一个能够模拟软土时效特性的简单弹黏塑性模型_英文_.pdf

第30卷 第6期 岩 土 工 程 学 报 Vol.30 No.6 2008 年 6 月 Chinese Journal of Geotechnical Engineering June, 2008 Modelling of time-dependent behaviour of soft soils using simple elasto-viscoplastic model YIN Zhen-yu1, 2, 3，ZHANG Dong-Mei2, 3，HICHER Pierre-yves1，HUANG Hong-wei2, 3 1. Research Institute in Civil and Mechanical Engineering, Ecole Centre of Nantes, Nantes 44321, France; 2. Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China ; 3 .Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Abstract The purpose of this study was to present the development of an elasto-viscoplastic constitutive model to describe the time-dependent behaviour of soft soils. The elasto-viscoplastic model was established within the framework of Perzyna’s overstress theory and the Modified Cam Clay model. The stress-strain relationship was solved by using an implicit backward Euler of stress with updated algorithm, and implemented in a finite element program. Different types of tests were simulated using the EVP-MCC model to simulate the time-dependent behaviour of soft soils under different loading conditions, such as the strain rate effects on preconsolidation pressure as well as on undrained shear strength; the primary, secondary consolidation behaviour and stress effects on secondary compression coefficient αe C; the creep and stress relaxation features under different stress levels. It was shown that the model could satisfactorily describe the time-dependent behaviour of normally consolidated or slightly overconsolidated clayey soils along different loading paths. Time-dependent properties behaved in multiple stages triaxial tests, and field and laboratory pressuremeter tests had also been successfully simualted by the proposed EVP-MCC model. Key words elasto-viscoplastic constitutive model; time-dependency; soft soils; strain rate; creep; relaxation CLC numberTU411 Document codeA Article ID1000–4548200806–0880–09 Biography YIN Zhen-yu 1975 – ， male, Ph.D. of Ecole Centre of Nantes and Tongji University. E-mail zhenyuyin. 一个能够模拟软土时效特性的简单弹黏塑性模型 尹振宇 1,2，张冬梅2，HICHER Pierre-yves1，黄宏伟2 （1. 南特中央理工大学土木工程与力学研究所，南特 44321，法国；2. 同济大学岩土及地下工程教育部重点实验室，上海 200092； 3.同济大学地下建筑与工程系，上海 200092 摘 要基于 Perzyna 超应力理论与修正剑桥模型，建立了一个能够模拟软土时效特性的简单的弹黏塑性本构模型，提 出了参数的实验室确定方法。以室内试验为基础，模拟了不同试验条件下软土的时效特性应变速率对先期固结压力 和不排水抗剪强度的影响；一维固结与次固结特征及竖向应力对次固结系数的影响；不同应力水平下的不排水蠕变特 征；不同应变水平下的应力松弛特征。通过实验数据与数值模拟的比较，对模型进行了验证，发现上述本构模型能够 较好地描述不同加载路径下的正常固结与微超固结土的时效特征。同时，通过对同一试样的多阶段加卸载三轴实验、 现场压力仪实验及实验室压力仪实验的模拟，发现此模型可以较好地拟合实验过程中复杂应力路径下软土的时效特征。 关键词弹黏塑性模型；时效特性；软土；应变速率；蠕变；应力松弛 0 Introduction The numerous constitutive models integrating the viscous behaviour of fine soils which have been proposed up to now can all be classified into three categories as empirical models, rheological models and general stress–strain–time models. The empirical models are mainly obtained by matching the equations with experimental results from constant strain rate, creep and stress relaxation tests[1-7]. These models are strictly ─────── Foundation item National Natural Science Foundation of China 50608058; Hi-tech Research and Development Program 863 Program of China 2006AA11Z118 Received date2007–07–02 第 6 期 YIN Zhen-yu, et al. Modelling of time-dependent behaviour of soft soils using simple elasto-viscoplastic model 881 limited to specific boundary and loading conditions. Rheological models usually describe uniaxial conditions and they are expressed in a differential or as closed- solutions. They are often used to obtain a conceptual understanding of time effects in soil[8-9]. According to Singh and Mitchell[3], the rheological models could be generalised from one to three dimensions, but practical calibration and application seem to be difficult to achieve. The general stress–strain–time models are three-dimensional models and they are usually expressed in incremental . Therefore, they are suitable to numerical implementation in a finite element program for example. Furthermore, these models are not limited to given boundary conditions and can be used to simulate all possible stress paths. In the last three decades, the majority of researchers proposed viscoplastic models based on the framework of Perzyna’s overstress theory. Some of them used the distance between the static yield surface and the dynamic loading surface as their scaling function[10-15], while others introduced the secondary compression coefficient αe C of one-dimensional tests into their scaling function [16-20]. However, the coefficient αe C varies during the compression for different stress levels[21]. Fig. 1 shows the evolution of αe C versus the applied stress from conventional oedometer tests pered on several natural clays Flumet clay[12], Shanghai clay and Yuhuan clay by authors. It is found that the value of αe C varies for a large domain of applied stress before reaching a stable value for these natural clays. Fig. 1 Evolution of the secondary compression coefficient versus applied vertical stress from conventional oedometer tests The purpose of this study is to develop a time-dependent constitutive model to provide a simple but realistic approach in modelling time-dependent behaviour of soils, so that it can be easily calibrated and used in geotechnical projects. In this paper, the authors present firstly the development of an elasto-viscoplastic model with its numerical derivation for stress-strain relationship and its implementation in a finite element program CESAR_LCPC. The model parameters identification is then proposed. The experimental verification of the proposed model for soft soils is finally undertaken, based on simulating different types of laboratory and in situ tests, along stress and strain paths such as constant strain rate loading, creep and stress relaxation. 1 Constitutive model 1.1 Constitutive model The Modified Cam Clay model[22] has been widely used for estimating the time-independent behaviour of soft clay. Because this model is ulated very simply and suited for finite element analysis, it is adopted in the present study as a basis for ulating a viscoplastic model. According to the assumption of Perzyna’s overstress theory[23-24], the viscous effects are negligible in the elastic region. In other words, the elastic strains are time independent, whereas the inelastic strains are time dependent. We define a static yield criterion s f which represents a reference yield surface elastic limit with null viscoplastic flow rule for the material. Its initial shape depends on the consolidation pressure s c p. The expansion of the static yield surface, which represents the hardening of the material, is expressed by the variation of the consolidation pressure due to the inelastic volumetric strain vp v ε. s ssvpvp0c ccvv * 1 ddd ep ppεε λκβ ⋅⋅⋅ − ， 1 where s c pmeans the static consolidation pressure, vp v εis the inelastic volumetric strain, * βis the compressibility index. A dynamic loading yield criterion d f is defined to represent the current state of stress and is expressed as 2 d dc 2 0 q fppp M ′′⋅− ， 2 where d c pis the dynamic consolidation pressure, M is the slope of the critical state line, p′is the effective mean stress, q is the deviatoric stress. Based on the values of s c pand d c p, the scaling function F φ⋅ is taken as an exponential 882 岩 土 工 程 学 报 2008 年 shown as Equation 3. It could control the amplitude of the viscoplastic strain rate and enlarge the domain of application of this model[12]. d c s c exp11 p FN p φ ⎛⎞⎡⎤ ⎛⎞ ⋅⋅⋅−−⎜⎟ ⎢⎥⎜⎟ ⎜⎟ ⎢⎥⎝⎠ ⎣⎦⎝⎠ ， 3 where, and N are the viscosity parameters of the model. For constant strain rate test, N controls the strain rate parameter 0α ρ, and controls the amplitude of stress as showed in details by Yin[25]. The flow rule for the viscoplastic strain rate, in a simple case of infinitesimal strain field, follows the proposed by Perzyna[24]. vpd ij ij f Fε φ σ ∂ ′∂ ， 4 where, the function of MacCauley is 0 0 0 . F F FF ≤⎧ ⎨ ⎩ ， The principles of the elasto-viscoplastic model called EVP-MCC model are illustrated by the effective stress path of an undrained triaxial test in pq ′− space see Fig. 2. The stress state “A” represents an initially normally 0 K consolidation state. Along the loading stress path “A-B-C” for constant strain rate test, viscoplastic volumetric strains occur during loading and cause the static yield surface to expand in the stress space. As point C approaches C’, corresponding to the critical state, the soil is subjected to a constant amount of overstress which provokes an increase of the deviatoric strain at constant strain rate, without any volumetric strain. Fig. 2 Schematic behaviour of the elasto-viscoplastic Modified .Cam Clay model during CAU triaxial compression and creep tests As for the creep stress path “B-D”, the static yield surface expands with the viscoplastic volumetric strain, function of the amount of overstress. If the static yield surface can reach the actual stress point, the equilibrium is obtained and the strain will be stabilized with time. If not, the effective stress will continue to evolve until it reaches the critical state at point D where it will stop because no viscoplastic volumetric strain will develop, but deviatoric strain will continue to increase. Taking into account the elastic stress-strain relations, the constitutive equations of the viscoplastic model for normally consolidated clays are derived as follows d c 2 3 2 233 ijijij ijij ss p Fpp GKM δ εδ φ ′′ ⎛⎞′ ′− ⎜⎟ ⎝⎠ ，5 where ij σ′ is the effective stress tensor, vp ij εis the inelastic strain rate tensor, ij s′ is the deviatoric stress tensor. An implicit backward Euler of stress update algorithm is used to integrate the viscoplastic stress-strain equations to a FEM code of CESAR-LCPC. 1.2 ology of parameter identification The proposed model involves the parameters of Cam Clay model{ } * c0 , ,,,EM pν β, and two additional parameters of viscosity{},N. The details about the sensitivity of the parameters of Cam Clay model will not be presented here since studies on them are well investigated and documented[26]. We can point out as 1 E , M , c0 p′have a significant effect on the stress-strain relationship; 2 ν has a very small effect on the stress-strain relationship; 3 * β has a very small effect on the stress-strain relationship for triaxial and pressuremeter tests but a significant effect for oedometer tests; 4 only E can change the initial slope of the strain-stress curve. Yin[26] found that the viscoplastic parameters of N and have an important effect on the computed stress, especially for520Nand 6 1 10 − . The parameters will be conducted with at least three different strain rates. Based on the parameters study[25], a general procedure to determine the soil parameters from laboratory tests is established and employed as follows 1 ν was taken equal to 0.3, which is a common value for clay. 2 E could be determined by the initial slope of the strain-stress curves. 3 * β could be determined from oedometer tests pered on the studied soil * 0 /1eβλκ−. 4 The slope M of the critical state line could be 第 6 期 YIN Zhen-yu, et al. Modelling of time-dependent behaviour of soft soils using simple elasto-viscoplastic model 883 Table 1 Values of EVP model and hydraulic parameters Sites w / IP / E /kPa ν β* M pc0 /kPa N s -1kPa k ms -1 Batiscan 80 21 5000 0.30 0.76 1.00 70 10 110 -9 1.010 -8 Burswood 83 4000 0.30 0.07 1.45 38 12 210 -7 3.310 -9 Flumet 4000 0.30 0.07 1.50 70 10 110 -8 1.010 -9 Fukakusa 22 25000 0.30 0.05 1.00 350 12 110 -10 Haney 18 35000 0.25 0.09 0.85 515 10 110 -7 Hongkong 52 32 20000 0.30 0.06 1.10 270 12 210 -8 Osaka 35000 0.30 0.14 1.20 500 12 510 -9 St-Herblain 121 42 2000 0.30 0.13 1.25 30 10 110 -9 1.010 -10 determined from triaxial tests. 5 c0 p′ could be determined by the fitting of several curves, knowing that its value was located between the initial consolidation pressure and the preconsolidation pressure measured by oedometer tests. However, its determination from oedometer tests is not straightforward, since its value depends on the loading rate, as it will be shown later. 6 N and could be determined from tests with more than two different strain rates, or tests with a relaxation or a creep stage. 7 k could be determined by fitting the pore pressure dissipation curves for tests with drained or partly drained conditions. 2 Modelling the time-dependent beh- avior of soft soils along different loading paths 2.1 General description To analyze the model capability of reproducing the time-dependent behaviour of soils, different types of tests found in the literatures are simulated, and the results were compared with the experimental data. For each type of test, the authors summarized the water content and the plastic index of the studied soil in Table 1. The permeability value was taken into account for tests with drained or partially drained conditions. 2.2 Effect of strain rate on preconsolidation stress Leroueil et al[5] presented constant strain rate oedometer tests on Batiscan clay with strain rate varying from test to test between 1.710 -8 s-1 and 410-5 s-1. The specimens were 19.0 mm high and 50.8 mm in diameter. The drainage was allowed only at the top of the specimens. The initial state was taken at v0 σ′ 65 kPa, a stress equal to the in situ vertical effective stress at the corresponding depth of the specimens. The values of the parameters determined from constant strain rate and creep tests are given in Table 1. The predictions and experimental results, shown in Fig. 3, are in good agreement for settlements less than 17. In particular, the EVP-MCC model can take into account the effect of strain rate on the preconsolidation pressure, which is more accurate than that described by the linear relationship Eq.6 between the preconsolidation pressure and the vertical strain rate assuming a constant ratio of αec /CC proposed by Leroueil et al[27], i.e., the EVP-MCC model improves the description of the effect of strain rate on the preconsolidation pressure compared with these models based on the ulation Eq. c p v11 v0 p v0 e C C v α σ ε εσ ′ ⎛⎞ ⎜ ⎟ ′ ⎝⎠ ， 6 where the index 0 ν, 1 νrepresent the two values corresponding to two different strain rates; αe C is the coefficient of secondary compression; c C is the compression index. However, for the settlements more than 17, there is a large difference between the measured and pr