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.     

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     

On simplified solutions of contact problems for elastic foundations

In the paper, a method of averaging displacements in a circular area lying in a linearly elastic transversally isotropic foundation is developed for the four modes of motion: vertical, horizontal, rocking and torsional. The corresponding formulae are constructed in a general form which does not depend on the kind of Green functions. For vertical and horizontal modes, uniform load distribution are applied and the simple integral mean is considered, whereas for rotational modes the load proportional to the distance from an axis of rotation is used, and angles of rotation for individual points are averaged with weight of the distance squared. Along with the case of equal radii of circles of loading and averaging, the case of different radii is studied, which allows one to consider contact problems for embedded axisymmetric foundations having the radius varying with depth. As examples the following contact problems are studied: static stiffness for a cone embedded in a homogeneous isotropic half-space in vertical motion, and dynamic stiffness for a disk on a layer resting on a homogeneous half-space for four modes of motion. Comparisons with the corresponding exact solutions are carried out.     

Two-dimensional sliding frictional contact of functionally graded materials

A multi-layered model for sliding frictional contact analysis of functionally graded materials (FGMs) with arbitrarily varying shear modulus under plane strain-state deformation has been developed. Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into several sub-layers and in each sub-layers the shear modulus is assumed to be a linear function while the Poisson's ratio is assumed to be a constant. In the contact area, it is assumed that the friction is one of Coulomb type. With this model the fundamental solutions for concentrated forces acting perpendicular and parallel to the FGMs layer surface are obtained. Then the sliding frictional contact problem of a functionally graded coated half-space is investigated. The transfer matrix method and Fourier integral transform technique are employed to cast the problem to a Cauchy singular integral equation. The contact stresses and contact area are calculated for various moving stamps by solving the equations numerically. The results show that appropriate gradual variation of the shear modulus can significantly alter the stresses in the contact zone.     

具有水平摩阻力的弹性地基上双层叠合梁的解

将双层叠合梁之间的接触状况拟合为一符合Goodman假设的弹性夹层, 并对 Winkler地基模型加以推广, 把地基视为具有水平和竖向反力的弹性支承体, 进而, 导出 具有水平摩阻力的弹性地基上双层叠合梁的微分方程组及其解析解. 随后, 提出了考虑梁截 面竖向拉压和水平剪切效应的夹层和地基广义反应模量的概念和计算式. 最后, 通过几个算 例来考察地基水平摩阻力对叠合梁挠度、截面总弯矩力和最大弯拉应力的影响. 结果表明, 地基水平摩阻力的影响将不能被忽略.     

Surface displacements due to elastic wave scattering by buried planar and non-planar cracks

The scattering of time-harmonic plane longitudinal, shear, and Rayleigh waves by a crack in two dimensions embedded in a semi-infinite homogeneous isotropic elastic half-space has been studied in this paper. Two problems have been considered: a straight crack and a Y-shaped crack. A hybrid numerical technique combining a multipolar representation of the scattered field in the half-space with the finite element method has been used to obtain the far-field displacements as well as the stress-intensity factors for the crack tips. Results for vertical displacement on the free surface of the half-space are presented in this paper.     

饱和土埋置力源的三维动力Lamb问题解答

基于一组弹性土波动方程，应用Fourier级数展开和Hankel积分变换，得到了三维问题饱和土骨架与孔隙水的应力及位移分量在变换域内的积分形式通解．考虑地基表面透水情形，由边界条件导出了半空间饱和土体在埋置力源作用下的三维动力Lamb问题的解答．给出了埋置水平力作用下地基表面竖向位移、径向位移及周向位移的数值解．该研究为运用边界元法求解饱和地基的动力响应课题奠定了理论基础．     

Scattering of antiplane shear waves by a submerged crack

The scattering of time harmonic antiplane shear waves by a crack situated parallel to the surface of a homogenious elastic half-space is considered. The problem is investigated for both clamped and stress-free conditions at the surface of the half-space. Singular integral equations are derived for each case and treated numerically by the Gauss-Chebyshev technique. Harmonic stress intensity factors and crack opening displacements are computed as functions of dimensionless frequency, submerged depth-to-crack width ratio, and angle of incidence. Results are obtained for a moderately wide range of frequencies and are set out in graphical form.     

Fundamental solutions to contact problems of a magneto-electro-elastic half-space indented by a semi-infinite punch

This paper presents the fundamental contact solutions of a magneto-electro-elastic half-space indented by a smooth and rigid half-infinite punch. The material is assumed to be transversely isotropic with the symmetric axis perpendicular to the surface of the half-space. Based on the general solutions, the generalized method of potential theory is adopted to solve the boundary value problems. The involved potentials are properly assumed and the corresponding boundary integral equations are solved by using the results in literature. Complete and exact fundamental solutions are derived case by case, in terms of elementary functions for the first time. The obtained solutions are of significance to boundary element analysis, and an important role in determining the physical properties of materials by indentation technique can be expected to play.     

Anti-plane shear Green’s functions for an isotropic elastic half-space with a material surface

This paper presents analytical Green’s function solutions for an isotropic elastic half-space subject to anti-plane shear deformation. The boundary of the half-space is modeled as a material surface, for which the Gurtin–Murdoch theory for surface elasticity is employed. By using Fourier cosine transform, analytical solutions for a point force applied both in the interior or on the boundary of the half-space are derived in terms of two particular integrals. Through simple numerical examples, it is shown that the surface elasticity has an important influence on the elastic field in the half-space. The present Green’s functions can be used in boundary element method analysis of more complicated problems.     

Green’s functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials

In this paper, we obtain Green’s functions of two-dimensional (2D) piezoelectric quasicrystal (PQC) in half-space and bimaterials. Based on the elastic theory of QCs, the Stroh formalism is used to derive the general solutions of displacements and stresses. Then, we obtain the analytical solutions of half-space and bimaterial Green’s functions. Besides, the interfacial Green’s function for bimaterials is also obtained in the analytical form. Before numerical studies, a comparative study is carried out to validate the present solutions. Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force, the line dislocation, the line charge, and the phason line force. As a result, the coupling effect among the phonon field, the phason field, and the electric field is prominent, and the butterfly-shaped contours are characteristic in 2D PQCs. In addition, the changes of material parameters cause variations in physical quantities to a certain degree.     

FINITE LAYER ANALYSIS FOR SEMI-INFINITE SOILS (Ⅰ)——GENERAL THEORY

In the present paper a finite layer method is studied for the elastodynamics of transverse isotropic bodies. With this method, semi-infinite soils can be considered as an transverse isotropic half-space, its material functions varying with depth. Dividing the half.space into a series of layers in the direction of depth the material fimetioms in each layer are simulated by exponential fumctions Consequently, the fundamental equations to be solved can be simplified if the fouricr transform with repsect to coordinates is used. We have obtained the relationship between the "layer forces" and "layer displacements". This finite layer method, in fact, can also be called a semi-analytical method. It possesses those advantages as the usual semi-analytical methods do, and can be used to analyse the problem of the interaction between soils and structures.     

Evaluation of Green’s function for vertical point-load excitation applied to the surface of a layered semi-infinite elastic medium

The topic of this paper is to show that the integrals of infinite extent representing the surface displacements of a layered half-space loaded by a harmonic, vertical point load can be reduced to integrals with finite integration range. The displacements are first expressed through wave potentials and the Hankel integral transform in the radial coordinate is applied to the governing equations and boundary conditions, leading to the solutions in the transformed domain. After the application of the inverse Hankel transform it is shown that the inversion integrands are symmetric/antimetric in the transformation parameter and that this characteristic is preserved for any number of layers. Based on this fact the infinite inversion integrals are reduced to integrals with finite range by choosing the suitable representation of the Bessel function and use of the fundamental rules of contour integration, permitting simpler analytical or numerical evaluation. A numerical example is presented and the results are compared to those obtained by the CLASSI program.     

Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

Thermoelastostatic equilibrium of a spatial quadrant: Green’s function and solution in integrals

In this study, a new Green??s function and a new Green-type integral formula for a 3D boundary value problem (BVP) in thermoelastostatics for a quarter-space are derived in closed form. On the boundary half-planes, twice mixed homogeneous mechanical boundary conditions are given. One boundary half-plane is free of loadings and the normal displacements and the tangential stresses are zero on the other one. The thermoelastic displacements are subjected by a heat source applied in the inner points of the quarter-space and by mixed non-homogeneous boundary heat conditions. On one of the boundary half-plane, the temperature is prescribed and the heat flux is given on the other one. When the thermoelastic Green??s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by ??-Dirac??s function. All results are obtained in elementary functions that are formulated in a special theorem. As a particular case, when one of the boundary half-plane of the quarter-space is placed at infinity, we obtain the respective results for half-space. Exact solutions in elementary functions for two particular BVPs for a thermoelastic quarter-space and their graphical presentations are included. They demonstrate how to apply the obtained Green-type integral formula as well as the derived influence functions of an inner unit point body force on volume dilatation to solve particular BVPs of thermoelasticity. In addition, advantages of the obtained results and possibilities of the proposed method to derive new Green??s functions and new Green-type integral formulae not for quarter-space only, but also for any canonical Cartesian domain are also discussed.     

The three-dimensional steady-state thermo-elastodynamic problem of moving sources over a half space

A procedure based on the Radon transform and elements of distribution theory is developed to obtain fundamental thermoelastic three-dimensional (3D) solutions for thermal and/or mechanical point sources moving steadily over the surface of a half space. A concentrated heat flux is taken as the thermal source, whereas the mechanical source consists of normal and tangential concentrated loads. It is assumed that the sources move with a constant velocity along a fixed direction. The solutions obtained are exact within the bounds of Biot’s coupled thermo-elastodynamic theory, and results for surface displacements are obtained over the entire speed range (i.e. for sub-Rayleigh, super-Rayleigh/subsonic, transonic and supersonic source speeds). This problem has relevance to situations in Contact Mechanics, Tribology and Dynamic Fracture, and is especially related to the well-known heat checking problem (thermo-mechanical cracking in an unflawed half-space material from high-speed asperity excitations). Our solution technique fully exploits as auxiliary solutions the ones for the corresponding plane-strain and anti-plane shear problems by reducing the original 3D problem to two separate 2D problems. These problems are uncoupled from each other, with the first problem being thermoelastic and the second one pure elastic. In particular, the auxiliary plane-strain problem is completely analogous to the original problem, not only with regard to the field equations but also with regard to the boundary conditions. This makes the technique employed here more advantageous than other techniques, which require the prior determination of a fictitious auxiliary plane-strain problem through solving an integral equation.     

The use of complex valued functions for the solution of three-dimensional elasticity problems

For the treatment of plane elasticity problems the use of complex functions has turned out to be an elegant and effective method. The complex formulation of stresses and displacements resulted from the introduction of a real stress function which has to satisfy the 2-dimensional biharmonic equation. It can be expressed therefore with the aid of complex functions. In this paper the fundamental idea of characterizing the elasticity problem in the case of zero body forces by a biharmonic stress function represented by complex valued functions is extended to 3-dimensional problems. The complex formulas are derived in such a way that the Muskhelishvili formulation for plane strain is included as a special case. As in the plane case, arbitrary complex valued functions can be used to ensure the satisfaction of the governing equations. Within the solution of an analytical example some advantages of the presented method are illustrated.     

The torsion of a non-homogeneous stratum with a hyperbolic variation of shear modulus

An exact formulation of the governing dual integral equations for the torsion of a non-homogeneous stratum due to a rigid circular body at its free surface is presented. The stratum varies in shear modulus according to the hyperbolic variation in a contemporary work [1]. It is shown that the unknown static stress distribution under the rigid body is governed by modified Bessel function of the first kind. By comparing the governing functions in the dual integral equations for five cases of elastic media: homogeneous half-space, and stratum, linearly non-homogeneous half-space and stratum and, finally, the present non-homogeneous stratum with hyperbolic variation, it is established that the surface shear modulus is the dominant parameter in the assessment of the stress and displacement fields in a non-homogeneous stratum where lateral variation of elastic properties is negligible.     

Pile analysis by simple integral equation methods

Two simple integral equation methods are proposed for the analysis of vertical loaded pile. One of them is; let the axisymmetrical loads formed by Mindlin's horizontal point forces be distributed along the axis z in [0, L] of the elastic half-space, and composed with the Boussinesq's point force. The other is: in addition to the above fictitious loads, the. Mindlin's vertical forces are distributed along the axis z in [0, L]. The former reduces the problem of a vertical loaded pile embedded in a half-space with the following boundary conditions.     

Coupled rocking and sliding vibration response of a structure to harmonic horizontal ground motion

For a strip wall erected on a rigid strip foundation and supported by the surface of the ground, the dynamic soil-structure interaction under the action of the horizontal ground motion is investigated. The ground motion is idealized as vertically propagating, horizontal steady-state motion. Because the horizontal ground motion brings about the sliding vibration of the foundation as well as the rocking vibration, the coupled rocking and sliding vibration of the soil-structure system is considered in the present paper. For the contact between the ground and foundation, the following assumptions are made: 1) the contact is assumed to be welded, that is to say, the motion of the foundation is consistent with the ground; 2) the horizontal translation at each point on the bottom surface of the foundation is equal to a constant; 3) the distribution of the normal displacements under the foundation remains to be linear in the rocking vibration. For comparison, the case of uncoupled vibration is considered also. The use of Fourier transform method yields dual integral equations (for the case without coupling effect) or simultaneous dual integral equations (for the case with coupling effect). Both of them are solved by means of infinite series of orthogonal functions, the Jacobi polynomials. The numerical results show that there is a significant difference between the displacements of the foundation, the relative displacements of the top of the wall with respect to its base, and the distribution of contact stresses beneath the foundation, for the cases with and without coupling effect.