A high strain‐rate constitutive model for sand and its application in finite‐element analysis of tunnels subjected to blast

The paper presents a constitutive model for simulating the high strain‐rate behavior of sands. Based on the concepts of critical‐state soil mechanics, the bounding surface plasticity theory and the overstress theory of viscoplasticity, the constitutive model simulates the high strain‐rate behavior of sands under uniaxial, triaxial and multi‐axial loading conditions. The model parameters are determined for Ottawa and Fontainebleau sands, and the performance of the model under extreme transient loading conditions is demonstrated through simulations of split Hopkinson pressure bar tests up to a strain rate of 2000/s. The constitutive model is implemented in a finite‐element analysis software Abaqus to analyze underground tunnels in sandy soil subjected to internal blast loads. Parametric studies are conducted to examine the effect of relative density and type of sand and of the depth of tunnel on the variation of stresses and deformations in the soil adjacent to the tunnels. Copyright © 2012 John Wiley & Sons, Ltd.     

考虑组构变化效应的饱和密砂循环本构模型

为了真实地描述饱和密砂在循环加载过程中的变形行为,需要引入考虑剪胀阶段组构变化的宏观参量。在已有的基于状态参量的本构模型基础上,引入反映组构变化的剪胀内变量,简称组构-剪胀内变量z。以相变线PTL作为参考线,采用基于相变的状态参量判断砂土在初始时刻和任意时刻体积变形的变化趋势,并通过z对剪胀比d的影响,考虑反向加载过程中塑性变形的累积,建立了一个针对饱和密砂的循环加载的弹塑性本构模型。该模型根据试验现象将已有模型中的塑性剪切模量区分为首次加载模量与再加载模量,能较好地模拟排水情况下砂土循环加载的胀-缩变化过程。最后,针对密砂的三轴排水情况,利用文中模型进行预测,并把预测结果与试验结果进行比较,结果表明该模型能够总体反映砂土循环加载的变形行为。     

Constitutive model for cyclic behaviour of cohesionless sands

This paper presents a constitutive model for describing the stress-strain response of sands under cyclic loading. The model, formulated using the critical state theory within the bounding surface plasticity framework, is an upgraded version of an existing model developed for monotonic behaviour of cohesionless sands. With modification of the hardening law, plastic volumetric strain increment and unloading plastic modulus, the original model was modified to simulate cyclic loading. The proposed model was validated against triaxial cyclic loading tests for Fuji River sand, Toyoura sand and Nigata sand. Comparison between the measured and predicted results suggests that the proposed modified model can capture the main features of cohesionless sands under drained and undrained cyclic loading.     

考虑颗粒破碎的钙质砂修正邓肯-张E-B模型

在常规应力水平下颗粒发生破碎是钙质砂有别于其他砂土的重要性质之一,且由于颗粒破碎的存在,使用传统的本构模型无法很好地模拟钙质砂的力学行为。因此本文以最为普及的本构模型之一--邓肯-张E-B模型为基础,对其进行颗粒破碎方面的修正以得到一个能用于钙质砂的本构模型。具体方法为:首先本文采用Hardin提出的相对破碎Br这一指标来度量颗粒破碎的大小。之后研究分析得出了颗粒破碎对邓肯-张模型参数(内摩擦角φ、割线模量E50及体积模量B)的影响规律。然后通过颗粒破碎与输入能量之间的关系将各状态下无法直接确定的相对破碎Br与可确定的应力-应变状态联系起来。最终得到了一个考虑颗粒破碎的钙质砂修正邓肯-张E-B模型。为验证模型的准确性及适用性,本文还使用该模型对4种不同粒径范围且试验围压不同的钙质砂的三轴排水行为进行了模拟。结果表明拟合效果较好,模型能适用于各种不同粒径范围的钙质砂,并且在颗粒破碎较大的情况下明显优于传统邓肯-张模型。     

A modification to dense sand dynamic simulation capability of Pastor–Zienkiewicz–Chan model

A modification to the nonlinear Pastor–Zienkiewicz–Chan (PZC) constitutive model without any change in the number of model parameters is introduced in order to simulate stiffness degradation of dense sands at dynamic loading. The PZC model is based on generalized plasticity and was verified by good prediction of liquefaction and undrained behavior of saturated sand. The PZC is a robust model that can predict drained dynamic behavior of sands, especially stiffness increase in loose sand at reloading of dynamic loading. Yet, this model does not show stiffness degradation of dense sand at reloading. The modification is made through modifying the stress memory factor, H _DM, which is multiplied by the plastic modulus, H _L. This modification does not influence reloading behavior of loose sand. The modified PZC model is verified via results of drained cyclic tests. Two cyclic triaxial tests on loose and dense specimens, along with two cyclic plane strain tests on dense sand are utilized for validation. The model simulation shows that the modified PZC model is able to predict the stiffness degradation of dense sand at reloading well.     

A constitutive model for sand with anisotropic hardening rule

An anisotropic hardening model for sand or other granular materials is presented. The state of initial densification is represented by a configuration surface and the state of loading by a loading surface in the stress space. The domain of elastic response may not exist and the irreversible strain is assumed to occur for both active and reverse loading. The derived constitutive equations are applicable both for dense and loose sands. Some model predictions are compared with available experimental data for triaxial compression and extension under undrained condition.     

A two-surface plasticity model for clay

This paper presents a two-surface plasticity constitutive model for clays based on critical-state soil mechanics. The model reproduces the mechanical response of clays under multi-axial loading conditions and predicts both drained and undrained behavior at small and large strains. The constitutive model also captures both the strain-rate-dependent behavior of clays and the drop in strength towards a residual value at very large shear strains using novel approaches. The paper also describes a hierarchical process for the determination of the model parameters relying more on simple curve fitting of model equations to experimental data points corresponding to specific soil states instead of trial-and-error simulations of entire experiments. Model parameter values are determined for London Clay, San Francisco Bay Mud, Boston Blue Clay and Lower Cromer Till, and the performance of the model in simulating mechanical response of clays is demonstrated for a variety of initial states and loading conditions.     

Development of a generalized model using a new plastic modulus based on bounding surface plasticity

Numerous attempts have been made to modify the generalized constitutive model and to introduce new constitutive models in the framework of generalized plasticity. The modified models can predict the behavior of sand fairly well, however, such models require many parameters and are difficult to calibrate. Moreover, it is highly desirable for a model to be able to reproduce soil behavior using a single set of parameters. In this paper, the constitutive model by Pastor and Zienkiewicz is further developed based on critical state and bounding surface models. The model is used to simulate the behavior of three types of sands under monotonic and cyclic loadings.     

Predictive potential of Perzyna viscoplastic modelling for granular geomaterials

This paper reappraises Perzyna-type viscoplasticity for the constitutive modelling of granular geomaterials, with emphasis on the simulation of rate/time effects of different magnitude. An existing elasto-plastic model for sands is first recast into a Perzyna viscoplastic formulation and then calibrated/validated against laboratory test results on Hostun sand from the literature. Notable model features include (1) enhanced definition of the viscous nucleus function and (2) void ratio dependence of stiffness and viscous parameters, to model the pycnotropic behaviour of granular materials with a single set of parameters, uniquely identified against standard creep and triaxial test results. The comparison between experimental data and numerical simulations points out the predicative capability of the developed model and the complexity of defining a unique viscous nucleus function to capture sand behaviour under different loading/initial/boundary and drainage conditions. It is concluded that the unified viscoplastic simulation of both drained and undrained response is particularly challenging within Perzyna's framework and opens to future research in the area. The discussion presented is relevant, for instance, to the simulation of multiphase strain localisation phenomena, such as those associated to slope stability problems in variably saturated soils.     

An AI-based model for describing cyclic characteristics of granular materials

Modelling cyclic behaviour of granular soils under both drained and undrained conditions with a good performance is still a challenge. This study presents a new way of modelling the cyclic behaviour of granular materials using deep learning. To capture the continuous cyclic behaviour in time dimension, the long short-term memory (LSTM) neural network is adopted, which is characterised by the prediction of sequential data, meaning that it provides a novel means of predicting the continuous behaviour of soils under various loading paths. Synthetic datasets of cyclic loading under drained and undrained conditions generated by an advanced soil constitutive model are first employed to explore an appropriate framework for the LSTM-based model. Then the LSTM-based model is used to estimate the cyclic behaviour of real sands, ie, the Toyoura sand under the undrained condition and the Fontainebleau sand under both undrained and drained conditions. The estimates are compared with actual experimental results, which indicates that the LSTM-based model can simultaneously simulate the cyclic behaviour of sand under both drained and undrained conditions, ie, (a) the cyclic mobility mechanism, the degradation of effective stress and large deformation under the undrained condition, and (b) shear strain accumulation and densification under the drained condition.     

A middle surface concept (MSC) model for saturated sands in general stress space

An elastoplastic constitutive model is proposed for saturated sands in general stress space using the middle surface concept (MSC). In MSC, different features of stress–strain response of a material are divided into different pseudo‐yield surfaces. The true‐yield surface representing the true response is established by using various links between the yield surfaces. In this MSC sand model, several well‐known features of sand response are represented by three different pseudo‐yield surfaces, which are developed in a simple and straightforward way. These features include the critical state behaviour, the effects of state parameter, unloading and reloading plastic deformation, the influence of fabric anisotropy, and phase transformation line related behaviour. Finally, the model predictions and test results are compared for two different types of sands under a variety of loading conditions and good comparisons are obtained. The application of MSC to saturated sand modelling shows the versatility of MSC as a general concept for modelling stress–strain response of materials. Copyright © 2006 John Wiley & Sons, Ltd.     

A Generalized Plasticity-Based Model for Sandstone Considering Time-Dependent Behavior and Wetting Deterioration

Based on the concept of generalized plasticity, this study proposes a constitutive model to describe the time-dependent behavior and wetting deterioration of sandstone. The proposed model (1) exhibits nonlinear elasticity under hydrostatic and shear loading, (2) follows the associated flow rule for viscoplastic deformation, (3) adopts a creep modulus that varies with the stress ratio, (4) considers the primary and secondary creep behaviors of rock, and (5) considers the effect of wetting deterioration. This model requires 13 material parameters, comprising 3 for elasticity, 7 for plasticity, and 3 for creep. All parameters can be determined easily by following the suggested procedures. The proposed model is first validated by comparison with triaxial tests of sandstone under different hydrostatic stress and cyclic loading conditions. In addition, the model is versatile in simulating time-dependent behavior through a series of multistage creep tests. Finally, to consider the effects of wetting deterioration, triaxial and creep tests under dry and water-saturated conditions are simulated. Comparison of the simulated and experimental data shows that the proposed model can predict the behavior of sandstone in dry and saturated conditions.     

Modeling of the simple shear deformation of sand: effects of principal stress rotation

The paper presents a simple constitutive model for the behavior of sands during monotonic simple shear loading. The model is developed specifically to account for the effects of principal stress rotation on the simple shear response of sands. The main feature of the model is the incorporation of two important effects of principal stress on stress–strain response: anisotropy and non-coaxiality. In particular, an anisotropic failure criterion, cross-anisotropic elasticity, and a plastic flow rule and a stress–dilatancy relationship that incorporate the effects of non-coaxiality are adopted in the model. Simulations of published experimental results from direct simple shear and hollow cylindrical torsional simple shear tests on sands show the satisfactory performance of the model. It is envisioned that the model can be valuable in modeling in situ simple shear response of sands and in interpreting simple shear test results.     

Modeling cyclic shearing of sands in the semifluidized state

Liquefaction is associated with the loss of mean effective stress and increase of the pore water pressure in saturated granular materials due to their contractive tendency under cyclic shear loading. The loss of mean effective stress is linked to loss of grain contacts, bringing the granular material to a “semifluidized state” and leading to development and accumulation of large cyclic shear strains. Constitutive modeling of the cyclic stress-strain response in earthquake-induced liquefaction and post-liquefaction is complex and yet very important for stress-deformation and performance-based analysis of sand deposits. A new state internal variable named strain liquefaction factor is introduced that evolves at low mean effective stresses, and its constitutive role is to reduce the plastic shear stiffness and dilatancy while maintaining the same plastic volumetric strain rate in the semifluidized state. This new constitutive ingredient is added to an existing critical state compatible, bounding surface plasticity reference model, that is well established for constitutive modeling of cyclic response of sands in the pre-liquefaction state. The roles of the key components of the proposed formulation are examined in a series of sensitivity analyses. Their combined effects in improving the performance of the reference model are examined by simulating undrained cyclic simple shear tests on Ottawa sand, with focus on reproducing the increasing shear strain amplitude as well as its saturation in the post-liquefaction response.     

Generalized plasticity state parameter‐based model for saturated and unsaturated soils. Part 1: Saturated state

The behavior of granular materials is known to depend on its loose or dense nature, which in turns depends both on density and confining pressure. Many models developed in the past require the use of different sets of constitutive parameters for the same material under different confining pressures. The purpose of this paper is to extend a basic generalized plasticity model for sands proposed by Pastor, Zienkiewicz and Chan by modifying the main ingredients of the model flow—rule, loading–unloading discriminating direction and plastic modulus—to include a dependency on the state parameter. The proposed model is tested against the available experimental data on three different sands, using for each of them a single set of material parameters, finding a reasonably good agreement between experiments and predictions. Copyright © 2010 John Wiley & Sons, Ltd.     

An experimental database for the development,calibration and verification of constitutive models for sand with focus to cyclic loading: part I—tests with monotonic loading and stress cycles

For numerical studies of geotechnical structures under earthquake loading, aiming to examine a possible failure due to liquefaction, using a sophisticated constitutive model for the soil is indispensable. Such a model must adequately describe the material response to a cyclic loading under constant volume (undrained) conditions, amongst others the relaxation of effective stress (pore pressure accumulation) or the effective stress loops repeatedly passed through after a sufficiently large number of cycles (cyclic mobility, stress attractors). The soil behaviour under undrained cyclic loading is manifold, depending on the initial conditions (e.g. density, fabric, effective mean pressure, stress ratio) and the load characteristics (e.g. amplitude of the cycles, application of stress or strain cycles). In order to develop, calibrate and verify a constitutive model with focus to undrained cyclic loading, the data from high-quality laboratory tests comprising a variety of initial conditions and load characteristics are necessary. The purpose of these two companion papers was to provide such database collected for a fine sand. The database consists of numerous undrained cyclic triaxial tests with stress or strain cycles applied to samples consolidated isotropically or anisotropically. Monotonic triaxial tests with drained or undrained conditions have also been performed. Furthermore, drained triaxial, oedometric or isotropic compression tests with several un- and reloading cycles are presented. Part I concentrates on the triaxial tests with monotonic loading or stress cycles. All test data presented herein will be available from the homepage of the first author. As an example of the examination of an existing constitutive model, the experimental data are compared to element test simulations using hypoplasticity with intergranular strain.     

Piles in fully liquefied soils with lateral spread

The paper provides a new analysis procedure for the assessment of the lateral response of isolated piles/drilled shafts in saturated sands as liquefaction and lateral soil spread develop in response to dynamic loading such as that generated by the earthquake shaking. The presented method accounts for: (1) the development of full liquefaction in the free-field soil that could trigger the lateral spread of the overlying crust layer; (2) the driving force exerted by the crust layer based on the interaction between the pile and the upper non-liquefied soil (crust) layer; and (3) the variation of the excess pore water pressure (i.e. post-liquefaction soil strength) in the near-field soil with the progressive pile deflection under lateral soil spread driving force. A constitutive model for fully liquefied sands under monotonic loading and undrained conditions is developed in order to predict the zone of post-liquefaction zero-strength of liquefied sand before it rebounds with the increasing soil strain in the near-field. The analytical and empirical concepts employed in the Strain Wedge (SW) model allow the modeling of such a sophisticated phenomenon of lateral soil spread that could accompany or follow the occurrence of seismic events without using modifying parameters or shape corrections to account for soil liquefaction.     

An experimental database for the development,calibration and verification of constitutive models for sand with focus to cyclic loading: part II—tests with strain cycles and combined loading

For numerical studies of geotechnical structures under earthquake loading, aiming to examine a possible failure due to liquefaction, using a sophisticated constitutive model for the soil is indispensable. Such model must adequately describe the material response to a cyclic loading under constant volume (undrained) conditions, amongst others the relaxation of effective stress (pore pressure accumulation) or the effective stress loops repeatedly passed through after a sufficiently large number of cycles (cyclic mobility, stress attractors). The soil behaviour under undrained cyclic loading is manifold, depending on the initial conditions (e.g. density, fabric, effective mean pressure, stress ratio) and the load characteristics (e.g. amplitude of the cycles, application of stress or strain cycles). In order to develop, calibrate and verify a constitutive model with focus to undrained cyclic loading, the data from high-quality laboratory tests comprising a variety of initial conditions and load characteristics are necessary. It is the purpose of these two companion papers to provide such database collected for a fine sand. Part II concentrates on the undrained triaxial tests with strain cycles, where a large range of strain amplitudes has been studied. Furthermore, oedometric and isotropic compression tests as well as drained triaxial tests with un- and reloading cycles are discussed. A combined monotonic and cyclic loading has been also studied in undrained triaxial tests. All test data presented herein will be available from the homepage of the first author. As an example of the examination of an existing constitutive model, the experimental data are compared to element test simulations using hypoplasticity with intergranular strain.     

Recognition of an Elastic–Plastic Constitutive Law by a Multiobjective Evolutionary Algorithm

This study presents the recognition of an elastic–plastic constitutive law by a multiobjective evolutionary algorithm (MOEA). This idea is illustrated by the identification of ellipse aspect ratio and plastic modulus of a reported bounding surface model. Based on the multi-goals of well predicting all available drained or undrained stress–strain behaviors simultaneously, the compromising solutions of these two parameters are found by a strength Pareto evolutionary algorithm 2 (SPEA2). Their fittest values are then determined by additionally introducing the Akaike information criterion. Experimental data for the Ottawa sand are used to test such processes. The results show that an MOEA is an efficient and automatic tool to identify the fittest form of an elastic–plastic constitutive law from a large amount of experimental data. However, sufficient data are required to determine the correct searching range of parameters to be identified.     

Unified 3D critical state bounding-surface plasticity model for soils incorporating continuous plastic loading under cyclic paths. Part II: Calibration and simulations

A systematic calibration procedure for the constitutive model gUTS using conventional triaxial data for all but one of the material constants is described. Typical ranges for the constants for clay and sand are specified together with default values. When all default values are adopted, just eight material constants need to be determined (gUTS-lite). A comprehensive series of 49 simulations on clays, silt and sands in loose and dense states under a wide range of monotonic and cyclic stress paths, initial states and drainage conditions provide very satisfactory agreement with experimental results.