横观各向同性饱和地基的三维动力响应

首先引入位移函数,将直角坐标系下横观各向同性饱和土Biot波动方程转化为2个解耦的六阶和二阶控制方程;然后基于双重Fourier变换,求解了Biot波动方程,得到以土骨架位移和孔隙水压力为基本未知量的积分形式的一般解,并用一般解给出了饱和土总应力分量的表达式．在此基础上系统研究了横观各向同性饱和半空间体的稳态动力响应问题,考虑表面排水和不排水两种情况,得到了半空间体在任意分布的表面谐振荷载作用下,表面位移的稳态动力响应,文末给出了算例．     

Dynamic response of a stratified transversely isotropic half-space with a poroelastic interlayer due to a buried moving source

Response of a ground subjected to dynamic sources is a classical problem in many fields. This paper develops a transversely isotropic (TI) half-space ground model which considers the alternating distribution of different media: a TI elastic surface layer, a TI poroelastic interlayer and a TI elastic half-space from top to bottom. The model can consider soils with different properties and the existence of groundwater. Dynamic responses of the model subjected to a buried moving source are investigated. General solutions to the TI poroelastic interlayer in the wavenumber domain are derived by Fourier transform and a potential function method in a Cartesian coordinate system. The layered system with different media is solved adopting an ‘adapted stiffness matrix method’ for a buried source scenario. Dynamic responses in the time-spatial domain are obtained by inverse fast Fourier transform (iFFT) and numerical integration. The performance of the method in this study is first tested by comparison with existing research, and the influences of the existence of pore water in the interlayer, transverse isotropy, and source features are comprehensively investigated. The effect of pore water is more significant for a TI ground compared with isotropic ground, and the model in this study cannot be simplified by an ‘equivalent’ layered TI elastic model. Peak values of displacement and pore pressure exist with increasing source speed. Source speeds corresponding to them are related to the wave speeds of the medium and affected by transverse isotropy.     

层状横观各向同性饱和地基上圆板的非轴对称振动

研究了层状横观各向同性饱和地基上弹性圆板的非轴对称振动问题．首先,通过方位角的Fourier变换,将圆柱坐标系下横观各向同性饱和土的三维动力方程转化为一阶常微分方程组,基于径向Hankel变换,建立问题的状态方程,求解状态方程后得到传递矩阵;其次,利用传递矩阵,结合层状饱和地基的边界条件、排水条件及层间接触和连续条件,给出了任意简谐激振力作用下层状横观各向同性饱和地基动力响应的通解;然后,按混合边值问题建立层状饱和地基上弹性圆板非轴对称振动的对偶积分方程,并将对偶积分方程化为易于数值计算的第二类Fredholm积分方程,并给出了算例．     

Assessing the mechanical responses for anisotropic multi-layered medium under harmonic moving load by Spectral Element Method (SEM)

The article is concerned with the mechanical responses of anisotropic multi-layered medium under harmonic moving load. An analytical solution for two-dimensional anisotropic multi-layered medium subjected to harmonic moving load is devoted via Spectral Element Method (SEM), while the anisotropic property is approximated as transverse isotropy. Starting with the constitutive equations of transversely isotropic body and the governing equations of motion based on the loading properties. The analytical spectral elements in the wavenumber domain are obtained according to the principle of wave superposition and Fourier transformation. Then, the spectral global stiffness matrix of the multi-layered medium is derived by assembling the nodded stiffness matrices of all layers depended on the different interlayer conditions between the adjacent layers, i.e. sliding and bonded. The corresponding analytical solutions are achieved by taking the Fourier series and Inverse Fast Fourier Transform (IFFT) algorithm. Finally, some examples are given to validate the accuracy of the proposed analytical solution, and to demonstrate the impact of both anisotropy, top layer thickness, interlayer conditions, and loading properties (velocity and natural frequency) on the mechanical response of the multi-layered medium.     

饱和地基上弹性基础的竖向振动特性研究

采用解析的方法研究了饱和地基上受一简谐竖向荷载作用下弹性基础的动力响应．在分析中,首先利用积分变换技术获得了饱和介质基本控制方程的变换解,然后基于基础-半空间完全放松接触、半空间表面完全透水或不透水的假设,建立了该动力混合边值问题的对偶积分方程,并把该对偶积分方程进一步化为易于数值求解的第二类Fredholm积分方程A·D2文末数值算例给出了动力柔度系数、位移和孔隙水压力随振动频域和土-基础体系物理力学参数特性的变化曲线．结果表明:饱和地基上弹性基础的动力响应完全不同于饱和地基上刚性圆板的动力响应．所用方法可用于研究波的传播、土-结构动力相互作用等许多问题．     

横观各向同性饱和弹性多孔介质非轴对称动力响应

应用Fourier展开和Hankel变换求解了简谐激励下横观各向同性饱和弹性多孔介质的非轴对称Biot波动方程,得到了一般解。用一般解给出了多孔介质总应力分量的表达式。最后对求解横观各向同性饱和弹性多孔介质非轴对称动力响应边值问题的方法作了系统说明,并且给出了数值分析特例。     

A modified scaled boundary finite element method for dynamic response of a discontinuous layered half-space

In this paper, a modified scaled boundary finite element method is proposed to deal with the dynamic analysis of a discontinuous layered half-space. In order to describe the geometry of discontinuous layered half-space exactly, splicing lines, rather than a point, are chosen as the scaling center. Based on the modified scaled boundary transformation of the geometry, the Galerkin's weighted residual technique is applied to obtain the corresponding scaled boundary finite element equations in displacement. Then a modified version of dimensionless frequency is defined, and the governing first-order partial differential equations in dynamic stiffness with respect to the excitation frequency are obtained. The global stiffness is obtained by adding the dynamic stiffness of the interior domain calculated by a standard finite element method, and the dynamic stiffness of far field is calculated by the proposed method. The comparison of two existing solutions for a horizontal layered half-space confirms the accuracy and efficiency of the proposed approach. Finally, the dynamic response of a discontinuous layered half-space due to vertical uniform strip loadings is investigated.     

Asymmetric Green's functions for a functionally graded transversely isotropic tri-material

This paper considers the elastic analysis of a functionally graded transversely isotropic tri-material solid under the arbitrary distribution of applied static loads. Using two displacement potential functions, for three-dimensional point-load and patch-load configurations, Green's functions for displacement and stress components are generated in the form of infinite line-integrals. These solutions are shown to be analytically reducible to the special cases of exponentially graded bi-material, exponentially graded half-space and homogeneous tri-material Green's functions. It also encompasses a functionally graded finite layer on a rigid base with surface loading with two cases of interfacial conditions, rigid-bonded and rigid-frictionless. Finally, for the special case of a functionally graded layer sandwiched between two homogeneous layers, using several numerical displays, the effect of material inhomogeneity on the responses is studied and the accuracy of numerical scheme is verified.     

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.     

Transient response of a transversely isotropic multilayered half-space due to a vertical loading

This paper investigates the transient response of a transversely isotropic multilayered half-space under vertical loadings. With the aid of a Laplace–Hankel transform, the global stiffness matrix for a multilayered half-space is acquired by assembling the analytical layer-element of each layer medium. The solutions for the displacements in the time domain are obtained by using the global stiffness matrix equations and a numerical inversion procedure. The accuracy of the proposed method is verified through comparisons with existing solutions for displacements induced by a step and rectangular pulse loading. In addition, selected numerical results for displacements induced by the buried loading are presented to illustrate the effect of transient loading type and material anisotropy on the transient response.     

Poroelastodynamic potential method for transversely isotropic fluid-saturated poroelastic media

By introduction of two scalar potentials, an analytical method is developed for the solution of poroelastodynamic boundary value problems in transversely isotropic fluid-saturated poroelastic media. The governing equations of motion are considered in the framework of Biot's complete model without any assumption or simplification. As a case of application, solutions in three dimensions for a transversely isotropic fluid saturated porous half space loaded by an arbitrary distribution of time harmonic tractions at the free surface is derived. The free surface of the half space may be considered either permeable or impermeable. As a particular solution, Green's functions for uniform vertical and horizontal circular patch loads are presented as semi-infinite integrals which may be evaluated by means of an appropriate numerical method proposed. The accuracy of the solutions is verified both analytically and numerically against the preceding solutions. Some numerical results are also presented to clarify the influence of different degrees of anisotropy and frequency of excitation on the response of the medium.     

Time-harmonic response of transversely isotropic and layered poroelastic half-spaces under general buried loads

A semi-analytical method is developed for solving the dynamic response of transversely isotropic, multilayered, and poroelastic half-spaces with different surface hydraulic conditions and subjected to time-harmonic vertical and horizontal loads buried in the layered half-space. The coupled governing equations of motion are presented in details in terms of the Biot's poroelastodynamic theory via the (u,p) formulation. The cylindrical system of vector functions is introduced to express the unknown primary quantities so that the coupled governing partial differential equations can be reduced and separated into two sets of first-order ordinary differential equations (i.e., the LM- and N-types). A recursive relation for the expansion coefficients among different layers is established by virtue of the stable and efficient dual variable and position method. Making use of the boundary and interface conditions, the fundamental solutions are obtained in terms of the vector-function system. The corresponding physical-domain solutions are then derived via an accurate semi-infinite integral algorithm. The developed fundamental solutions are carefully checked with existing solutions, and numerical examples are further presented to demonstrate the effect of material anisotropy, loading depth, material layering, and surface hydraulic condition on the dynamic response, which should be useful to design engineers. These solutions could be further served as benchmarks for future numerical methods.     

横观各向同性饱和地基与中厚圆板的非轴对称动力相互作用

研究横观各向同性饱和土地基上中厚弹性圆板的非轴对称振动问题,即首先利用Fourier展开和Hankel变换技术,求解了简谐激励下横观各向同性饱和土地基的非轴对称Biot波动方程,然后按混合边值问题建立地基与弹性中厚圆板非轴对称动力相互作用的对偶积分方程,并将对偶积分方程化为易于数值计算的第二类Fredholm积分方程．文末给出了算例．数值结果表明,在一定频率范围内,地基表面的位移幅值随激振频率增加而增大,随距离的增大以振荡形式衰减变化．     

横观各向同性饱和地基上刚性圆板的扭转振动

通过解析方法研究了横观各向同性饱和半空间上刚性圆板在简谐扭转荷载作用下的振动问题．运用Hankel变换求解了横观各向同性饱和土的动力控制方程,结合混合边界条件得出了刚性基础的扭转对偶积分方程,并将对偶积分方程转化为第二类Fredholm积分方程求解了基础的扭转振动问题,同时给出了动力柔度系数,基础的角位移幅值和基底接触剪应力的表达式．通过数值算例研究了地基的各向异性程度对基础扭转振动的影响．     

Combined Fourier-wavelet transforms for studying dynamic response of anisotropic multi-layered flexible pavement with linear-gradual interlayers

The objective of this study is to develop an analytical solution for studying dynamic response of anisotropic multi-layered flexible pavement with linear-gradual interlayers under a harmonic moving load. In this study, the flexible pavement structure is simplified as one multi-layered flexible system, which is assumed to be a semi-infinite medium. A new approach combining the Fourier transform with the wavelet transform for solving the dynamic analytical solution. The wavelet transforms for solving inverse Fourier transform, in solving the solution in the physical domain, is superior to the conventional inverse Fourier transform. The linear-gradual interlayer between the adjacent layers is defined using the shear spring model, and the anisotropic property is simplified as transverse isotropy. Also, in the dynamic analytical solving processes, the motion in the transversely isotropic medium is decoupled into in-plane motion and out-of-plane motion because of the propagation of the waves in a transversely isotropic medium with coupling phenomena. The corresponding analytical solution is entered into a MATLAB-based computer program, which can compute the dynamic responses of an anisotropic multi-layered medium at different interlayer conditions. The accuracy of this program is confirmed through comparison with the results from the examples from the references. The influence analyses of linear-gradual interlayers and anisotropic properties of structural layers are illustrated. It is concluded that the proposed analytical solution-based computer program could be used in the multi-layered flexible pavement structural design and risk management in civil engineering.     

The interaction of punches on a transversely isotropic half-space

Three-dimensional contact problems on the interaction of two similar punches on an elastic transversely isotropic half-space (five elastic constants) are investigated, when the isotropy planes are perpendicular to the boundary of the half-space. In this connection the stiffness of the half-space boundary depends on the direction. The kernel of the integral equation of the contact problems is represented in a quadrature-free form using the theory of generalized functions. This form of the kernel enables it to be regularized at singular points and enables Galanov's method to be used to solve the contact problem with an unknown contact area.     

Behavior of laterally and vertically loaded piles in multi-layered transversely isotropic soils

In this paper, a coupled approach of the finite element method (FEM) and the analytical layer-element method (ALEM) is proposed to conduct a research on vertically and laterally loaded piles. The FEM is used to model the pile, and the ALEM is utilized to solve the multi-layered transversely isotropic soils. Then with the assumption of force equilibrium and deformation compatibility, the interaction equation of pile and soils is obtained. Finally, the behavior of piles simultaneously subjected to lateral and vertical loads in layered transversely isotropic soils is investigated by considering the influence of lateral–vertical loads interaction and soft soil stratum.     

横观各向同性弹性层点力解

本文根据弹性层状结构的传递矩阵法思想，由横观各向同性弹性力学基本方程，导出了含应力和位移两类变量的混合方程，利用Ｆｏｕｒｉｅｒ变换和文献［７］的位移函数通解，以及计算机代数软件，得到了横观各向同性层的点力解，这个点力解可直接退化到各同性情形的解．     

Green's function approach to study the propagation of SH‐wave in piezoelectric layer influenced by a point source

Green's function technique serves as a powerful tool to find the particle displacements due to SH‐wave propagation in layer of a shape different from the space between two parallel planes. Therefore, the present paper undertook to study the propagation of SH‐wave in a transversely isotropic piezoelectric layer under the influence of a point source and overlying a heterogeneous substrate using Green's function technique. The coupled electromechanical field equations are solved with the aid of Green's function technique. Expression for displacements in both layer and substrate, scalar potential and finally the dispersion relation is obtained analytically for the case when wave propagates along the direction of layering. Numerical computations are carried out and demonstrated with the aid of graphs for six different piezoelectric materials namely PZT‐5H ceramics, Barium titanate (BaTiO₃) ceramics, Silicon dioxide (SiO₂) glass, Borosilicate glass, Cobalt Iron Oxide (CoFe₂O₄), and Aluminum Nitride (AlN). The effects of heterogeneity, piezoelectric and dielectric constants on the dispersion curve are highlighted. Moreover, comparative study is carried out taking the phase velocity for different piezoelectric materials on one hand and isotropic case on the other. Dispersion relation is reduced to well‐known classical Love wave equation with a view to illuminate the authenticity of problem. Copyright © 2017 John Wiley & Sons, Ltd.     

Numerical generation of transparent boundary conditions on the side surface of a vertical transverse isotropic layer

We consider elastodynamics in transversely isotropic media with vertical symmetry axis. The governing equations are the two-dimensional second-order system for displacements. A numerical method for generating transparent boundary conditions on the cylindrical surface is proposed. The correspondent operator is non-local in both z-direction and time: it handles low-frequency spatial harmonics of the solution convolving their Fourier coefficients with sums-of-exponentials kernels with respect to time. Test calculations show high accuracy, efficiency, and stability of the proposed non-reflecting conditions even for those media parameters where PML fails.