求解饱和半空间上弹性圆板固结沉降的积分方程

本文采用解析方法分析了弹性圆板在饮和半空间上的固结沉降。考虑弹性圆板与饮和半空间的接触面上无摩擦力,且饱和半空间表面为全部透水的。运用Ｂｉｏｔ固结理论和积分方程技术,在Ｌａｐｌａｃｅ变换域上建立了弹性圆板固结沉降的对偶积分方程,并化此对偶积分方程为第二类Ｆｒｅｄｈｏｌｍ积分方程。通过对其核函数的有效数值发得到第二类Ｆｒｅｄｈｏｌｍ积分方程的解,再利用Ｌａｐａｃｅ反演技术获得弹性板在时间域中的固结沉     

Semi-analytical solution to one-dimensional consolidation in unsaturated soils

In this paper, a series of semi-analytical solutions to one-dimensional consolidation in unsaturated soils are obtained. The air governing equation by Fredlund for unsaturated soils consolidation is simplified. By applying the Laplace transform and the Cayley-Hamilton theorem to the simplified governing equations of water and air, Darcy＇s law, and Fick＇s law, the transfer function between the state vectors at top and at any depth is then constructed. Finally, by the boundary conditions, the excess pore-water pressure, the excess pore-air pressure, and the soil settlement are obtained under several kinds of boundary conditions with the large-area uniform instantaneous loading. By the Crump method, the inverse Laplace transform is performed, and the semi-analytical solutions to the excess pore-water pressure, the excess pore-air pressure, and the soils settlement are obtained in the time domain. In the case of one surface which is permeable to air and water, comparisons between the semi-analytical solutions and the analytical solutions indicate that the semi-analytical solutions are correct. In the case of one surface which is permeable to air but impermeable to water, comparisons between the semi-analytical solutions and the results of the finite difference method are made, indicating that the semi-analytical solution is also correct.     

Fracture analysis of circular-arc interface cracks in piezoelectric materials

The anti-plane problem of N arc-shaped interfacial cracks between a circular piezoelectric inhomogeneity and an infinite piezoelectric matrix is investigated by means of the complex variable method. Cracks are assumed to be permeable and then explicit expressions are presented, respectively, for the electric field on the crack faces, the complex potentials in media and the intensity factors near the crack-tips. As examples, the corresponding solutions are obtained for a piezoelectric bimaterial system with one or two permeable arc-shaped interfacial cracks, respectively. Additionally, the solutions for the cases of impermeable cracks also are given by treating an impermeable crack as a particular case of a permeable crack. It is shown that for the case of permeable interfacial cracks, the electric field is jumpy ahead of the crack tips, and its intensity factor is always dependent on that of stress. Moreover all the field singularities are dependent not only on the applied mechanical load, but also on the applied electric load. However, for the case of a homogeneous material with permeable cracks, all the singular factors are related only to the applied stresses and material constants.     

Elasto-plastic consolidation of soil

A theory of the behaviour of an ideal mass of two-phase soil is developed. This theory unites the usually separately considered aspects of analytical soil mechanics of settlement and deformation; time dependent consolidation; and yielding leading to collapse of the mass. The theory assumes the soil to be an elasto-plastic permeable material and a finite element approximation of the resulting equations is derived through the principle of virtual work. Two example problems are treated numerically.     

On a moving interface crack with a contact zone in a piezoelectric bimaterial

An inplane problem for a crack moving with constant subsonic speed along the interface of two piezoelectric materials is considered. A mechanically frictionless and electrically permeable contact zone is assumed at the right crack tip whilst for the open part of the crack both electrically permeable and electrically insulated conditions are considered. In the first case a moving concentrated loading is prescribed at the crack faces and in the second case an additional electrical charge at the crack faces is prescribed as well. The main attention is devoted to electrically permeable crack faces. Introducing a moving coordinate system at the leading crack tip the corresponding inhomogeneous combined Dirichlet–Riemann problem is formulated and solved exactly for this case. All electromechanical characteristics at the interface are presented in a closed form for arbitrary contact zone lengths, and further, the transcendental equation for the determination of the real contact zone length is derived. As a particular case of the obtained solution a semi-infinite crack with a contact zone is considered. The numerical analysis performed for a certain piezoelectric bimaterial showed an essential increase of the contact zone length and the associated stress intensity factor especially for the near-critical speed region. Similar investigations have been performed for an electrically insulated crack and the same behavior of the above mentioned parameters is observed.     

Analytical solution to one-dimensional consolidation in unsaturated soils

This paper presents an analytical solution of the one-dimensional consolidation in unsaturated soil with a finite thickness under vertical loading and confinements in the lateral directions. The boundary contains the top surface permeable to water and air and the bottom impermeable to water and air. The analytical solution is for Fredlund＇s one-dimensional consolidation equation in unsaturated soils. The transfer relationship between the state vectors at top surface and any depth is obtained by using the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy＇s law and Fick＇s law. Excess pore-air pressure, excess pore-water pressure and settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial conditions and boundary conditions. By performing inverse Laplace transforms, the analytical solutions are obtained in the time domain. A typical example illustrates the consolidation characteristics of unsaturated soil from an- alytical results. Finally, comparisons between the analytical solutions and results of the finite difference method indicate that the analytical solution is correct.     

A CONSOLIDATION MODEL OF GROUP WELLS ON COMPOSITE FOUNDATION1)

针对复合地基固结问题,考虑井阻和土体径竖向渗流的影响,建立 以"桩-土-桩"为单元的复合地基群井计算模型,采用解析解法,推导出瞬时载荷作用下复合地基群井固结解析解,讨论了复合地基固结特点和孔压等的变化规律。研究表明,复合地基群井固结模型为复合地基固结理论的研究提供了一种新思路,在井径比大于5 时,基于中心桩、边桩流量相等假设,复合地基群井固结理论所求固结度和传统固结理论所求固结度相差很小,可忽略不计;桩体渗透系数、内外扰动区土体径向渗透系数、土体竖向渗透系数越大,复合地基固结速率越快;堆载越大,最终达到稳定时的沉降量和竖向应变越大;地基表层土体的孔压消散速率和竖向应变速率大于地基深部土体。     

层状饱和土Biot固结问题状态空间法

针对饱和多孔介质空间非轴对Biot固结问题，引入状态变量，构造了两组相比独立的状态变量方程，利用Fourier级数和Laplace-Hankel变换，将状态变量方程转换为两组一阶常微分方程组，提出了均质饱和多孔介质空间非轴对称Biot固结问题的传递矩阵，得到以状态变量和传递矩阵乘积的形式表示的均质饱和多孔介质空间非轴对称Biot固结问题的解，利用层间完全接触的条件，可得到N层饱和多孔介质空间非轴对称Biot固结问题的一般解析表达式，文中考虑几种不同的边界条件，分析了两个算例，数值结果表明该方法具有较高的计算精度和良好的计算稳定性。     

一维大应变固结理论的一类解析解

简述了一维固结理论的发展，研究了不同描述方法下固结系数的定义，运用变换群的方法求得了某些条件下一维大应变固结方程的完整解析解答，在此基础上与相同条件下的小应变固结理论作了比较，并通过对实例的计算，分析了不同假设条件下一维固结理论之间的差异．计算结果表明，在固结过程中，由大应变固结理论所得出的沉降量要大于由小应变理论所得出的沉降量，而两种固结理论所得出的最终沉降量相同．     

On the influence of the electric permeability on an interface crack in a piezoelectric bimaterial compound

An interface crack between two semi-infinite piezoelectric spaces under the action of remote mixed mode loading and electric flux is considered. The properties of the materials, loading and crack geometry admit to consider a two-dimensional problem in the plane perpendicular to the crack front. The crack is assumed to be free from mechanical loading and the limited permeable electric condition holds true. Assuming the electric flux is constant along the crack area, using the known presentations of all electromechanical fields via a piecewise holomorphic vector function, the problem is reduced to a vector Hilbert problem and solved in an analytical way. Clear analytical expressions for stresses and electric displacement as well as for stress and electric intensity factors are derived. As a particular case, a crack in a homogeneous piezoelectric material is considered and exact analytical formulae are presented for this case. The numerical analysis of the obtained formulae showed that for small values of the electric flux the model of a completely permeable crack can be used for any real crack permeability’s. The validity of such an approximation decreases with increase in the mechanical loading and especially of the electric flux.     

Plane problem of filtration consolidation for an elastic half-space with discontinuous initial conditions

The plane consolidation problem for an elastic half-space under an instantaneously applied vertical load is solved in the general case assuming that the Terzaghi hypothesis holds.     

Antiplane crack at the interface of two bonded dissimilar graded piezoelectric materials

The problem of an antiplane crack situated in the interface of two bonded dissimilar graded piezoelectric half-spaces is considered under the permeable crack assumption. The mechanical and electrical properties of the half-spaces are considered for a class of functional forms for which the equilibrium equation has analytic solutions. By using an integral transform technique, the problem is reduced to dual integral equations which are transformed into a Fredholm integral equation by introducing an auxiliary function. The stress intensity factors are obtained in explicit form in terms of auxiliary functions. By solving the Fredholm integral equation numerically, the numerical results for stress intensity factors are obtained which have been displayed graphically to show the influence of the graded piezoelectric materials.     

考虑在地基固结和流变的上部结构和地基基础的共同作用问题

本文提出一种能同时考虑地基的瞬时变形,团结变形和流变的地基基础与上部结构共同作用的一种新的计算方法,在这种方法中,土的计算考虑了Biot固结理论和流变,上部结构利用子结构法进行凝聚可得边界刚度矩阵和边界力,利用位移协讯条件则可以得到共同作用的控制方程,对上述方程进行Laplace变换,可以得到Laplace变换域内的控制方程,在Laplace变换域内对上述方程进行数值求解并进行相应的Laplace逆变换则可以得到时间境内任意时刻的解。     

Dynamic interaction of multiple arbitrarily oriented planar cracks in a piezoelectric space: A semi-analytic solution

The problem of determining the electro-elastic fields around arbitrarily oriented planar cracks in an infinite piezoelectric space is considered. The cracks which are acted upon by a transient load are either electrically impermeable or permeable. A semi-analytic method based on the theory of exponential Fourier transformation is proposed for solving the problem in the Laplace transform domain. The Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces are determined by solving a system of hypersingular integral equations. Once these displacement and electric potential jumps are obtained, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be computed with the help of a suitable algorithm for inverting Laplace transforms. The stress and electric displacement intensity factors are computed for some specific cases of the problem.     

The axisymmetric double contact problem for a frictionless elastic layer

The general axisymmetric double contact problem for an elastic layer pressed against a half space by an elastic stamp is considered. The problem is solved under the assumptions that the three materials have different elastic properties, the contact along the interfaces is frictionless and only compressive normal tractions can be transmitted across the interfaces, and, in the case of the elastic stamp, the local radius of curvature of the stamp is large compared to the stamp-layer contact radius. The problem is reduced to a system of singular integral equations in which the contact pressures are the unknown functions. The solution is obtained and extensive numerical results are given for three stamp geometries, namely, rigid and elastic spherical stamps, and a flat-ended rigid cylindrical stamp. The results show that in the case of a flat-ended rigid cylindrical stamp the radius b of the contact area between the layer and the subspace is independent of the magnitude P of the total transmitted load and in all other cases b will depend on P.     

垂直受荷群桩在半空间饱和土中的稳态反应

研究了均质半空间饱和土中群桩在垂直稳态荷载作用下的动力反应问题。半空间饱和土采用Biot提出的三维波动原理。将桩看作是一维的弹性杆单元。利用圆形简谐载荷作用下的Biot固结方程的基本解和桩土之间的变形协调条件得到饱和土中群桩的第二类Fredholm积分方程,群桩的动力阻抗采用动力相互作用因子的方法。分析桩距等参数对群桩阻抗、轴力和孔压等的影响。本文的方法可以用于计算层状饱和土中群桩的动力反应问题。     

热压电介质中的渗透型周期裂纹问题

利用Stroh公式,研究了含共线周期裂纹热的压电介质的广义二维问题。该工作有两个特征：一是裂纹被建模为具有渗透表面的缝隙,并假设为跨越上下表面时,电场的切向分量和电位移的法向分量是连续的;另一个特征是,机－电载荷和热载荷被假设作用在无限远处,而不是在裂纹表面。基于这两个假设,我们获得了有关场强因子,以及裂纹内电场的相当简洁的表达式。结果表明：①在裂纹内电场是线性变化的,②电位移的奇异性总是取决于应力的奇异性.③所有场的奇异性与所加的电载荷无关。     

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.     

ONE-DIMENSIONAL NONLINEAR CONSOLIDATION THEORY FOR SOFT GROUND WITH CONSIDERATION OF SECONDARY CONSOLIDATION1)

将软黏土变形分为有效应力变化引起的变形和次固结引起的变形,推导了软土新型应力应变关系。然后结合Davis固结理论,建立了考虑次固结的一维非线性固结控制方程,并对其进行解析求解。通过与数值方法对比,验证了解析解的可靠性。在此基础上,分析了次固结对软土地基固结沉降的影响。结果表明：考虑次固结的孔压消散速度与固结速度较不考虑次固结的慢;忽略次固结将低估软黏土地基的工后沉降。     

Nonlinear and Elasto-Plasticity Consolidation Models of Unsaturated Soil and Applications

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