allows a fit to the 1982–1985 levelling data as good as, or better than, in the continuous half-space case, with quite a different source depth which fits the actual magma chamber position as seen from seismic waves. These results show the importance of volcanic structures, and mainly of caldera collapses, in ground deformation episodes.

Three-dimensional Green's functions of steady-state motion in anisotropic half-spaces and bimaterials

Three-dimensional Green's functions (GFs) of steady-state motion in linear anisotropic elastic half-space and bimaterials are derived within the framework of generalized Stroh formalism and two-dimensional Fourier transforms. The present study is limited to the subsonic case where the sextic equation has six complex eigenvalues. If the source and field points reside in the same material, the GF is expressed in two parts: a singular part that corresponds to the infinite-space GF, and a complementary part that corresponds to the reflective effects of the interface in the bimaterial case and of the free surface in the half-space case. The singular part in the physical domain is calculated analytically by applying the Radon transform and the residue theorem. If the source and field points reside in different materials (in the bimaterial case), the GF is a one-term solution. The physical counterparts of the complementary part in the half-space case and of the one-term solution in the bimaterial case are derived as a one-dimensional integral by analytically carrying out the integration along the radial direction in the Fourier-inverse transform. When the source and field points are both on the interface in the bimaterial case or on the surface in the half-space case, singularities appear in the Fourier-inverse transform of the GF. The singularities are treated explicitly using a method proposed recently by the authors. Numerical examples are presented to demonstrate the effects of wave velocity on the stress fields, which may be of interest in various engineering problems of steady-state motion. Furthermore, these GFs are required in the steady-state boundary-integral-equation formulation of anisotropic elasticity.

Critical exponent for semilinear damped wave equations in the N-dimensional half space

We generalize a previous result of Ikehata (Math. Methods Appl. Sci., in press), which studies the critical exponent problem of a semilinear damped wave equation in the one-dimensional half space, to the general N-dimensional half space case. That is to say, one can show the small data global existence of solutions of a mixed problem for the equation u tt − Δu+ u t =| u| p with the power p satisfying p ∗(N)=1+2/(N+1)<p⩽N/[N−2] + if we deal with the problem in the N-dimensional half space.