半空间饱和土在内部简谐水平力作用下的Ge函数

根据积分变换方法得出了半空间内部作用简谐水平力时的Gree函数．首先，利用Hankel积分变换方法，直接对频域内的Biot波动方程进行求解，得出Biot波动方程的通解；利用通解和半空间内部作用水平力时边界上的边界条件，以及力作用面上的连续性条件，可以得出上述边值问题的解；对于边值问题在变换域内的解进行相应的逆变换，就可以得出频域内的Gree函数．本文得到的线弹性退化解与文献中的结果吻合．最后，文中给出了两个算例．      

Solution of dynamic Green׳s function for unsaturated soil under internal excitation

The closed form three-dimensional Green׳s function of a semi-infinite unsaturated poroelastic medium subjected to an arbitrary internal harmonic loading is derived, with consideration of capillary pressure and dynamic shear modulus varying with saturation. By applying the Fourier expansion techniques and Hankel integral transforms to the circumferential and radial coordinates, respectively, the general solution for the governing partial differential equations is obtained in the transformed domain. A corresponding boundary value problem is formulated. The integral solutions for the induced displacements, pore pressure and net stress are then determined considering the continuity conditions. The formulas are compared with the degenerated solution of saturated soils and confirmed. Numerical results reveal that the response of the unsaturated half-space depends significantly on the saturation by altering dynamic shear modulus to account for the effects of matric suction on soil stiffness. Slight differences between the results occur if only the saturation is taken into account. Moreover, a large source-depth results in a pronounced contribution to the reduction of surface displacement amplitudes. The analytical solutions concluded in the study offer a broader application to dynamic response associated with axi-symmetric and asymmetric conditions.     

Scattering of elastic waves by a general anisotropic basin. Part 2: a 3D model

Scattering of plane harmonic waves by a three‐dimensional basin of arbitrary shape embedded within elastic half‐space is investigated by using an indirect boundary integral equation approach. The materials of the basin and the half‐space are assumed to be the most general anisotropic, homogeneous, linearly elastic solids without any material symmetry (i.e. triclinic). The unknown scattered waves are expressed in terms of three‐dimensional triclinic time harmonic full‐space Green's functions. The results have been tested by comparing the surface response of semi spherical isotropic and transversely isotropic basins for which the numerical solutions are available. Surface displacements are presented for a semicircular basin subjected to a vertical incident plane harmonic pseudo‐P‐, SV‐, or SH‐wave. These results are compared with the motion obtained for the corresponding equivalent isotropic models. The results show that presence of the basin may cause significant amplification of ground motion when compared to the free‐field displacements. The peak amplitude of the predominant component of surface motion is smaller for the anisotropic basin than for the corresponding isotropic one. Anisotropic response may be asymmetric even for symmetric geometry and incidence. Anisotropic surface displacement generally includes all three components of motion which may not be the case for the isotropic results. Furthermore, anisotropic response strongly depends upon the nature of the incident wave, degree of material anisotropy and the azimuthal orientation of the observation station. These results clearly demonstrate the importance of anisotropy in amplification of surface ground motion. Copyright © 2003 John Wiley & Sons, Ltd.     

Dynamic response of a flexible plate on saturated soil layer

An analytical approach is developed to study the dynamic response of a flexible plate on single-layered saturated soil. The analysis is based on Biot's two-phased theory of poroelasticity and also on the classical thin-plate theory. First, the governing differential equations for saturated soil are solved by the use of Hankel transform. The general solutions of the skeleton displacements, stresses, and pore pressures, derived in the transformed domain, are subsequently incorporated into the imposed boundary conditions, which leads to a set of dual integral equations describing the corresponding mixed boundary value problem. These governing integral equations are finally reduced to the Fredholm integral equations of the second kind and solved by standard numerical procedures. The accuracy of the present solution is validated via comparisons with existing solutions for an ideal elastic half-space. Furthermore, some numerical results are presented to show the influences of the layer depth, the plate flexibility, and the soil porosity on the dynamic compliances.     

Exact analytical solutions for two-dimensional advection-dispersion equation in cylindrical coordinates subject to third-type inlet boundary condition

Exact analytical solutions for two-dimensional advection-dispersion equation (ADE) in cylindrical coordinates subject to the third-type inlet boundary condition are presented in this study. The finite Hankel transform technique in combination with the Laplace transform method is adopted to solve the two-dimensional ADE in cylindrical coordinates. Solutions are derived for both continuous input and instantaneous slug input. The developed analytical solutions are compared with the solutions for first-type inlet boundary condition to illustrate the influence of the inlet condition on the two-dimensional solute transport in a porous medium system with a radial geometry. Results show significant discrepancies between the breakthrough curves obtained from analytical solutions for the first-type and third-type inlet boundary conditions for large longitudinal dispersion coefficients. The developed solutions conserve the solute mass and are efficient tools for simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment or an in situ infiltration test with a tracer.     

Quasi-static Planar Deformation in a Medium Composed of Elastic and Thermoelastic Solid Half Spaces Due to Seismic Sources in an Elastic Solid

A two-dimensional problem of quasi static deformation of a medium consisting of an elastic half space in welded contact with thermoelastic half space, caused due to seismic sources, is studied. Source is considered to be in the elastic half space. The basic equations, governed by the coupled theory of thermoelasticity, are used to model for thermoelastic half space. The analytical expressions for displacements, strain and stresses in the two half spaces are obtained first for line source and then for dip slip fault. The results for two particular cases, adiabatic conditions and isothermal conditions, are also obtained. Numerical results for displacements, stresses and temperature distribution have also been computed and are shown.     

Forced vertical vibration of rigid circular disc buried in an arbitrary depth of a transversely isotropic half space

This paper is concerned with the investigation of the vertical vibration of a rigid circular disc buried at an arbitrary depth in a transversely isotropic half space in such a way the axis of material symmetry of the half space is normal to the surface of it and parallel to the vibration direction. By using the Hankel integral transforms, the mixed boundary-value problem is transformed to a pair of integral equations called dual integral equations, which generally can be reduced to a Fredholm integral equation of the second kind. With the aid of complex variable or contour integration, the governing integral equation is numerically solved in the general dynamic case. Two degenerated cases (i) the disc is buried in a transversely isotropic full space, and (ii) rigid circular disc is attached on the surface of the half space are discussed. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure and the static impedance/compliance function are explicitly found. It is shown that the vertical pressure and the compliance function reduced for isotropic half space are identical to the previous solutions reported in the literature. The dynamic contact pressure under the disc and the impedance function are numerically evaluated in general dynamic case and graphically shown that the singularity exists in the contact pressure at the edge of the disc is the same as the static case. In addition, the impedance functions evaluated here for the isotropic domain are collapsed on the solution given by Luco and Mita. To show the effect of different material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared.     

Impedance matrices for circular foundation embedded in layered medium

A numerical scheme is developed in the paper for calculating torsional, vertical, horizontal, coupling and rocking impedances in frequency domain for axial-symmetric foundations embedded in layered media. In the scheme, the whole soil domain is divided into interior and exterior domains. For the exterior domain, the analytic solutions with unknown coefficients are obtained by solving three-dimensional (3D) wave equations in cylindrical coordinates satisfying homogeneous boundary conditions. For the interior domain, the analytical solutions are also obtained by solving the same 3D wave equations satisfying the homogeneous boundary conditions and the prescribed boundary conditions. The prescribed conditions are the interaction tractions at the interfaces between embedded foundation and surrounding soil. The interaction tractions are assumed to be piecewise linear. The piecewise linear tractions at the bottom surface of foundation will be decomposed into a series of Bessel functions which can be easily fitted into the general solutions of wave equations in cylindrical coordinates. After all the analytic solutions with unknown coefficients for both interior and exterior domains are found, the variational principle is employed using the continuity conditions (both displacements and stresses) at the interfaces between interior and exterior domains, interior domain and foundation, and exterior domain and foundation to find impedance functions.     

Transient elastodynamic analysis by time domain bem in cylindrical coordinates

A time domain boundary element in a cylindrical coordinate system is developed for the analysis of wave propagation in a half space. The integral formulation is based on Graffi's dynamic reciprocal theorem and Stokes' fundamental solutions. The field quantities (displacements and tractions) are expressed as products of Fourier series in the tangential direction and linear polynomials in the other spatial directions. Gaussian integration is used to integrate the non-singular parts of the integral equations, whereas the integration of the singular components, which are either of order 1/r or 1/r², is handled by special numerical schemes. In the time marching aspect, the field quantities are assumed to vary linearly in the temporal direction as well. Examples for wave propagation due to various forms of surface excitations are reported to demonstrate the accuracy of the method.     

横观各向同性饱和地基中埋置荷载的非轴对称瞬态响应

基于孔隙介质的Biot理论,首先利用Laplace变换,给出圆柱坐标系下横观各向同性饱和弹性多孔介质在变换域上的波动方程;将波动方程解耦后,根据方位角的Fourier展开和径向Hankel变换,求解了Biot波动方程,得到以土骨架位移、孔隙水压力和土介质总应力分量的积分形式的一般解;借助一般解,建立了有限厚度饱和土层和饱和半空间的精确动力刚度矩阵,并由土层的层间界面连续条件建立三维非轴对称层状饱和地基的总刚度方程;在此基础上,系统研究了横观各向同性饱和半空间体在内部集中荷载激励下的动力响应,并给出了问题的瞬态解答.该研究为运用边界元法求解饱和地基动力响应奠定了理论基础.     

Transverse response of underground cavities and pipes to incident SV waves

The transverse response of underground cylindrical cavities to incident SV waves is investigated. Analytical solutions are derived for unlined cavities embedded within an elastic half‐space using Fourier–Bessel series and a convex approximation of the half‐space free surface. The computed displacements at the half‐space free surface and the tangential stresses on the cavity are compared with the results of previous investigations. The analytical solutions are extended to formulate approximate solutions for assessing hoop stresses within cavity liners impinged by low‐frequency waves having wavelengths much longer than the cavity diameter. The approximate solutions are compared to existing numerical solutions, and used to evaluate the dynamic response of a flexible buried pipe shaken by the 1994 Northridge earthquake. The proposed approximate model for cavity liners is useful for the seismic analysis of underground pipes and small‐diameter tunnels. Copyright © 2001 John Wiley & Sons, Ltd.     

In-plane dynamic Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic half-space

The dynamic stiffness method combined with the Fourier transform is utilized to derive the in-plane Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic(TI)half-space.The loaded layer is fixed to obtain solutions restricted in it and the corresponding reactions forces,which are then applied to the total system with the opposite sign.By adding solutions restricted in the loaded layer to solutions from the reaction forces,the global solutions in the wavenumber domain are obtained,and the dynamic Green’s functions in the space domain are recovered by the inverse Fourier transform.The presented formulations can be reduced to the isotropic case developed by Wolf(1985),and are further verified by comparisons with existing solutions in a uniform isotropic as well as a layered TI halfspace subjected to horizontally distributed loads which are special cases of the more general problem addressed.The deduced Green’s functions,in conjunction with boundary element methods,will lead to significant advances in the investigation of a variety of wave scattering,wave radiation and soil-structure interaction problems in a layered TI site.Selected numerical results are given to investigate the influence of material anisotropy,frequency of excitation,inclination angle and layered on the responses of displacement and stress,and some conclusions are drawn.     

Vertical vibration of an embedded rigid foundation in a poroelastic soil

This paper considers time-harmonic vertical vibration of an axisymmetric rigid foundation embedded in a homogeneous poroelastic soil. The soil domain is represented by a homogeneous poroelastic half space that is governed by Biot's theory of poroelastodynamics. The foundation is subjected to a time-harmonic vertical load and is perfectly bonded to the surrounding half space. The contact surface can be either fully permeable or impermeable. The dynamic interaction problem is solved by employing an indirect boundary integral equation method. The kernel functions of the integral equation are the influence functions corresponding to vertical and radial ring loads, and a ring fluid source applied in the interior of a homogeneous poroelastic half space. Analytical techniques are used to derive the solution for influence functions. The indirect boundary integral equation is solved by using numerical quadrature. Selected numerical results for vertical impedance of rigid foundations are presented to demonstrate the influence of poroelastic effect, foundation geometry, hydraulic boundary condition along the contact surface and frequency of excitation.     

Dynamic response of a multi-layered poroelastic medium

An exact stiffness matrix method is presented to evaluate the dynamic response of a multi-layered poroelastic medium due to time-harmonic loads and fluid sources applied in the interior of the layered medium. The system under consideration consists of N layers of different properties and thickness overlying a homogeneous half-plane or a rigid base. Fourier integral transform is used with respect to the x-co-ordinate and the formulation is presented in the frequency domain. Fourier transforms of average displacements of the solid matrix and pore pressure at layer interfaces are considered as the basic unknowns. Exact stiffness (impedance) matrices describing the relationship between generalized displacement and force vectors of a layer of finite thickness and a half-plane are derived explicitly in the Fourier-frequency space by using rigorous analytical solutions for Biot's elastodynamic theory for porous media. The global stiffness matrix and the force vector of a layered system is assembled by considering the continuity of tractions and fluid flow at layer interfaces. The numerical solution of the global equation system for discrete values of Fourier transform parameter together with the application of numerical quadrature to evaluate inverse Fourier transform integrals yield the solutions for poroelastic fields. Numerical results for displacements and stresses of a few layered systems and vertical impedance of a rigid strip bonded to layered poroelastic media are presented. The advantages of the present method when compared to existing approximate stiffness methods and other methods based on the determination of layer arbitrary coefficients are discussed.     

A general approach to advective–dispersive transport with multirate mass transfer

A numerical solution that is significantly more general than other semi-analytical solutions is presented for governing equations describing advective–dispersive transport with multirate mass transfer between mobile and immobile domains. The new solution approach is general in the sense that it does not impose any restrictive assumption on the spatial or temporal variability of advective and dispersive processes in the mobile domain. A single integro-differential equation (IDE) is developed for the concentration in the mobile domain by separating the concentration in the immobile domain from the set of two partial differential equations. The solution to the IDE requires the evaluation of a temporal integral of the concentration in the mobile domain, which is a function of the Laplace transform of the distribution of the mass transfer rate coefficient. The Laplace transform is not limited to flow fields with known constant velocities. The solutions for one- and two-dimensional examples obtained using the new approach agree with those obtained by existing semi-analytical and numerical approaches.     

Axisymmetric dynamic response of the multi-layered transversely isotropic medium

By virtue of the precise integration method (PIM) and the technique of mixed variable formulations, solutions for the dynamic response of the multi-layered transversely isotropic medium subjected to the axisymmetric time-harmonic forces are presented. The planes of cross anisotropy are assumed to be parallel to the horizontal surface of the stratified media. Four kinds of vertically acting axisymmetric loads are prescribed either at the external surface or in the interior of the soil system. Thicknesses and number of the medium strata are not limited. Employing the Hankel integral transform in cylindrical coordinate, the axisymmetric governing equations in terms of displacements of the multi-layered media are uncoupled. Applying mixed variable formulations, more concise first-order ordinary differential matrix equations from the uncoupled motion equations can be obtained. Solutions of the ordinary differential matrix equations in the transformed domain are acquired by utilizing the approach of PIM. Since PIM is highly accurate to solve the sets of first-order ordinary differential equations, any desired accuracy of the solutions can be achieved. All calculations are based on the corresponding algebraic operations and computational efforts can be reduced to a great extent. Comparisons with the existing numerical solutions are made to confirm the accuracy of the present solutions proposed by this procedure. Several examples are illustrated to explore the influences of the type and degree of material anisotropy, the frequency of excitation and loading positions on the dynamic response of the stratified medium.     

Surface vibration due to a sequence of high speed moving harmonic rectangular loads

The transmission of vibrations over the surface of the ground, due to high-speed moving, vertical harmonic rectangular loads, is investigated theoretically. The problem is three-dimensional and the interior of the ground is modelled as an elastic half-space or a multilayered ground. The transformed solutions are obtained using the Fourier transform on the space variable. A new damping model in the spatial wavenumber domain, presented in Lefeuve-Mesgouez et al. [J. Sound. Vibr. 231 (2000) 1289] is used. Numerical results for the displacements on the surface are presented for loads moving with speeds up to and beyond the Rayleigh wave speed of the half-space.     

Transient foundation solution of saturated soil to impulsive concentrated loading

The transient dynamic response of saturated soil under suddenly applied normal and horizontal concentrated loading is studied in this paper. The behavior of saturated soil is governed by Biot's consolidation theory. The general solutions for Biot equations of equilibrium are derived in terms of displacements and variations of fluid volume, using Laplace–Hankel integral transforms. The solutions in the time domain can be evaluated by numerical inverse Laplace–Hankel transforms. Selected numerical results for displacements, stresses, and pore pressures are presented. Comparisons with existing closed-form solutions for the elastic half-space are made to confirm the accuracy of the present solutions. The solutions can be used to study a variety of transient wave propagation problems and dynamical interactions between saturated soil and structures.     

Scattering of elastic waves by a general anisotropic basin. Part 1: a 2D model

Scattering of incident plane harmonic pseudo P‐, SH‐, and SV‐waves by a two‐dimensional basin of arbitrary shape is investigated by using an indirect boundary integral equation approach. The basin and surrounding half‐space are assumed to be generally anisotropic, homogeneous, linearly elastic solids. No material symmetries are assumed. The unknown scattered waves are expressed as linear combinations of full‐space time‐harmonic two‐dimensional Green functions. Using the Radon transform, the Green functions are obtained in the form of finite integrals over a unit circle. An algorithm for the accurate and efficient numerical evaluation of the Green functions is discussed. A detailed convergence and parametric analysis of the problem is presented. Excellent agreement is obtained with isotropic results available in the literature. Steady‐state surface ground motion is presented for semi‐circular basins with generally anisotropic material properties. The results show that surface motion strongly depends upon the material properties of the basin as well as the angle of incidence and frequency of the incident wave. Significant mode conversion can be observed for general triclinic materials which are not present in isotropic models. Comparison with an isotropic basin response demonstrates that anisotropy is very important for assessing the nature of surface motion atop basins. Copyright © 2001 John Wiley & Sons, Ltd.     

Rocking vibrations of rigid disk on saturated poroelastic medium

The dynamic response of a rigid disk on a saturated poroelastic half space and subjected to harmonic rocking excitation is studied. The mixed boundary-value problem for the case of relaxed contact condition between the disk and the poroelastic half space is reduced to a Fredholm integral equation of the second kind, which is solved numerically. The dynamic compliance coefficient for the rocking vibration of a rigid disk on a poroelastic half space is presented.