A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

轴对称一维六方准晶圆柱的精化理论

将轴对称圆柱的精化分析推广到一维六方准晶中轴对称圆柱的研究当中。利用准调和函数的Bessel算子函数表示以及一维六方准晶中的通解,在不做任何预先假设的情况下,给出了一维六方准晶中轴对称圆柱的精化理论。首先,根据准调和函数的Bessel算子函数表示,利用三个一维待定函数,表示出声子场和相位子场的位移场和应力分量。再根据非齐次边界条件,推导出柱面受径向外载时的精化方程。通过舍弃高阶项,推导出了在径向方向受到柱面载荷的近似解。     

具有混合边界透水条件的多孔饱和半空间上刚性圆板的垂直振动

用积分变换和积分方程研究多孔饱和半空间上刚性圆板的垂直振动问题。首先应用逐次解耦方法求解多孔饱和固体的动力基本方程－Ｂｉｏｔ波动方程。然后考虑混合边界透水条件（半空间表面与圆板的接触面是不透水的,而其余表面是透水的）,建立子多孔饱和半空间上刚性圆板垂直振动的对偶积分方程,并化对偶积分方程为第二类Ｆｒｄｄｈｏｌｍ积分方程。     

3-D DYNAMIC RESPONSE OF TRANSVERSELY ISOTROPIC SATURATED SOILS

Introduction Thedynamicanalysisofsaturatedporoelasticmediaisofbroadmeaninginmanyaspectssuchasseismology,earthquakeengineering,soilmechanicsandgeophysics,etc.Thetransverselyisotropicsaturatedsoilsissolid fluidcoupledtwophasemediainwhichthe soilskeletondisp…     

Compressible and incompressible constituents in anisotropic poroelasticity: The problem of unconfined compression of a disk

The governing equations for the theory of anisotropic poroelastic materials with incompressible constituents undergoing small deformations are developed from the theory of anisotropic poroelastic materials without the constituent incompressibility constraint. The development of the constituent specific incompressibility constraint is accomplished by restricting the elastic constants rather than by introducing Lagrange multipliers, the conventional method. The advantage of the approach is insight into the nature of the elastic response that characterizes incompressible poroelasticity. An application of the theory to the unconfined compression of a circular porous disk is presented to illustrate the effects of compressibility vs. incompressibility and transverse isotropy vs. isotropy.     

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.     

Approximate theories for wave propagation and vibrations in elastic rings and helical coils of small pitch

From the equations of linear elasticity, three levels of approximate theories are derived for in-plane deformation and motion of thin, circular rings. The accuracy of each theory is determined by comparison with harmonic wave solutions of the elasticity solution. Boundary conditions for uniqueness are established. The results may also be applied to helical coils of small pitch and to cylindrical shells when the equations are converted to plane strain.     

Complex variable function method for the scattering of plane waves by an arbitrary hole in a porous medium

Based on the complex variable function method, a new approach for solving the scattering of plane elastic waves by a hole with an arbitrary configuration embedded in an infinite poroelastic medium is developed in the paper. The poroelastic medium is described by Biot's theory. By introducing three potentials, the governing equations for Biot's theory are reduced to three Helmholtz equations for the three potentials. The series solutions of the Helmholtz equations are obtained by the wave function expansion method. Through the conformal mapping method, the arbitrary hole in the physical plane is mapped into a unit circle in the image plane. Integration of the boundary conditions along the unit circle in the image plane yields the algebraic equations for the coefficients of the series solutions. Numerical solution of the resulting algebraic equations yields the displacements, the stresses and the pore pressure for the porous medium. In order to demonstrate the proposed approach, some numerical results are given in the paper.     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

直角坐标下横观各向同性饱和半空间体动力响应问题的解析解

基于双重Fourier变换技术，成功求解了直角坐标系下，横观各向同性弹性饱和多孔介质的三维Biot动力方程，得到了以固体骨架位移分量和孔隙流体压力为基本未知量的积分形式一般解．进而，用一般解给出了饱和介质的总应力分量表达式。在此基础上，研究了在任意分布的表面竖向和水平谐振力作用下，横观各向同性饱和半空间体的动力响应问题。数值结果表明，采用各向同性饱和介质的动力学模型，不能准确描述具有明显各向异性特性的饱和土地基的动力性能。     

A simple BEM formulation for poroelasticity via particular integrals

A simple particular integral formulation is presented for poroelastic analysis. The elastostatics and steady-state potential flow equations are used as the complementary solution. A set of global shape functions is considered to approximate the pore pressure loading term in the poroelastic equation, the transient terms of pore pressure and displacements in the pore fluid flow equation to obtain the particular integrals for displacement, traction, pore pressure and flux.Numerical results for four plane problems of soil consolidation are given and compared with their analytical solutions to demonstrate the accuracy of the present formulation. Generally, agreement among all of those results is satisfactory if a few interior points are added to the usual boundary elements.     

A dynamic analysis of elastic circular plate on saturated poroelastic half space

In this paper the low frequency vibrations of an elastic circular plate on a saturated poroelastic half space are studied by the analytical method. First the governing equations of the dynamic problem for a saturated poroelastic medium are solved by means of Hankel transform. Then the dual integral equations of vertical forced vibration of an elastic plate on saturated poroelastic half space are established according to the mixed boundary-valued condition. By applying Abel transform the dual integral equations are reduced to a Fredholm integral equation of the second kind. Numerical examples are given at the end of the paper.     

横观各向同性饱和地基上中厚圆板的非轴对称振动

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。基于横观各向同性饱和介质Biot波动方程的一般解,按混合边值问题建立了饱和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,并将对偶积分方程转化为易于计算的第二类Fredholm积分方程;采用数值方法求解该积分方程。数值算例结果表明,当h/a>0.05时,饱和半空间体上中厚度圆板在不同频率下的振动特性与相应频率下的刚性板的振动特性基本相同,当h/a<0.05时,板中心的位移将随h/a的减小而增大。     

Dynamic interaction between elastic thick circular plate and transversely isotropic saturated soil ground

IntroductionThe vibration of the plate on the porous saturated building foundation is a complicateddynamic contact problem.Its consideration is very important in both earthquake and geo-technical engineering.Lots of research works have been done in recent…     

功能梯度中厚圆/环板轴对称弯曲问题的解析解

基于一阶剪切变形板理论，导出了热/机载荷作用下，位移形式的功能梯度 中厚圆/环板轴对称弯曲问题的控制方程，获得了问题的位移和内力的一般解析解. 作为特 例，分别研究了边缘径向固定和可动的夹紧和简支的4种实心功能梯度圆板，给出了它们的 解，并分析了热/机载荷作用下解的形态，讨论了横向剪切变形、材料梯度常数和边界条件， 对板的轴对称弯曲行为的影响.     

Frequency domain fundamental solutions for a poroelastic half-space

In frequency domain, the fundamental solutions for a poroelastic half-space are re-derived in the context of Biot＇s theory. Based on Biot＇s theory, the governing field equations for the dynamic poroelasicity are established in terms of solid displacement and pore pressure. A method of potentials in cylindrical coordinate system is proposed to decouple the homogeneous Biot＇s wave equations into four scalar Helmholtz equations, and the general solutions to these scalar wave equations are obtained. After that, spectral Green＇s functions for a poroelastic full-space are found through a decomposition of solid displacement, pore pressure, and body force fields. Mirror-image technique is then applied to construct the half-space fundamental solutions.Finally, transient responses of the half-space to buried point forces are examined.     

Singular boundary method for 2D dynamic poroelastic problems

This paper extends a strong-form meshless boundary collocation method, named the singular boundary method (SBM), for the solution of dynamic poroelastic problems in the frequency domain, which is governed by Biot equations in the form of mixed displacement–pressure formulation. The solutions to problems are represented by using the fundamental solutions of the governing equations in the SBM formulations. To isolate the singularities of the fundamental solutions, the SBM uses the concept of the origin intensity factors to allow the source points to be placed on the physical boundary coinciding with collocation points, which avoids the auxiliary boundary issue of the method of fundamental solutions (MFS). Combining with the origin intensity factors of Laplace and plane strain elastostatic problems, this study derives the SBM formulations for poroelastic problems. Five examples for 2D poroelastic problems are examined to demonstrate the efficiency and accuracy of the present method. In particular, we test the SBM to the multiply connected domain problem, the multilayer problem and the poroelastic problem with corner stress singularities, which are all under varied ranges of frequencies.     

移动简谐力作用下三维多孔饱和半空间的动力问题

研究了移动荷载作用下多孔饱和地基的动力问题.应用Fourier变换求解该问题的控制偏微分方程,考虑了荷载的移动速度及振动频率对多孔饱和地基动力响应的影响,重点研究了移动速度达地基表面波速时多孔饱和半空间的振动问题(马赫效应),并与相应的弹性介质的解答进行了比较.结果显示当移动速度与多孔饱和半空间的表面波速相近时,地基会产生很大的振动;当移动速度大于表面波速时,多孔饱和半空间的动力响应与弹性半空间的动力响应有较大的差别.     

多孔饱和半空间上弹性圆板垂直振动的积分方程

应用新的方法求解多孔饱和固体的动力基本方程－Ｂｉｏｔ波动方程,首先把Ｂｉｏｔ波动方程化为仅有土骨架位移和孔隙水压力的偏微分方程组,并且逐次解耦方法（不引入位移势函数）求解此偏微分方程组,然后按混合边值条件建立多孔饱和半空间上弹性圆板垂直振动的对偶积分方程,用Ａｂｅｌ变换化对偶积分方程为第二类Ｆｒｅｄｈｏｌｍ积分方程。文中考虑两种孔隙流体的表面边界条件：（ａ）半空间表面（包括圆板与半空间的接触面）是     

受移动简谐力作用的多孔弹性半平面问题

研究了匀速移动的振动荷载作用下半无限多孔饱和固体中产生的应力和孔隙水压力．应用Fourier变换求解该问题的控制偏微分方程,考虑了荷载的移动速度及振动频率对多孔饱和固体中应力与孔隙水压力的影响,并与相应的弹性介质的解答进行了比较．结果显示多孔饱和半平面中应力和孔隙水压力随荷载的移动速度与振动频率的增加而增大,多孔饱和固体在移动荷载下的动力响应与相应的单相弹性固体的动力响应有较大的差别。