Finite element formulation and algorithms for unsaturated soils. Part II: Verification and application

The finite‐element formulation and integration algorithms developed in Part I are used to analyse a number of practical problems involving unsaturated and saturated soils. The formulation and algorithms perform well for all the cases analysed, with the robustness of the latter being largely insensitive to user‐defined parameters such as the number of coarse time steps and error control tolerances. The efficiency of the algorithms, as measured by the CPU time consumed, does not depend on the number of coarse time steps, but may be influenced by the error control tolerances. Based on the analyses presented here, typical values for the error control tolerances are suggested. It is also shown that the constitutive modelling framework presented in Part I can, by adjusting one constitutive equation and one or two material parameters, be used to simulate soils that expand or collapse upon wetting. Treating the suction as a strain variable instead of a stress variable proves to be an efficient and robust way of solving suction‐dependent plastic yielding. Moreover, the concept of the constitutive stress is a particularly convenient way of handling the transition between saturation and unsaturation. Copyright © 2003 John Wiley & Sons, Ltd.     

非饱和土水-力本构模型及其隐式积分算法

在已有工作基础上建立了水力-力学耦合的非饱和土本构模型,在硬化方程中考虑饱和度的影响,同时在土水特征曲线中考虑了塑性体变的影响,从而使模型可以反映非饱和土中的毛细现象与土中弹塑性变形现象的耦合行为。采用隐式积分方法,建立了非饱和土耦合模型的数值模型,并推导了得到了水力-力学耦合的非饱和土的一致切线模量。利用该算法编制了本构模型计算的子程序,使其能向外输出切线刚度矩阵,用于有限元计算。为了验证该算法和程序的正确性,用所编制程序对不同路径下的土体行为进行了预测。通过预测结果与试验结果相对比,表明程序预测结果与试验数据相吻合,模型可以较好地模拟土体的水力-力学耦合行为特性。     

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。     

非饱和土弹塑性模型参数的试验确定及有限元法

简要介绍广泛应用的非饱和土Alonso模型的原理与发展,并通过3组共7个试样的非饱和土三轴试验,得到模型的11个参数。在饱和土Cam本构关系的基础上,推导出一个基于非饱和土Alonso模型的应力-应变增量方程的计算公式,该方程与饱和土的本构方程形式相同。编制了能够考虑净应力和吸力二者对土体硬化规律影响的有限元程序,从而为非饱和土弹塑性计算提供一条途径,便于进一步应用于实际工程。     

Equivalent stress approach in modelling unsaturated soils

This article presents an equivalent stress approach that can be used in many elastoplastic constitutive models for unsaturated soils. The use of the equivalent stress leads to a modified yield locus that is independent of the suction. In addition, the equivalent stress becomes the major stress variable, with suction required only as an additional variable in calculations. The model on the basis of equivalent stress predicts exactly the same soil behaviour, with the sole difference being the use of equivalent stress instead of original stress variables. This article also presents the equivalent stress formulations of several constitutive models for unsaturated soils, including the Barcelona Basic Model. The predictions from these models remain unchanged, with the only difference being in their implementation. Finally, the equivalent stress approach and the net stress approach are compared for the Barcelona Basic Model. Copyright © 2011 John Wiley & Sons, Ltd.     

A kinematic hardening based model for unsaturated soils considering different hydraulic conditions

At present, several of the existing elastoplastic constitutive models are adapted for describing the stress–strain behavior of unsaturated soils. However, most of them present certain limitations in this field. These limitations can be related to the basic model and/or added unsaturated state variables and formulations. In this regard, inability to model the hydro‐mechanical behavior in constant water (CW) conditions is an example of these limitations. In this paper, an advanced version of CJS model is selected for adaptation to the unsaturated states. Adaptation to unsaturated states is achieved in the framework of effective stress approach. Effective stress equation and unsaturated state variables are selected based on the recent research existing in the literature. The developed model is capable of describing the complex behavior of unsaturated soil in the CW condition in addition to predicting the behavior at failure and post–failure, nonlinear elastoplastic behavior at low levels of stress and strain (by selecting a very small elastic domain), as well as wetting and collapse behaviors. In order to validate the model, results of triaxial tests in CD and CW conditions are used. The validation results indicate the good capability of the proposed model. Behavior of the unsaturated soils during wetting is an important issue. For this reason, the model is also evaluated based on the results of wetting and collapse triaxial tests. A comparison between the tests and simulation results shows that the model is able to predict the soil behavior under the wetting path. Copyright © 2016 John Wiley & Sons, Ltd.     

A plasticity constitutive model for unsaturated,anisotropic, nonexpansive soils

The paper describes and evaluates an incremental plasticity constitutive model for unsaturated, anisotropic, nonexpansive soils (CMUA). It is based on the modified Cam-Clay (MCC) model for saturated soils and enhances it by introducing anisotropy (via rotation of the MCC yield surface) and an unsaturated compressibility framework describing a double dependence of compressibility on suction and on the degree of saturation of macroporosity. As the anisotropic and unsaturated features can be activated independently, the model is downwards compatible with the MCC model. The CMUA model can simulate effectively: the dependence of compressibility on the level of developed anisotropy, uniqueness of critical state independent of the initial anisotropy, an evolving compressibility during constant suction compression, and a maximum of collapse. The model uses Bishop's average skeleton stress as its first constitutive variable, favouring its numerical implementation in commercial numerical analysis codes (eg, finite element codes) and a unified treatment of saturated and unsaturated material states.     

Dynamics of unsaturated soils using various finite element formulations

Unsaturated soils are three‐phase porous media consisting of a solid skeleton, pore liquid, and pore gas. The coupled mathematical equations representing the dynamics of unsaturated soils can be derived based on the theory of mixtures. Solution of these fully coupled governing equations for unsaturated soils requires tremendous computational resources because three individual phases and interactions between them have to be taken into account. The fully coupled equations governing the dynamics of unsaturated soils are first presented and then two finite element formulations of the governing equations are presented and implemented within a finite element framework. The finite element implementation of all the terms in the governing equations results in the complete formulation and is solved for the first time in this paper. A computationally efficient reduced formulation is obtained by neglecting the relative accelerations and velocities of liquid and gas in the governing equations to investigate the effects of fluid flow in the overall behavior. These two formulations are used to simulate the behavior of an unsaturated silty soil embankment subjected to base shaking and compared with the results from another commonly used partially reduced formulation that neglects the relative accelerations, but takes into account the relative velocities. The stress–strain response of the solid skeleton is modeled as both elastic and elastoplastic in all three analyses. In the elastic analyses no permanent deformations are predicted and the displacements of the partially reduced formulation are in between those of the reduced and complete formulations. The frequency of vibration of the complete formulation in the elastic analysis is closer to the predominant frequency of the base motion and smaller than the frequencies of vibration of the other two analyses. Proper consideration of damping due to fluid flows in the complete formulation is the likely reason for this difference. Permanent deformations are predicted by all three formulations for the elastoplastic analyses. The complete formulation, however, predicts reductions in pore fluid pressures following strong shaking resulting in somewhat smaller displacements than the reduced formulation. The results from complete and reduced formulations are otherwise comparable for elastoplastic analyses. For the elastoplastic analysis, the partially reduced formulation leads to stiffer response than the other two formulations. The likely reason for this stiffer response in the elastoplastic analysis is the interpolation scheme (linear displacement and linear pore fluid pressures) used in the finite element implementation of the partially reduced formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

Effective stress concept in unsaturated soils: Clarification and validation of a unified framework

The effective stress principle, conventionally applied in saturated soils, is reviewed for constitutive modelling purposes. The assumptions for the applicability of Terzaghi's single effective stress are recalled and its advantages are inventoried. The possible stress frameworks applicable to unsaturated soil modelling are reassessed in a comparative manner, specifically the Bishop's single effective stress, the independent stress variables approach and the generalized stress framework. The latter considerations lead to the definition of a unified stress context, suitable for modelling soils under different saturation states. In order to qualify the implications brought by the proposed stress framework, several experimental data sets are re‐examined in the light of the generalized effective stress. The critical state lines (CSLs) at different saturation states tend to converge remarkably towards a unique saturated line in the deviatoric stress versus mean effective stress plane. The effective stress interpretation is also applied to isotropic paths and compared with conventional net stress conception. The accent is finally laid on a second key feature for constitutive frameworks based on a unified stress, namely the sufficiency of a unique mechanical yield surface besides the unique CSL. Copyright © 2007 John Wiley & Sons, Ltd.     

Use of coupled finite element analysis in unsaturated soil problems

This paper presents a variation of Biot's consolidation theory for analysing problems involving unsaturated soils, and implemented using the finite element method. The numerical method is applied to a few geotechnical problems as examples and the results obtained are compared to some published data. The illustrative examples show how the numerical method can be used to analyse seepage and consolidation problems associated with unsaturated soils and demonstrate the flexibility and applicability of the presented method. Copyright © 2000 John Wiley & Sons, Ltd.     

一个高饱和度非饱和土的本构模型

高饱和度的非饱和土中由于气体处于封闭状态,其内部气压的变化必将对土体的行为产生影响。首先,对高饱和度非饱和土特性进行探讨和研究,随后,在已有非饱和土模型框架基础上,采用广义有效应力原理,建立一个适用于高饱和度条件下的非饱和土的弹塑性本构模型。模型中引入气相耗散的影响,在硬化方程中考虑封闭气体压力改变的影响。最后,利用已有的试验结果来对模型进行验证,并将模型预测结果与前人模型进行对比,表明模型预测可以很好地预测土体的行为,尤其是在高饱和度条件下其结果比其他模型更加接近实际情况。     

Generalized plasticity state parameter‐based model for saturated and unsaturated soils. Part II: Unsaturated soil modeling

The aim of this paper is to extend the generalized plasticity state parameter‐based model presented in part 1 to reproduce the hydro‐mechanical behavior of unsaturated soils. The proposed model is based on two pairs of stress–strain variables and a suitable hardening law taking into account the bonding—debonding effect of suction and degree of saturation. A generalized state parameter for unsaturated state is proposed to reproduce soil behavior using a single set of material parameters. Generalized plasticity gives a suitable framework to reproduce not only monotonic stress path but also cyclic behavior. The hydraulic hysteresis during a drying—wetting cycle and the void ratio effect on the hydraulic behavior is introduced. Comparison between model simulations and a series of experimental data available, both cohesive and granular, are given to illustrate the accuracy of the enhanced generalized plasticity equation. Copyright © 2010 John Wiley & Sons, Ltd.     

考虑残余含气量的非饱和土的水力耦合本构模型

大量的非饱和土干湿循环试验表明,当土体处于吸湿过程直至吸力降低为0 kPa时,土体并不能达到完全饱和状态,还存在一定的残余气体。在高饱和度时,由于残余气体以封闭气泡的形式分布在土体中,土体呈现较大的压缩性,使其与饱和土的性质不同。在这种状态下,现有的非饱和土本构模型预测到的土饱和度为100%,与试验结果存在一定的偏差。为了使本构模型在高饱和度状态时具有较高的精度,对非饱和土的毛细滞回和塑性变形耦合本构模型进行了修正,使其能够考虑残余含气量的影响。通过预测与实测结果比较,证明了新模型能够有效地模拟残余含气量对非饱和土力学特性的影响。     

A hierarchical thermo‐hydro‐plastic constitutive model for unsaturated soils and its numerical implementation

Unsaturated soils are highly heterogeneous 3‐phase porous media. Variations of temperature, the degree of saturation, and density have dramatic impacts on the hydro‐mechanical behavior of unsaturated soils. To model all these features, we present a thermo‐hydro‐plastic model in which the hydro‐mechanical hardening and thermal softening are incorporated in a hierarchical fashion for unsaturated soils. This novel constitutive model can capture heterogeneities in density, suction, the degree of saturation, and temperature. Specifically, this constitutive model has 2 ingredients: (1) it has a “mesoscale” mechanical state variable—porosity and 3 environmental state variables—suction, the degree of saturation, and temperature; (2) both temperature and mechanical effects on water retention properties are taken into account. The return mapping algorithm is applied to implement this model at Gauss point assuming an infinitesimal strain. At each time step, the return mapping is conducted only in principal elastic strain space, assuming no return mapping in suction and temperature. The numerical results obtained by this constitutive model are compared with the experimental results. It shows that the proposed model can simulate the thermo‐hydro‐mechanical behavior of unsaturated soils with satisfaction. We also conduct shear band analysis of an unsaturated soil specimen under plane strain condition to demonstrate the impact of temperature variation on shear banding triggered by initial material heterogeneities.     

Work input for unsaturated soils considering interfacial effects

In this research, the interfacial energy is taken into account in the deformation work for unsaturated soils. Based on porous media theory, the thermodynamic balance equations for each phase and the interface are used to derive the work input for unsaturated soils. The work input equation serves as the basis and starting point for the choice of stress state variables, based on which the conjugate stresses and strain increments are derived. The influences of the interfaces on the effective stress and the constitutive law for the liquid phase are then discussed based on the work input equation. The effective stress can be expressed as Bishop's type, and the effective stress parameter is shown to be a function of both the degree of saturation and the interfacial area. The constitutive law for the liquid phase under dynamic condition is also presented. The relationship among interfacial area, saturation, and capillary pressure is proposed to calculate the value of the effective stress. Experimental data obtained from literature are used to validate the proposed model equations. Results show that our findings are in accordance with the existing research. Unlike the phenomenal study, our research has a rigorous theoretical basis, which lays a foundation for further research of unsaturated soils considering the interfacial effects.     

Implicit integration under mixed controls of a breakage model for unsaturated crushable soils

This paper discusses a series of stress point algorithms for a breakage model for unsaturated granular soils. Such model is characterized by highly nonlinear coupling terms introduced by breakage‐dependent hydro‐mechanical energy potentials. To integrate accurately and efficiently its constitutive equations, specific algorithms have been formulated using a backward Euler scheme. In particular, because implementation and verification of unsaturated soil models often require the use of mixed controls, the incorporation of various hydro‐mechanical conditions has been tackled. First, it is shown that the degree of saturation can be replaced with suction in the constitutive equations through a partial Legendre transformation of the energy potentials, thus changing the thermomechanical state variables and enabling a straightforward implementation of a different control mode. Then, to accommodate more complex control scenarios without redefining the energy potentials, a hybrid strategy has been used, combining the return mapping scheme with linearized constraints. It is shown that this linearization strategy guarantees similar levels of accuracy compared with a conventional strain–suction‐controlled implicit integration. In addition, it is shown that the use of linearized constraints offers the possibility to use the same framework to integrate a variety of control conditions (e.g., net stress and/or water‐content control). The convergence profiles indicate that both schemes preserve the advantages of implicit integration, that is, asymptotic quadratic convergence and unconditional stability. Finally, the performance of the two implicit schemes has been compared with that of an explicit algorithm with automatic sub‐stepping and error control, showing that for the selected breakage model, implicit integration leads to a significant reduction of the computational cost. Such features support the use of the proposed hybrid scheme also in other modeling contexts, especially when strongly nonlinear models have to be implemented and/or validated by using non‐standard hydro‐mechanical control conditions. Copyright © 2015 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.     

ACMEG‐TS: A constitutive model for unsaturated soils under non‐isothermal conditions

This paper introduces an unconventional constitutive model for soils, which deals with a unified thermo‐mechanical modelling for unsaturated soils. The relevant temperature and suction effects are studied in light of elasto‐plasticity. A generalized effective stress framework is adopted, which includes a number of intrinsic thermo‐hydro‐mechanical connections, to represent the stress state in the soil. Two coupled constitutive aspects are used to fully describe the non‐isothermal behaviour. The mechanical constitutive part is built on the concepts of bounding surface theory and multi‐mechanism plasticity, whereas water retention characteristics are described using elasto‐plasticity to reproduce the hysteretic response and the effect of temperature and dry density on retention properties. The theoretical formulation is supported by comparisons with experimental results on two compacted clays. Copyright © 2008 John Wiley & Sons, Ltd.     

A thermo‐hydro‐mechanical model for unsaturated soils based on the effective stress concept

A simple thermo‐hydro‐mechanical (THM) constitutive model for unsaturated soils is described. The effective stress concept is extended to unsaturated soils with the introduction of a capillary stress. This capillary stress is based on a microstructural model and calculated from attraction forces due to water menisci. The effect of desaturation and the thermal softening phenomenon are modelled with a minimal number of material parameters and based on existing models. THM process is qualitatively and quantitatively modelled by using experimental data and previous work to show the application of the model, including a drying path under mechanical stress with transition between saturated and unsaturated states, a heating path under constant suction and a deviatoric path with imposed suction and temperature. The results show that the present model can simulate the THM behaviour in unsaturated soils in a satisfactory way. Copyright © 2010 John Wiley & Sons, Ltd.