A tri-material elastodynamic solution for a transversely isotropic full-space

A linear elastic full-space composed of an upper half-space, a lower half-space and a layer of three different transversely isotropic materials under an internal load is considered. The axes of symmetry of the different regions are assumed to be normal to the planar interfaces of the regions and are thus parallel. An arbitrary load in the frequency domain is allowed on a finite patch located at the interface of the upper half-space and the adjacent layer. By means of the complete displacement potentials, the displacements and stresses in the three regions are determined in Fourier–Hankel space in the form of line integrals. The solution can be degenerated to the solution for (i) a full-space under an arbitrary buried load, (ii) a half-space contain a layer bonded to the top of it under an arbitrary surface force, (iii) a half-space under an arbitrary surface load, (iv) a two layer half-space under an arbitrary force applied at the interface of two regions, (v) a half-space under an arbitrary buried force, (vi) a layer of finite thickness fixed at the bottom and under an arbitrary surface load, and (vii) a bi-material full-space under an arbitrary load at the interface of two materials. Examples of the displacements and stresses are obtained numerically and compared to existing solutions.     

An analytical solution of an axisymmetric problem of deforming an isotropic half-space with an elastic fixed boundary under the action of a distributed load

An analytical solution to the axisymmetric problem on the action of a distributed load on an isotropic half-space when the load is given by a function dependent on the radial coordinate is obtained. The surface of the half-space is elastically fixed outside the circular domain of load application, the shear stresses are absent along the entire boundary, and the stresses vanish at infinity. At the boundary and inside the elastic half-space, the solutions are represented by the formulas for the stress tensor components and for the displacement vector components.     

Earth surface response to a load moving in a tunnel

Using the solution of the problem on an elastic half-space subjected to a load uniformly propagating on the surface of a circular cylindrical cavity along its generator parallel to the free boundary of the half-space, we study the stress-strain state of the Earth surface over the tunnel under the action of normal axisymmetric periodic and aperiodic loads moving in the tunnel. Numerical results are analyzed on the basis of the tables and graphs presented in the paper.     

Seepage consolidation of elastic half-space under an axisymmetric load

The process of seepage consolidation of elastic saturated half-space under the action of a normal load on its surface is investigated assuming both incompressibility of fluid and skeleton grains and independence of the total skeleton stresses on time. Analytic representations for the fluid pressure and the half-space surface settlement are found when the half-space is loaded by a concentrated force. The maximum settlement is also found for a uniform loading of the surface over the circle area.     

Application of the reciprocity theorem to scattering of surface waves by a cavity

Scattering of surface waves by a cylindrical cavity at the surface of a homogenous, isotropic, linearly elastic half-space is analyzed in this paper. In the usual manner, the scattered field is shown to be equivalent to the radiation from a distribution of tractions, obtained from the incident wave on the surface of the cavity. For the approximation used in this paper, these tractions are shifted to tractions applied to the projection of the cavity on the surface of the half-space. The radiation of surface waves from a normal and a tangential line load, recently determined by the use of the reciprocity theorem, is employed to obtain the field scattered by the cavity from the superposition of displacements due to the distributed surface tractions. The vertical displacement at some distance from the cavity is compared with the solution of the scattering problem obtained by the boundary element method (BEM) for various depths and widths of the cavity. Comparisons between the analytical and BEM results are graphically displayed. The limitations of the approximate approach are discussed based on the comparisons with the BEM results.     

STEADY-STATE RESPONSE OF A TIMOSHENKO BEAM ON AN ELASTIC HALF-SPACE UNDER A MOVING LOAD

By introducing the equivalent stiffness of an elastic half-space interacting with a Timoshenko beam, the displacement solution of the beam resting on an elastic half-space subjected to a moving load is presented. Based on the relative relation of wave velocities of the half-space and the beam, four cases with the combination of different parameters of the half-space and the beam, the system of soft beam and hard half-space, the system of sub-soft beam and hard half-space, the system of sub-hard beam and soft half-space, and the system of hard beam and soft half-space are considered. The critical velocities of the moving load are studied using dispersion curves. It is found that critical velocities of the moving load on the Timoshenko beam depend on the relative relation of wave velocities of the half-space and the beam. The Rayleigh wave velocity in the half-space is always a critical velocity and the response of the system will be infinite when the load velocity reaches it. For the system of soft beam and hard half-space, wave velocities of the beam are also critical velocities. Besides the shear wave velocity of the beam, there is an additional minimum critical velocity for the system of sub-soft beam and hard half-space. While for systems of (sub-) hard beams and soft half-space, wave velocities of the beam are no longer critical ones. Comparison with the Euler-Bernoulli beam shows that the critical velocities and response of the two types of beams are much different for the system of (sub-) soft beam and hard half-space but are similar to each other for the system of (sub-) hard beam and soft half space. The largest displacement of the beam is almost at the location of the load and the displacement along the beam is almost symmetrical if the load velocity is smaller than the minimum critical velocity (the shear wave velocity of the beam for the system of soft beam and hard half-space). The largest displacement of the beam shifts behind the load and the asymmetry of the displacement along the beam increases with the increase of the load velocity due to the damping and wave radiation. The displacement of the beam at the front of the load is very small if the load velocity is larger than the largest wave velocity of the beam and the half space. The results of the present study provide attractive theoretical and practical references for the analysis of ground vibration induced by the high-speed train.     

Using complex potentials to determine the reaction of a prestressed two-layer elastic half-space to a moving load

Complex potentials in common form for compressible and incompressible elastic bodies are used to formulate and solve the problem of stationary motion of a prestressed two-layer elastic half-space under a moving surface load. The results presented are similar to those obtained earlier using the Fourier transform __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 3–15, May 2008.     

Exterior 3D lamb problem: Harmonic load distributed over a surface

The solutions of the exterior Lamb problem with a distributed harmonic surface load acting on the boundary of an elastic half-space are studied. A load normal to the surface and distributed over the surface as the Poisson kernel is considered. The solution is constructed with the use of integral transforms and the finite-element method.     

非饱和弹性半空间地基的稳态响应半解析法研究

应用半解析法研究简谐荷载下非饱和弹性半空间地基的稳态响应。基于非饱和土的动力控制方程以及非饱和弹性半空间的边界条件,建立地基层单元的半解析函数,应用加权残数法得到在简谐荷载下非饱和弹性半空间地基的稳态响应半解析方程。对半解析方程求解,得到了竖向简谐荷载作用下非饱和弹性地基水平位移和竖向位移幅值,数值分析了饱和度和地基深度等参数对孔压和位移幅值的影响。研究结果表明,应用本文方法研究非饱和弹性半空间地基的稳态响应是切实有效的。     

A dynamic problem for a prestressed compressible layered half-space

The linearized theory of elasticity for prestressed bodies is used to solve a stationary plane problem for a prestressed two-layer half-space under a surface load moving with constant velocity. The half-space is assumed to be compressible and to have an arbitrary elastic potential. The Fourier transform is used to obtain the fundamental solution of the problem for different contact conditions and load velocities. A compressible material with a harmonic elastic potential is considered as an example __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 35–55, April 2008.     

Numerical analysis of the isolation of the vibration due to Rayleigh waves by using pile rows in the poroelastic medium

The isolation of the vibration due to harmonic Rayleigh waves using pile rows embedded in a saturated poroelastic half-space is investigated in this study. Based on Biot’s theory and the potential function method, the free field solution for Rayleigh waves along the surface of the poroelastic half-space is derived first. The fundamental solution for a harmonic circular patch load applied in the poroelastic half-space are obtained in terms of Biot’s theory and the integral transform method. Using Muki’s method and the fundamental solution for the circular patch load as well as the Rayleigh waves solution for the poroelastic half-space, the second kind of Fredholm integral equations in the frequency domain for pile rows are derived. Numerical solution of the integral equations yields the dynamic response of the pile–soil system to incident Rayleigh waves. Influences of various parameters on the vibration isolation effect of piles rows are investigated numerically. Numerical results suggest that for the same vibration source, the same pile rows will produce a better vibration isolation effect for the poroelastic medium than for a single phase elastic medium. Also, stiffer piles tend to have better vibration isolation effect than flexible piles. Moreover, the pile length and the spacing between neighboring piles in each pile row have significant influence on the vibration isolation effect of pile rows.     

Dynamic response of a half-space to radial shear surface loadings

Summary The axisymmetric response of an elastic half-space to the sudden application of radial shear surface loadings is investigated in this paper. Exact expressions for the surface displacement components are developed by means of integral transforms and complex functions. Numerical results are presented in diagrams to show the wave propagation on the surface due to the dynamic loadings, and the wave front singularities in the displacement components are particularly discussed. It is found that, for a given point on the half-space surface, Rayleigh surface waves issued from the nearest and farthest disturbances give rise to a jump or a two-sided logarithmic singularity in the displacement components.     

Green functions for an incompressible linearly nonhomogeneous half-space

Summary Time-harmonic vibrations of an incompressible half-space having shear modulus linearly increasing with depth are studied. The half-space is subjected to a surface load which has vertical or hovizontal direction. The general solution of the time-harmonic, in the vertical direction nonhomogeneous problem is constructed for arbitrary angular distribution in the horizontal plane. Numerical results concerning surface displacements due to a point force are given for the case of nonzero shear modulus at the surface. These results show that nonhomogeneity can considerably increase amplitudes at large distances from the applied force.     

Dynamics of an elastic half-space with a reinforced cylindrical cavity under moving loads

Partial separation of variables and reexpansion of cylindrical and plane waves are used to find the solution describing the uniform motion of a load along a thin circular cylindrical shell in an elastic half-space with the free surface parallel to the axis of the shell. This is a model problem for studying the dynamics of tunnels and shallow-buried pipelines under transport loads. Dispersion curves for the cases of sliding and tight contact between the shell and the half-space are plotted and analyzed. The effect of the shell parameters on the stress–strain state of the half-space is examined     

Elastic Displacement in a Half-Space Under the Action of a Tensor Force. General Solution for the Half-Space with Point Forces

The elastic displacement in an isotropic elastic half-space with free surface is calculated for a point tensor force which may arise from the seismic moment of seismic sources concentrated at an inner point of the half-space. The starting point of the calculation is the decomposition of the displacement by means of the Helmholtz potentials and a simplified version of the Grodskii-Neuber-Papkovitch procedure. The calculations are carried out by using generalized Poisson equations and in-plane Fourier transforms, which are convenient for treating boundary conditions. As a general result we compute the displacement in the isotropic elastic half-space with free surface caused by point forces with arbitrary structure and orientation, localized either beneath the surface (generalized Mindlin problem) or on the surface (generalized Boussinesq-Cerruti problems). The inverse Fourier transforms are carried out by means of Sommerfeld-type integrals. For forces buried in the half-space explicit results are given for the surface displacement, which may exhibit finite values at the origin, or at distances on the surface of the order of the depth of the source. The problem presented here may be viewed as an addition to the well-known static problems of elastic equilibrium of a half-space under the action of concentrated loads. The application of the method to similar problems and another approach to the starting point of the general solution are discussed.     

Unbonded contact between a circular plate and an elastic half-space

The paper concerns the unbonded contact between a thin circular plate of finite radius, governed by Kirchhof or Reissner theory, pressed by means of rotationally symmetric distributed load and its own weight against the surface of an elastic half-space. The contact is assumed frictionless and unbonded. A Hankel transform solution is used for the half-space and the plate deflection is found by inverting the plate equation. The coefficients in a power expansion are obtained by equating plate and half-space deflections at a number of points in the contact region. The variation of contact radius with plate radius, the radius of the uniformly applied load, and the relative stiffness of plate and foundation, is displayed in a series of figures.     

A four-part boundary value problem in elasticity — indentation by a discontinuously concave punch

A solution is given for the frictionless indentation of an elastic half-space by a flat-ended cylindrical punch with a central circular recess, when the load is large enough to establish a circular region of contact in the recess. The problem is reduced to two simultaneous Fredholm equations using the method of complex potentials due to Green and Collins. Results are presented for the relationship between load, contact radius and penetration for various punch geometries.     

Analysis of nonlinear acoustoelastic effect of surface acoustic waves in laminated structures by transfer matrix method

The propagation of surface acoustic waves in a layered half-space is investigated in this paper, where a thin cubic Ge film is perfectly bonded to an isotropic elastic Si half-space. Application of the transfer matrix and by solving the coupled field equations, solutions to the mechanical displacements are obtained for the film and elastic substrate, respectively. The phase velocity equations for surface acoustic waves are obtained. Effects of the homogeneous initial stresses induced by the mismatch of the film and substrate are discussed in detail. The results are useful for the design of acoustic surface wave devices.     

The transient elastodynamic field excited by trans-Rayleigh trains

The elastic wave field due to a surface load in motion over an elastic half-space is investigated. The model serves as a canonical solution for the modelling of high speed ‘trans-Rayleigh’ trains. The analysis presented leads to closed form expressions for the particle displacement, conical waves and Rayleigh waves as separate contributions. The linearized elastodynamic equations are mapped into a proper form in order to apply the Cagniard-de Hoop technique and find closed form time domain solutions for the particle displacement in the subsonic state, transonic state and supersonic state. A special transformation is used that yields closed form space-time domain expressions for the Conical wave as well as the Rayleigh wave contributions. Attention is focussed on surface source speeds in the neighbourhood of the Rayleigh wave speed and speeds that exceed the wave speed of the shear wave. Numerical results for the conical wave field and Rayleigh wave field are presented at observation points just below the surface showing the enormous effects of the Rayleigh wave at source speeds in the near vicinity of the Rayleigh wave speed.     

Green functions for a compressible linearly non homogeneous half-space

Summary The paper presents a study of time-harmonic vibration of a half-space possessing a shear modulus linearly increasing with depth. Completing the previous paper [1], where the time-harmonic vibration of an incompressible half-space has been considered, the problem is now solved for a compressible as well as an incompressible material. The half-space is subjected to a vertical or horizontal surface load. The solution is represented in terms of Fourier-Bessel integrals containing functions of depth coordinate that are expressed through confluent hypergeometric functions. Numerical results concerning surface displacements due to a point force are given for a wide range of frequency variations and degree of non-homogeneity. The results show that, as compared to the homogeneous case, non-homogeneity can considerably increase vibration amplitudes at large distances from the applied force. Received 19 August 1996; accepted for publication 16, December 1996