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.

General semi-analytical solutions to one-dimensional consolidation for unsaturated soils

This paper presents general semi-analytical solutions to Fredlund and Hasan＇s one-dimensional （1D） consolidation equations for unsaturated soils subject to different initial conditions, homogeneous boundaries and time-dependent loadings. Two variables are introduced to transform the two-coupled governing equations of pore-water and poreair pressures into an equivalent set of partial differential equations （PDEs）, which are solved with the Laplace transform method. The pore-water and pore-air pressures and settlement are obtained in the Laplace transform domMn. The Crump＇s method is used to perform inverse Laplace transform to obtain the solutions in the time domain. The present solutions are more general in practical applications and show good agreement with the previous solutions in the literature.

ONE-DIMENSIONAL CONSOLIDATION OF LAYERED AND VISCO-ELASTIC SOILS UNDER ARBITRARY LOADING

Based on the layered visco-elastic soil model, according to the Terzaghi's one dimensional consolidation theory, by the method of Laplace transform and matrix transfer technique, the problems about the consolidation of layered and saturated visco-elastic soils under arbitrary loading were solved. Through deductions, the general solution, in the terms of layer thickness, the modulus and the coefficients of permeability and Laplacian transform's parameters was obtained. The strain and deformation of the layered and saturated visco-elastic soils under arbitrary loading can be calculated by Laplace inversion. According to the results of several numerical examples, the consolidation of visco-elastic soils logs behind that of elastic soils. The development of effective stress and the displacement is vibrant process under cyclic loading. Finally, an engineering case is studied and the results prove that the methods are very effective.

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.

Analysis of one-dimensional consolidation of soft soils with non-Darcian flow caused by non-Newtonian liquid

Based on non-Darcian flow caused by non-Newtonian liquid, the theory of one-dimensional (1D) consolidation was modified to consider variation in the total vertical stress with depth and time. The finite difference method (FDM) was adopted to obtain numerical solutions for excess pore water pressure and average degree of consolidation. When non-Darcian flow is degenerated into Darcian flow, a comparison between numerical solutions and analytical solutions was made to verify reliability of finite difference solutions. Finally, taking into account the ramp time-dependent loading, consolidation behaviors with non-Darcian flow under various parameters were analyzed. Thus, a comprehensive analysis of 1D consolidation combined with non-Darcian flow caused by non-Newtonian liquid was conducted in this paper.

ONE-DIMENSIONAL CONSOLIDATION OF LAYERED SOILS WITH IMPEDED BOUNDARIES UNDER TIME-DEPENDENT LOADINGS

On the basis of Terzaghi's one-dimensional consolidation theory, the variation of effective stress ratio in layered saturated soils with impeded boundaries under time-dependent loading was studied. By the method of Laplace transform, the solution was presented. Influences of different kinds of cyclic loadings and impeded boundaries conditions were discussed. Through numerical inversion of Laplace transform, useful illustrations were given considering several common time-dependent loadings. Pervious or impervious boundary condition is just the special case of the problem considered here. Compared with average index method,the results from the method illustrated are more accurate.

Exact Solution of Terzaghi’s Consolidation Equation and Extension to Two/Three-Dimensional Cases

The differential equation by Terzaghi and Fr?hlich, better known as Terzaghi’s one-dimensional consolidation equation, simulates the visco-elastic behaviour of soils depending on the loads applied as it happens, for example, when foundations are laid and start carrying the weight of the structure. Its application is traditionally based on Taylor’s solution that approximates experimental results by introducing non-dimensional variables that, however, contradict the actual behaviour of soils. The proposal of this research is an exact solution consisting in a non-linear equation that can be considered correct as it meets both mathematical and experimental requirements. The solution proposed is extended to include differential equations relating to two/three dimensional consolidation by adopting a transversally isotropic model more consistent with the inner structure of soils.

Rheological catastrophic model for soft clays

A brief review of the former studies on the mechanisms of soil rheology and microstructure is presented. Then a microstructure model and a set of rheological constitutive relations for soft clays, which describe how the rheological consolidation settlement develops, are established in the framework of the catastrophe theory. The validity of this model is verified by a series of rheological consolidation experiments with different loading rates. The experimental data show that creep deformation can be clearly observed in these tests, and the consolidation settlement is loading rate dependent. The characteristics of the deformation can be explained and reproduced successfully using the model. It can be concluded that only the biggest set of voids would collapse for one load increment. Parameters in the model, k and η, are gained by curve fitting. With only two free parameters, good fits of the data are achieved.

基于广义Voigt模型的半渗透边界隧道周围饱和软土固结分析

引入衬砌半渗透边界条件,使用广义Voigt黏弹性模型描述隧道周围饱和软黏土的流变特性,基于Terzaghi-Rendulic理论建立以超孔隙水压力为变量的软土二维流变固结控制方程。采用保角变换和分离变量方法,得到超孔隙水压力关于时间和空间变量的两个独立方程,依次使用Laplace变换法与部分分式展开法得到时域内超孔隙水压力消散解。将广义Voigt模型退化为已有四元件Burgers模型和五元件模型并与其对比,验证解答的可靠性。基于建立的解答,分析衬砌透水程度、Kelvin体数量、独立Newton黏壶黏滞系数、隧道埋深等因素对超孔隙水压力消散与分布的影响规律。结果表明:衬砌与土体的相对渗透系数比越大,超孔隙水压力的消散起始时间越早,消散速度越快。Kelvin体数量越多,超孔隙水压力消散越慢;独立黏壶黏滞系数越大,其产生的黏滞变形越小,超孔隙水压力消散速率越快;隧道埋深越大,土体渗流路径越长,超孔隙水压力消散越慢;当相对渗透系数比为0.01时,超孔隙水压力随着到隧道外壁距离的增大而逐渐减小,当相对渗透系数比为1.00时,超孔隙水压力随着距离的增大而出现先增大后减小的现象;随着相对渗透系数比的增加,Kelvin体数量对超孔压消散的影响越小,不同独立黏滞系数带来的超孔压消散差异逐渐减小,隧道衬砌作为土层内部新透水边界的作用越突出。

水泥-粉煤灰胶结矸石充填材料的流变特性与固结机理

将粉煤灰、水泥、煤矸石和水作为充填材料进行充填,既能实现固体废弃物的资源化利用,还能有效控制矿山地表沉降。研究针对胶结矸石充填材料进行了流变测试、力学性能测试以及水化特性试验,分别研究了矸石充填材料的流变特性、抗压强度、水化产物以及微观结构。结果表明:当325普通硅酸盐水泥含量为15%、低钙粉煤灰掺量超20%、料浆质量浓度保持在77%以上时,所制备的矸石充填试样28 d龄期的最小抗压强度超4.229 MPa,可以为井下充填开采提供良好的顶板支撑。通过流变试验探究了粉煤灰掺量和料浆质量浓度变化对矸石充填料浆流变性能的影响,并得到了充填料浆所对应的流变参数以及流变方程拟合。此外,对矸石胶结充填材料的矿物成分、水化产物以及微观结构进行了研究。研究可为大宗固体废弃物在矿山充填领域应用提供一定的理论指导。

饱和软土非平行等效时间线流变一维固结分析

在饱和软土非平行等效时间线流变模型基础上,综合考虑软土地基成层性和非线性特点,利用半解析法对地基一维固结控制方程进行求解分析。采用Fortran语言编制相关计算程序,将退化后的计算结果与已有的平行等效时间线下的流变固结解进行对比,验证该解答的正确性。利用该程序分析饱和软土地基流变固结性状,分析讨论了非平行线参数ζ对土体次固结系数的影响,以及非平行等效时间线流变固结下有效应力定义的固结度和地表总沉降的增长规律。分析结果表明:非平行等效时间线模型的次固结系数随自重应力和有效应力的增大而减小,且减小速率取决于ζ值大小;非平行等效时间线下的流变特性主要对主固结后期和次固结阶段影响显著,ζ值越大,主固结完成越早,后期次固结沉降量越小。

基于分数阶模型的隧道周围土体非线性流变固结分析

考虑隧道周围饱和软土压缩性和渗透性非线性变化,建立二维非线性固结控制方程,引入分数阶Merchant模型考虑土体流变特性影响,采用Douglas-Jone格式的交替隐式差分法对方程进行求解.通过与现有解析解进行对比,验证了本研究解的正确性.利用得到的差分解进行参数分析,研究隧道周围饱和软土的非线性流变固结特性.结果表明,隧道渗漏模式、初始渗透系数和渗透指数对土体固结速率影响较大,透水通道越多、初始渗透系数越大、渗透指数越小,土体固结速率越快,而压缩指数对固结速率的影响较小;渗透各向异性系数对固结速率有较大影响,在进行固结性状分析时,应当充分考虑土体不同方向渗透系数的差异;当考虑土体流变特性时,土体固结速率显著减小,但当初始渗透系数增大到一定程度时,土体流变特性对土层固结速率的影响可以忽略.

软土的次固结

计算软土次固结的常用方法没有反映荷载变化的影响,实际上只适用于正常固结土。而工程中所遇到的大都是超固结土。笔者所做试验表明,当荷载不大时,即处于超固结状态,次压缩明显与荷载有关。本文在Bjerrum等时e-lgp曲线理论的基础上,提出了相对时间坐标系与绝对时间坐标系的概念,建议在次压缩计算中采用绝对时间坐标系,即以正常固结状态的加荷时刻为时间零点的坐标系,从而建立了一种新的次固结计算方法。通过等时e-lgp曲线建立起时间与荷载的关系,使次压缩不仅仅与时间有关,而且也与荷载有关。它可以用于正常固结土和超固结土。这一计算方法可以清楚解释许多试验现象和工程问题,如次压缩随荷载的变化,超载预压后的次压缩,反复荷载对次压缩的影响等。该方法更符合次压缩的变形机理。

软土的一维次固结双曲线流变模型研究

软土的次固结对于软土的变形量非常重要。在软土次固结计算中,通常以次固结系数作为计算参数,但该系数不能反映荷载变化的影响,而且只适用于正常固结土。由于工程中所遇到的大都为超固结土,对超固结土的次固结计算还缺少合理的方法。通过室内试验开展了软土次固结沉降相关研究,进行了多组原状软土试样的分级加载次固结试验,试验结果表明,对于软土次固结计算,建议从次固结应变与时间关系的角度,采用双曲线形式进行拟合,并分别对正常固结和超固结状态的次固结应变参数进行分析,建立了一维次固结的经验模型公式,提出了一种考虑压力对次压缩影响的次固结沉降计算方法,并将该方法应用于现场工程项目,验证了该方法可以适用于正常固结土和超固结土,使得软土次固结沉降量的计算能够更好地反映实际工程问题。

宁波软土流变试验及经验模型

为了研究宁波软土的流变特性及经验模型,进行一维固结流变试验和三轴排水流变试验,并采用西原模型拟合模型参数.试验对比分析表明,固结试验的侧限作用会阻碍试样流变的发展,应力-应变等时曲线偏向应力轴,而三轴流变试验试样的流变能够得到充分地发展,应力-应变等时曲线偏向应变轴,能够更好地体现软黏土的流变性状.将一维西原模型应力-应变关系扩展到三维,采用修正剑桥模型屈服函数描述材料的塑性屈服.推导荷载恒定时土样的轴向应变计算公式,根据三轴排水流变试验结果进行曲线拟合,确定西原模型参数.结果表明,该模型能够较好地符合宁波软土的流变规律.

沥青混合料蠕变柔量的一种实用模型及其应用

通过分析沥青混合料加载时的“固结效应”和卸载时的“永久变形”等流变学特征 ,提出了一个较为实用的流变学模型 ,并给出了模型参数的确定方法 ,该模型参数数目可多可少 ,且能满足Blotzmann叠加原理 ,可应用于应力随时间变化以及其它各种复杂应力状态的情况 .

循环荷载作用下考虑非达西渗流的软黏土一维流变固结分析

基于Hansob渗流理论和Merchant流变模型,将微分型流变方程耦合到一维固结理论中,建立了循环荷载作用下双变量表述的软黏土一维流变固结耦合方程组。采用Flex PDE软件对方程组进行了求解,通过与已有文献结果对比,验证了算法的可靠性,在该基础上研究了循环荷载作用下饱和软黏土的一维流变固结特性,分析了Hansob渗流参数、Merchant流变模型参数以及循环荷载类型对饱和软黏土固结特性的影响。研究结果表明,按超孔压定义的平均固结度U_p和按变形定义的平均固结度U_s均呈现循环变化特征,且U_p>U_s;当循环周次较大时,U_p进行稳定循环状态,由于流变效应,U_s峰值仍呈逐渐增大趋势;随着Hansob渗流参数m和1I的逐渐增大,地基固结速率逐渐降低,同一循环周次内U_p、U_s的峰值也随之降低;随着流变参数F的逐渐增大,同一循环周次内U_p的峰值逐渐降低,U_s的峰值显著提高;随着b的逐渐增大,同一循环周次内U_p、U_s的峰值逐渐降低。循环荷载作用下地基的平均固结度始终处于循环状态,无法达到固结完成状态,其最大平均固结度也远小于线性荷载的情况。

双面半透水边界饱和土层在循环荷载作用下的一维粘弹性固结解析解

研究了在循环荷载作用下考虑土的流变特性时具有双面半透水边界的饱和粘土层的一维固结问题,根据瞬时加载条件下的解,通过积分的方法,得到了循环荷载条件下的解析解。并且根据解析解的结果,通过电算程序作出了几个主要影响因素对固结过程的影响曲线图,经过分析得出了一些重要的结论。

变荷载下软土一维流变固结解析理论

研究了变荷载作用下考虑土的流变特性时的饱和软土层的一维固结问题。通过 Laplace 变换得到了瞬时加载条件下的解析解,进而根据瞬时加载条件下的解,通过积分的方法,得到了变荷载条件下的解析解。并且根据解析解的结果,通过编程计算作出了几个主要影响因素对固结过程的影响曲线图。结果表明:η_1的取值对固结曲线的形状有显著影响:随η_0的减小,固结度减小,固结速率减缓。

基于分数阶流变模型的饱和软土一维流变固结

在太沙基一维固结理论的基础上,引入分数阶导数Merchant流变模型来描述土体的变形特性,通过对微分型流变方程和一维固结方程进行Laplace变换,推导得到了Laplace变换域内饱和软黏土的一维流变固结方程,采用FlexPDE软件对方程进行求解,并通过Durbin反Laplace变换获得了平均固结度的时域解答。在此基础上,研究了分数阶导数流变模型的微积分阶数和流变模型参数对饱和软土一维流变固结特性的影响规律。计算结果表明:基于整数阶和基于分数阶导数流变模型的流变固结理论在固结前期基本相同,但在固结后期两者差异显著,基于分数阶导数流变模型的软黏土Up要高于整数阶的情况,而Us的规律则截然相反。此外,微分阶数β对两种定义的平均固结度均有较大影响,β越大,Up越小而Us就越大,软黏土的流变效应越显著。随着流变模型参数F的逐渐增大,同一时刻软黏土的Up逐渐减小,而Us则逐渐增大;随着参数β1的逐渐增大,同一时刻软黏土的Up和Us均随之减小。