单层不可压缩饱和多孔介质一维瞬态响应精确解

基于Biot理论,考虑惯性、黏滞和机械耦合作用,假定固体颗粒和流体均不可压缩,得到了表面任意竖向荷载作用下单层饱和多孔介质一维瞬态响应的精确解。导出了以固体骨架位移表示的无量纲控制方程,并将边界条件齐次化。求解对应无黏滞耦合作用的特征值问题,得到一组满足齐次边界条件、关于空间坐标的正交函数基。利用变异系数法和基函数的正交性,得到一系列相互解耦的、关于时间的二阶常微分方程及相应的初始条件,并采用状态空间法求解常微分方程,得到位移分量。对整体平衡方程关于空间坐标积分,根据边界条件可确定总应力,并进而求得孔隙压力。通过算例验证所得解法的正确性     

混合非齐次边界下非饱和土一维固结解析解

非饱和土的固结研究对道路工程、软土地基处理工程等具有十分重要的意义。基于Fredlund和Hasan提出的非饱和土一维固结理论,给出了土体内孔隙水压力和孔隙气压力变化的控制方程,并给出了单层非饱和土的初始条件与一类随时间变化的混合非齐次边界条件,构成了非饱和土一维固结的定解问题。通过非齐次边界条件齐次化和特征函数展开法,得到了土体内孔隙水压力和孔隙气压力消散的精确时域解析解。最后,通过对比验证了解析方法的正确性,并分析了边界条件指数变化对非饱和土体内孔隙水压力和孔隙气压力的消散以及土体沉降的影响。结果表明,边界处孔压或孔压梯度随时间的指数变化对非饱和土固结过程有明显影响。     

采用Laplace变换分析桩基应力波传播问题

将土体中的混凝土桩基简化为黏滞介质中的一维黏弹性杆,桩顶部受锤头冲击产生内部应力波。根据杆内微元应力平衡条件建立杆中一维黏弹性应力波传播的控制方程,结合桩顶锤头冲击条件和桩底弹-黏性约束条件给出桩基两端的耦合边界条件。对控制方程和定解条件作Laplace变换并求解变换后的常微分方程,得到变换域的应力像函数解析解。采用数值反变换技术将像函数转变为时间域的应力波形。应用此方法可以较方便地分析桩基中应力波的产生、传播、反射及相互作用过程。     

横观各向同性含液饱和多孔介质黏弹性瞬态反应

在考虑横观各向同性含液饱和多孔介质固体骨架和流体可压缩性以及固体骨架的黏弹性特征下,基于横观各向同性含液饱和多孔介质u-w形式的三维动力控制方程,以固相位移u、液相相对位移w为基本未知量,综合运用Laplace变换、双重Fourier变换等方法,在直角坐标系下通过引入中间变量,将六元2阶动力控制方程组化为两组各含4个未知变量的常微分方程组,给出了直角坐标系下横观各向同性含液饱和多孔介质三维黏弹性动力反应的积分形式一般解;作为理论推导的验证,通过引入初始条件和边界条件,对横观各向同性含液饱和多孔介质半空间黏弹性瞬态反应问题进行了求解。解答的退化验证表明,所推导的理论解是正确的。     

半透水边界下分数阶黏弹性饱和土一维固结特性分析

为描述饱和土体的流变特性,引入分数阶导数Kelvin-Voigt黏弹性模型,采用解析方法对半透水边界下的分数阶黏弹性饱和土一维固结特性进行了研究。分别对骤加恒载下饱和土一维固结微分方程和分数阶Kelvin-Voigt黏弹性本构方程进行Laplace变换,并联立求解得到了双边半透水边界条件下分数阶黏弹性饱和土在Laplace变换域内的解析表达式。通过Crump方法实现Laplace数值反演,得到时间域内的半解析解。将所得到的解分别退化为分数阶黏弹性饱和土一维固结半解析解和双边半透水黏弹性饱和土一维固结半解析解,结果与已有文献半解析解相同,验证了提出的双边半透水边界条件下分数阶黏弹性饱和土一维固结解的可靠性。通过算例考察了半透水边界条件和分数阶黏弹性饱和土参数对一维固结特性的影响。研究表明,双边半透水边界下分数阶黏弹性饱和土一维固结发展过程与半透水边界条件、分数阶次和黏滞系数有关,且土体的压缩模量对饱和土一维固结最终沉降量有显著影响。     

任意荷载下双面半透水边界分数阶导数黏弹性饱和土层一维固结

基于Terzaghi一维固结理论,分析了考虑半透水边界条件的分数阶导数黏弹性饱和土层在随时间变化的任意荷载作用下一维固结问题。首先,应用Laplace变换联立求解饱和土层一维固结微分方程和分数阶Kelvin-Voigt黏弹性本构方程,推导出有效应力和沉降在Laplace变换域内的解析解,采用Crump方法进行Laplace逆变换,得到了时间域内的半解析解。然后将本文得到的半解析解分别退化为半透水边界条件下基于黏弹性假设的一维固结半解析解和双面透水边界条件下基于分数阶黏弹性假设的一维固结半解析解,结果与已有文献的半解析解相同,验证了本研究所提出解的可靠性。最后通过算例分别考察了半透水边界参数、分数阶黏弹性模型参数和荷载参数对饱和土层固结沉降的影响。研究表明,半透水边界条件参数、分数阶次与黏滞系数主要影响饱和土层固结的发展快慢,而饱和土层的最终沉降量主要受到土层压缩模量的影响;另外,饱和土层的固结规律与外荷载变化规律一致。     

非饱和土渗流与变形耦合问题的有限元分析

基于多孔介质力学原理,建立能模拟非饱和土两相流动与变形耦合问题的理论模型。利用Galerkin法对控制方程进行离散,得到控制方程的有限元计算格式。在此基础上,自主开发了有限元计算程序U-DYSAC2,并对Liakopoulos两相流动试验这一经典算例以及重非亲水相流体（DNAPL）在饱和多孔介质中迁移的离心模型试验进行了数值模拟。计算结果表明,理论预测与试验结果基本吻合,验证了所提出的分析方法在模拟非饱和土渗流以及变形问题时的有效性,从而为定量研究饱和-非饱和渗流以及变形问题提供了一条有效途径。     

非饱和土应力场与渗流场耦合一维解析求解与参数分析

由于基质吸力的存在,非饱和土应力场与渗流场耦合问题的解析求解变得非常困难。为有效分析该耦合场的耦合基本规律,本文采用Reynolds输运定律、Darcy定律,并结合多孔介质流体运动规律,推导了非饱和土应力场与渗流场耦合场的控制方程。在该耦合场控制方程中,非饱和土的渗透系数是基质吸力的函数,使得非饱和土耦合场控制方程具有强非线性的特征,而同伦分析法在求解强非线性微分方程时在选择线性算子以及辅助参数上具有明显的优势。因此,通过选取适当的初始猜测解与辅助参数,应用同伦分析法将非线性方程转换为线性的微分方程组,最后求解微分方程组得到耦合问题的解析解。在非饱和土应力场与渗流场耦合问题3阶变形近似解析解中,重点分析了辅助参数h、去饱和系数a、拟重对基质吸力分布的影响。研究结果表明:辅助参数h影响着解析解的收敛性,且随着h的增大,基质吸力总体上呈现减少趋势;基质吸力随着去饱和系数a的增大而减少,且基质吸力随着拟重的增大而呈现减少趋势。     

基于同伦分析法的非饱和土层固结问题的级数解

由于非饱和土的渗透系数是基质吸力的函数,使得控制方程带有强非线性的特征,进而使得控制方程的解析求解变得十分困难。同伦分析法对级数基函数和辅助线性算子的选择具有更大的自由性、灵活性,且收敛性的控制和调节更加容易实现,求解强非线性微分方程时在选择线性算子以及辅助参数上具有明显的优势。因此,针对非饱和土固结方程的非线性特征,对于处于地表浅层的非饱和土层,假设孔隙气压力为大气压力,在Richard经验公式与非饱和土一维固结理论的基础上,推导了非饱和一维固结无量纲控制方程;应用同伦分析法,通过选取适当的初始猜测解与辅助参数,将该非线性方程转换为线性的微分方程组并求解得到固结问题的级数解。此外,以压实高岭土为研究对象,在收集相关试验参数基础之上,将由同伦分析法求得的固结问题的近似解析解与有限差分法数值结果相对比,分析结果验证了解析解的正确性。     

非饱和土中圆柱形衬砌隧道的瞬态动力响应

在以往关于圆柱形衬砌隧道的瞬态动力响应中,衬砌周围土体大多假定为弹性介质或饱和介质。然而,自然界中的土体大多为非饱和介质。考虑土体与衬砌结构的动力相互作用及动荷载引起的附加质量密度的影响,研究了瞬态荷载作用下非饱和土中无限长深埋圆柱形衬砌隧道的动力响应。基于多孔介质混合物理论和连续介质力学理论,建立了非饱和土中圆柱形衬砌隧道受到瞬态荷载作用时衬砌及周围土体的控制方程,利用Durbin数值反演法得到了衬砌及土体在时间域的动力响应。数值分析了饱和度对瞬态荷载下径向位移、径向应力、环向应力和孔隙水压力的影响。结果表明：饱和度对衬砌及周围土体的瞬态响应影响显著;饱和度对径向位移沿径向的衰减影响较小,对环向应力和孔隙压力沿径向的衰减影响较大。     

Exact solutions for the one-dimensional transient response of unsaturated single-layer porous media

The one-dimensional transient response of unsaturated single-layer porous media is studied based on the theory of unsaturated porous media proposed by Zienkiewicz et al., and exact time-domain solutions are obtained for three types of nonhomogeneous boundary conditions. During the solution procedure, the nonhomogeneous boundary conditions are transformed into homogeneous boundary conditions. Then, the eigenfunction expansion method is utilised to obtain the exact solutions for these new boundary conditions. Several numerical examples are provided to investigate the propagation of compressional waves, and it is verified that three types of compressional waves exist in unsaturated porous media that contain two immiscible fluids.     

Semi-analytical solutions to one-dimensional consolidation for unsaturated soils with symmetric semi-permeable drainage boundary

This paper presents semi-analytical solutions to Fredlund and Hasan’s one-dimensional consolidation for unsaturated soils under symmetric semi-permeable drainage boundary conditions. Two variables are introduced to transform two coupled governing equations of pore-air and pore-water pressures into an equivalent set of partial differential equations, which are easily solved by the Laplace transform. Then, the pore-air and pore-water pressures, and soil settlement are obtained in the Laplace domain. Crump’s method is adopted to perform the inverse Laplace transform in order to obtain semi-analytical solutions in time domain. It is shown that the present solution is more applicable to various types of drainage boundary conditions, and in a good agreement with existing solutions from the literature. Furthermore, several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with traditional drainage boundary (single or double), and single-sided and double-sided semi-permeable drainage boundaries. Finally, it illustrates the changes in pore-air and pore-water pressures and soil settlement with time at different values of symmetric semi-permeable drainage boundary conditions parameters. In addition, parametric studies are conducted by the variations of pore-air and pore-water pressures at different ratios of air-water permeability coefficient and the depth.     

A semi-analytical solution for the transient response of one-dimensional unsaturated single-layer poroviscoelastic media

This paper develops a semi-analytical solution for the transient response of an unsaturated single-layer poroviscoelastic medium with two immiscible fluids by using the Laplace transformation and the state-space method. Using the elastic–viscoelastic correspondence principle, we first introduce the Kelvin–Voigt model into Zienkiewicz’s unsaturated poroelastic model. The vibrational response for unsaturated porous material can be obtained by combining these two models and assuming that the wetting and non-wetting fluids are compressible, the solid skeleton and solid particles are viscoelastic, and the inertial and mechanical couplings are taken into account. The Laplace transformation and state-space method are used to solve the basic equations with the associated initial and boundary conditions, and the analytical solution in the Laplace domain is developed. To evaluate the responses in the time domain, Durbin’s numerical inverse Laplace transform method is used to obtain the semi-analytical solution. There are three compressional waves in porous media with two immiscible fluids. Moreover, to observe the three compressional waves clearly, we assume the two immiscible fluids are water and oil. Finally, several examples are provided to show the validity of the semi-analytical solution and to assess the influences of the viscosity coefficients and dynamic permeability coefficients on the behavior of the three compressional waves.     

An explicit one-dimensional consolidation solution with semi-permeable drainage boundary for unsaturated soil

Existing solutions for analyzing one-dimensional (1-D) consolidation of unsaturated soil are only derived to cater to two extreme drainage conditions (fully drained and undrained). This study presents a new explicit solution for 1-D consolidation of unsaturated soil with semi-permeable drainage boundary. Based on the assumptions of two independent stress variables and the governing equations proposed by Fredlund, the eigenfunction expansion method is adopted to develop an explicit analytical solution to calculate excess pore-water and pore-air pressures in an unsaturated soil when it is subjected to external loads. The developed general solutions are expressed in terms of depth, z, and time, t. For the semi-permeable drainage boundary, eigenvalues and eigenfunctions in the space domain are developed. The technique of Laplace transform is used to solve the coupled ordinary differential equations in the time domain. The newly derived explicit solution is verified with the existing semi-analytical method in the literature, and an excellent agreement is obtained. Compared with the semi-analytical solution, the newly derived analytical solution is more straightforward and explicit so that this solution is relatively easier to be implemented into a computer program to carry out a preliminary assessment of 1-D consolidation of unsaturated soil.     

频域内饱和多孔介质的参数反演分析

基于Biot理论,假定固体颗粒和孔隙内流体均不可压缩,建立了以固体骨架位移表示的的控制方程.考虑单层饱和多孔介质在竖向简谐荷载作用下一维动力响应,通过理论推导获得了骨架位移、应力以及孔隙流体压力等物理量的解析表达式.基于饱和土的简谐动力模型试验数据,与所得到的理论解答相结合,将饱和多孔介质材料参数反演问题归结为非线性多峰函数的最优化问题.全局最优解的求解采用了遗传算法和模拟退火算法,并通过试验和数值算例验证了所得材料参数的正确性.     

Exact solutions for one‐dimensional transient response of fluid‐saturated porous media

Based on the Biot theory, the exact solutions for one‐dimensional transient response of single layer of fluid‐saturated porous media and semi‐infinite media are developed, in which the fluid and solid particles are assumed to be compressible and the inertial, viscous and mechanical couplings are taken into account. First, the control equations in terms of the solid displacement u and a relative displacement w are expressed in matrix form. For problems of single layer under homogeneous boundary conditions, the eigen‐values and the eigen‐functions are obtained by means of the variable separation method, and the displacement vector u is put forward using the searching method. In the case of nonhomogeneous boundary conditions, the boundary conditions are first homogenized, and the displacement field is constructed basing upon the eigen‐functions. Making use of the orthogonality of eigen‐functions, a series of ordinary differential equations with respect to dimensionless time and their corresponding initial conditions are obtained. Those differential equations are solved by the state‐space method, and the series solutions for three typical nonhomogeneous boundary conditions are developed. For semi‐infinite media, the exact solutions in integral form for two kinds of nonhomogeneous boundary conditions are presented by applying the cosine and sine transforms to the basic equations. Finally, three examples are studied to illustrate the validity of the solutions, and to assess the influence of the dynamic permeability coefficient and the fluid inertia to the transient response of porous media. Copyright © 2010 John Wiley & Sons, Ltd.     

Semi-analytical solutions of two-dimensional plane strain consolidation in unsaturated soils subjected to the lateral semipermeable drainage boundary

The study presents semi-analytical solutions of two-dimensional plane strain consolidation problem in unsaturated soils incorporating the lateral semipermeable drainage boundary by adopting Fourier sine series and Laplace transform. The two-dimensional plane strain consolidation equations in the form of two-order partial differential equations with three variables are firstly converted to two-order partial differential equations with two variables, which are similar to those of one-dimensional consolidation problem. The four-order ordinary differential equations about excess pore-air and excess pore-water pressures are got by applying Laplace transform and the substitution method. Then, the solutions of excess pore pressures and settlement are achieved in the Laplace transform domain. Afterwards, on the basis of Crump's method, the inverse Laplace transform is conducted to obtain the analytical solutions in time domain. The comparison is conducted to verify the exactness of the obtained solutions, and the two-dimensional plane strain consolidation property with the lateral semipermeable drainage boundary is illustrated and discussed. Parametric studies are demonstrated for the excess pore pressures and normalized settlement with the change of the boundary parameters, air-water and lateral-vertical permeability coefficients, and the distance and depth. It can be found that the lateral semipermeable drainage boundary impedes the consolidation rate obviously, and when different investigated parameters are adopted, the consolidation property is similar to each other under the later permeable and semipermeable drainage boundary conditions.     

Exact solutions for one‐dimensional consolidation of single‐layer unsaturated soil

Based on the Fredlund consolidation theory of unsaturated soil, exact solutions of the governing equations for one‐dimensional consolidation of single‐layer unsaturated soil are presented, in which the water permeability and air transmission are assumed to be constants. The general solution of two coupled homogeneous governing equations is first obtained. This general solution is expressed in terms of two functions psi₁ and ψ₂, where ψ₁ and ψ₂, respectively, satisfy two second‐order partial differential equations, which are in the same form. Using the method of separation of variables, the two partial differential equations are solved and exact solutions for three typical homogeneous boundary conditions are obtained. To obtain exact solutions of nonhomogeneous governing equations with three typical nonhomogeneous boundary conditions, the nonhomogeneous boundary conditions are first transformed into homogeneous boundary conditions. Then according to the method of undetermined coefficients and exact solutions of homogenous governing equations, the series form exact solutions are put forward. The validity of the proposed exact solutions is verified against other analytical solutions in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

Unsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed‐form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.