A non‐coaxial constitutive model for sand deformation under rotation of principal stress axes

A constitutive model for the simulation of non‐coaxiality, an aspect of anisotropic behavior of sand subjected to the rotation of the principal stress axes, is presented in this paper. Experimental studies have shown that non‐coaxiality or non‐coincidence of principal plastic strain increments with principal stress axes under loadings involving the rotation of principal stress axes may be considerable. Besides, the rotation of the principal stress axes results in dramatic effects on stiffness and dilatant behavior of sand. Therefore, the consequences of principal stress axes rotation on deformational behavior, dilatancy and soil stiffness must be taken into account in theoretical and practical problems. To this aim, the following steps are taken: (1) A general relationship for flow direction with respect to possibility of non‐coaxial flow is developed. Moreover, special circumstances linking non‐coaxiality to instantaneous interaction between loading and soil fabric are proposed. (2) Proposing novel expressions for plastic modulus and dilatancy function, the model is enforced to provide realistic simulations when sand is subjected to the rotation of the principal stress axes. Finally, with numerous examples and comparisons, the model capabilities are shown under various stress paths and drainage conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

A critical state sand plasticity model accounting for fabric evolution

Fabric and its evolution need to be fully considered for effective modeling of the anisotropic behavior of cohesionless granular sand. In this study, a three‐dimensional anisotropic model for granular material is proposed based on the anisotropic critical state theory recently proposed by Li & Dafalias [2012], in which the role of fabric evolution is highlighted. An explicit expression for the yield function is proposed in terms of the invariants and joint invariants of the normalized deviatoric stress ratio tensor and the deviatoric fabric tensor. A void‐based fabric tensor that characterizes the average void size and its orientation of a granular assembly is employed in the model. Upon plastic loading, the material fabric is assumed to evolve continuously with its principal direction tending steadily towards the loading direction. A fabric evolution law is proposed to describe this behavior. With these considerations, a non‐coaxial flow rule is naturally obtained. The model is shown to be capable of characterizing the complex anisotropic behavior of granular materials under monotonic loading conditions and meanwhile retains a relatively simple formulation for numerical implementation. The model predictions of typical behavior of both Toyoura sand and Fraser River sand compare well with experimental data. Copyright © 2013 John Wiley & Sons, Ltd.     

Necessary and sufficient conditions for reaching and maintaining critical state

According to classical critical state theory (CST) of granular mechanics, two analytical conditions on the ratio of stress invariants and the void ratio are postulated to be necessary and sufficient for reaching and maintaining critical state (CS). The present work investigates the sufficiency of these two conditions based on the results of a virtual three-dimensional discrete element method experiment, which imposes continuous rotation of the principal axes of stress with fixed stress principal values at CS. Even though the fixity of the stress principal values satisfies the two analytical CST conditions at the initiation of rotation, contraction and abandonment of CS occur, which proves that these conditions may be necessary but are not sufficient to maintain CS. But if fixity of stress and strain rate directions in regard to the sample is considered at CS, the two analytical conditions of CST remain both necessary and sufficient. The recently proposed anisotropic critical state theory (ACST) turned this qualitative requirement of fixity into an analytical condition related to the CS value of a fabric anisotropy variable A defined in terms of an evolving fabric tensor and the plastic strain rate direction, thus, enhancing the two CST conditions by a third. In this way, the three analytical conditions of ACST become both necessary and sufficient for reaching and maintaining CS. In addition, the use of A explains the observed results by relating the stress-strain response, in particular the dilatancy, to the evolution of fabric by means of the relevant equations of ACST.     

软黏土的各向异性临界状态模型

剑桥模型只适用于正常固结软黏土,不能描述不等向固结土的应力-应变行为的各向异性特性。基于剑桥模型,在其椭圆屈服面中引入各向异性张量和一个形状参数,建立了一个各向异性屈服面,提出了一个适用于等向和不等向固结软黏土的本构模型。各向异性张量的初始值由初始固结应力状态确定,其演化过程由一个与塑性剪应变和塑性体应变都有关的硬化法则描述。形状参数的引入保证了各向异性屈服面的灵活性和适应性。通过对Boston Blue黏土、高岭土和Otaniemi黏土的三轴试验结果的模拟,验证了模型的模拟能力。     

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.     

宏细观结合考虑主应力轴旋转的砂土破坏特性

采用宏细观结合各向异性破坏准则对主应力轴旋转条件下砂土的破坏特性进行分析。该准则是加载应力、组构各向异性程度和应力与组构几何关系3个因素的函数,可描述细观特性对任意应力旋转角度条件下破坏特性的影响。根据空心圆柱扭剪试验的特点推导一般正交坐标系下主应力轴旋转条件下的破坏关系式,考虑应力与砂土细观组构的几何关系,推导的关系式即可分析该条件下破坏特性。材料为各向异性时,主应力轴旋转造成破坏特性发生变化,细观各向异性程度越小变化越小;材料为各向同性时,则不会造成砂土破坏特性的变化。该式表明主应力轴旋转条件下不同破坏特性存在的根本原因是砂土各向异性的存在。采用空心圆柱试验结果进行验证,结果表明建立的关系式能较好描述不同应力加载角度条件下砂土的破坏特性。初步验证了由于砂土各向异性的存在使得主应力轴旋转造成了不同的破坏规律。     

CASM: a unified state parameter model for clay and sand

The purpose of this paper is to present a simple, unified critical state constitutive model for both clay and sand. The model, called CASM (Clay And Sand Model), is formulated in terms of the state parameter that is defined as the vertical distance between current state (v, p′) and the critical state line in v–ln p′ space. The paper first shows that the standard Cam-clay models (i.e. the original and modified Cam-clay models) can be reformulated in terms of the state parameter. Although the standard Cam-clay models prove to be successful in modelling normally consolidated clays, it is well known that they cannot predict many important features of the behavior of sands and overconsolidated clays. By adopting a general stress ratio-state parameter relation to describe the state boundary surface of soils, it is shown that a simple, unified constitutive model (CASM) can be developed for both clay and sand. It is also demonstrated that the standard Cam-clay yield surfaces can be either recovered or approximated as special cases of the yield locus assumed in CASM. The main feature of the proposed model is that a single set of yield and plastic potential functions has been used to model the behaviour of clay and sand under both drained and undrained loading conditions. In addition, it is shown that the behaviour of overconsolidated clays can also be satisfactorily modelled. Simplicity is a major advantage of the present state parameter model, as only two new material constants need to be introduced when compared with the standard Cam-clay models. © 1998 John Wiley & Sons, Ltd.     

A hypoplastic model for granular soils incorporating anisotropic critical state theory

It is well known that soil is inherently anisotropic and its mechanical behavior is significantly influenced by its fabric anisotropy. Hypoplasticity is increasingly being accepted in the constitutive modeling for soils, in which many salient features, such as nonlinear stress-strain relations, dilatancy, and critical state failure, can be described by a single tensorial equation. However, within the framework of hypoplasticity, modeling fabric anisotropy remains challenging, as the fabric and its evolution are often vaguely assumed without a sound basis. This paper presents a hypoplastic constitutive model for granular soils based on the newly developed anisotropic critical state theory, in which the conditions of fabric anisotropy are concurrently satisfied along with the traditional conditions at the critical state. A deviatoric fabric tensor is introduced into the Gudehus-Bauer hypoplastic model, and a scalar-valued anisotropic state variable signifying the interplay between the fabric and the stress state is used to characterize its impact on the dilatancy and strength of the soils. In addition, fabric evolution during shearing can explicitly be addressed. Modifications have also been undertaken to improve the performance of the undrained response of the model. The anisotropic hypoplastic model can simulate experimental tests for sand under various combinations of principle stress direction, intermediate principal stress (or mode of shearing), soil densities, and confining pressures, and the associated drastic effect of different principal stress orientations in reference to the material axes of anisotropy can be well captured.     

An anisotropic elastoplastic model for soft clays based on logarithmic contractancy

A new constitutive model for soft structured clays is developed based on an existing model called S‐CLAY1S, which is a Cam clay type model that accounts for anisotropy and destructuration. The new model (E‐SCLAY1S) uses the framework of logarithmic contractancy to introduce a new parameter that controls the shape of the yield surface as well as the plastic potential (as an assumed associated flow rule is applied). This new parameter can be used to fit the coefficient of earth pressure at rest, the undrained shear strength or the stiffness under shearing stress paths predicted by the model. The improvement to previous constitutive models that account for soil fabric and bonding is formulated within the contractancy framework such that the model predicts the uniqueness of the critical state line and its slope is independent of the contractancy parameter. Good agreement has been found between the model predictions and published laboratory results for triaxial compression tests. An important finding is that the contractancy parameter, and consequently the shape of the yield surface, seems to change with the degree of anisotropy; however, further study is required to investigate this response. From published data, the yield surface for isotropically consolidated clays seems ‘bullet’ or ‘almond’ shaped, similar to that of the Cam clay model; while for anisotropically consolidated clays, the yield surface is more elliptical, like a rotated and distorted modified Cam clay yield surface. © 2015 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons Ltd.     

中主应力系数影响下的冻结砂土损伤本构模型

人工冻结法是饱水砂层开挖过程中常用的止水和临时支护方法,通过冻土损伤特性研究为冻土力学特性和冻结体稳定性分析奠定基础。为研究冻结砂土的损伤力学特性,在-5℃下进行了不同中主应力系数的冻结砂土三维室内试验。从冻土微元破坏服从Weibull随机分布的特点出发,将Drucker-Prager强度准则作为冻土微元统计分布变量,利用应变等价性假说,建立了三维应力状态下冻结砂土损伤本构模型;在此基础上,讨论模型参数F₀与m和中主应力系数的关系,对模型参数进行合理修正,建立中主应力系数影响下的冻结砂土损伤本构模型,并与试验结果进行对比。分析结果表明：参数F₀和m随着中主应力系数的增大呈现先减小后增大的趋势;参数F₀反映了冻结砂土的强度特性,参数m代表了冻结砂土的延性及脆性特征,考虑中主应力系数影响的冻结砂土损伤本构模型能很好地模拟冻结砂土应力-应变全过程曲线。研究成果为人工冻结法工程设计提供一定的理论依据。移动阅读     

循环荷载作用下黏土改进边界面模型

为克服边界面模型应用不便、不能反映卸载过程中土体的弹塑性性质的缺陷,提出了一个循环荷载作用下可考虑弹塑性加、卸载过程的重塑黏土改进边界面模型。模型采用了简单的边界面形式,可大大简化该模型理论的推导与计算;采取了不预先定义边界面大小的方式,有效地降低了改进边界面模型在应用过程中的经验性;加入边界面胀缩规则,使模型可考虑土样卸载过程中的弹塑性性质,进而可反映土样的滞回特性。通过对重塑黏土进行循环三轴数值模拟试验,并与真实试验结果进行对比分析,验证了改进边界面模型的合理性和有效性。数值验证结果表明,改进边界面模型具有物理意义明确、参数易于确定、形式相对简单的特点,且该模型计算精度较高,计算结果与试验结果吻合得较好。     

A bounding surface model for anisotropically overconsolidated clay incorporating thermodynamics admissible rotational hardening rule

A bounding surface model is formulated to simulate the behavior of clays that are subject to an anisotropic consolidation stress history. Conventional rotational hardening is revisited from the perspective of thermodynamics. As the free energy cannot be accumulated infinitely upon critical state failure, the deviatoric back stress must vanish. This requires the rotated yield surface to be turned back to eventually align on the hydrostatic axis in the stress plane. Noting that most of the previous propositions violate this restriction, an innovative rotational hardening rule is formulated that is thermodynamically admissible. The bounding surface framework that employs the modified yield surface is applied to simulate elastoplastic deformations for overconsolidated clays, with which the overprediction of strength on the “dry” side can be greatly improved with reasonable results. Other important features, including contractive or dilative response and hardening or softening behavior, can also be well-captured. It has been shown that the model can simulate three types of reconstituted clays that are sheared with initial conditions over a wide range of anisotropic consolidation stress ratios and overconsolidation ratios under both triaxial undrained and drained conditions. Limitations and potential improvement of the model regarding the fabric anisotropy at critical state have been discussed.     

结构性粘土的边界面砌块体模型

把天然粘土的变形看作是由结构的破坏引起的。总的变形由结构性粘土的弹性变形、结构面的滑移塑性变形和结构体破损引起的的损伤塑性变形组成。把滑移屈服面看作可以扩大、旋转的运动硬化面,并作为边界面,通过内插塑性模量来描述滑移塑性变形。采用损伤边界面,通过规定加荷或反向加荷产生损伤变形,卸荷不产生损伤变形来描述损伤塑性变形。建立了结构性粘土的边界面砌块体模型,并用试验进行了验证。     

Formulation of a unified constitutive model for clays and sands

This paper presents a new generalized effective stress model, referred to as MIT-S1, which is capable of predicting the rate independent, effective stress–strain–strength behaviour of uncemented soils over a wide range of confining pressures and densities. Freshly deposited sand specimens compressed from different initial formation densities approach a unique condition at high stress levels, referred to as the limiting compression curve (LCC), which is linear in a double logarithmic void ratio, e, mean effective stress space, p′. The model describes irrecoverable, plastic strains which develop throughout first loading using a simple four-parameter elasto-plastic model. The shear stiffness and strength properties of sands in the LCC regime can be normalized by the effective confining pressure and hence can be unified qualitatively, with the well-known behaviour of clays that are normally consolidated from a slurry condition along the virgin consolidation line (VCL). At lower confining pressures, the model characterizes the effects of formation density and fabric on the shear behaviour of sands through a number of key features: (a) void ratio is treated as a separate state variable in the incrementally linearized elasto-plastic formulation: (b) kinematic hardening describing the evolution of anisotropic stress–strain properties: (c) an aperture hardening function controls dilation as a function of ‘formation density’; and (d) the use of a single lemniscate-shaped yield surface with non-associated flow. These features enable the model to describe characteristic transitions from dilative to contractive shear response of sands as the confining pressure increases. This paper summarizes the procedures used to select input parameters for clays and sands, while a companion paper compares model predictions with measured data to illustrate the model capability for describing the shear behaviour of clays and sands. Copyright © 1999 John Wiley & Sons, Ltd.     

Explicit integration of bounding surface model for the analysis of earthquake soil liquefaction

This paper presents a new plasticity model developed for the simulation of monotonic and cyclic loading of non‐cohesive soils and its implementation to the commercial finite‐difference code FLAC, using its User‐Defined‐Model (UDM) capability. The new model incorporates the framework of Critical State Soil Mechanics, while it relies upon bounding surface plasticity with a vanished elastic region to simulate the non‐linear soil response. Stress integration of constitutive relations is performed using a recently proposed explicit scheme with automatic error control and substepping, which so far has been employed in the literature only for constitutive models aiming at monotonic loading. The overall accuracy of this scheme is evaluated at element level by simulating cyclic loading along complex stress paths and by using iso‐error maps for paths involving change of the Lode angle. The performance of the new constitutive model and its stress integration scheme in complex boundary value problems involving earthquake‐induced liquefaction is evaluated, in terms of accuracy and computational cost, via a number of parametric analyses inspired by the successful simulation of the VELACS centrifuge Model Test No. 2 studying the lateral spreading response of a liquefied sand layer. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Bayesian model selection for sand with generalization ability evaluation

Current studies have focused on selecting constitutive models using optimization methods or selecting simple formulas or models using Bayesian methods. In contrast, this paper deals with the challenge to propose an effective Bayesian-based selection method for advanced soil models accounting for the soil uncertainty. Four representative critical state-based advanced sand models are chosen as database of constitutive model. Triaxial tests on Hostun sand are selected as training and testing data. The Bayesian method is enhanced based on transitional Markov chain Monte Carlo method, whereby the generalization ability for each model is simultaneously evaluated, for the model selection. The most plausible/suitable model in terms of predictive ability, generalization ability, and model complexity is selected using training data. The performance of the method is then validated by testing data. Finally, a series of drained triaxial tests on Karlsruhe sand is used for further evaluating the performance.     

基于广义塑性力学的土体次加载面循环塑性模型(Ⅰ): 理论与模型

简要地介绍了次加载面理论的基本思想、假设及其物理解释。在广义塑性力学的框架内，引入次加载面的思想，把常规的椭圆-抛物线双屈服面模型，扩展为次加载面循环塑性模型，以反映循环荷载作用下土体的曼辛效应与棘轮效应。模型能考虑塑性应变增量对应力增量的相关性，既能反映土体的循环加载特性，又能反映正常固结土和超固结土的单调加载特性。