A generalised D matrix for anisotropic elastic granular media

A matrix relating stress and elastic strain tensors for anisotropic particulate materials has been derived. The magnitude of the matrix depends on the state of the material anisotropy. Anisotropy in granular materials depends on strain because normal and tangential particle contact forces, as well as the spatial distribution of the contacts, vary with stress and strain. However, the rotation tensor and the strain tensor cannot be independent; they must satisfy certain constraints to meet the requirement for macroscopic stress tensor symmetry. These conditions and constraints lead to the derivation of the matrix presented in this article. The principal directions of the stress tensor and strain tensor are generally not coincident, and the values of deformation parameters, Young's modulus and Poisson's ratio, are direction dependent; these two aspects are also discussed in this paper. Whereas this matrix can be used in static numerical analyses for elastic problems, we note that this relationship can also be used as a basis upon which to derive a fully incremental stress–strain relationship for anisotropic granular materials in the plastic state, where the anisotropy is evolving with strain.     

从各向异性的角度解释和模拟土的非共轴特性

从发挥面的角度出发,分析论证各向异性是引起岩土材料出现非共轴现象的根本原因,得到与材料力学一致的结论。当共轭的两发挥面与沉积面的夹角不相等时,主应力面上将出现塑性应变增量的切向分量,所以塑性应变增量的主方向与应力的主方向非共轴。按照这一结论,对非共轴的数值模拟,也应当根据各向异性本构模型进行。为考虑各向异性影响新近提出的各向异性变换应力法,改变了各应力分量的相对大小,得到的各向异性变换应力张量与真实应力张量的主方向不一致,因此也能反映非共轴。利用各向异性变换应力法,能够在现有的弹塑性本构模型的框架下,描述土的非共轴现象。以各向异性UH模型为例,预测各种加载条件下的非共轴变形,验证了该方法的有效性。     

基于内变量和张量函数表示定理的本构方程

针对各向同性材料,基于张量函数表示定理,建立了本构关系的张量不变性表示,其中,3个不可约基张量取决于应力的0～2次幂,且相互正交,3个系数由塑性应变增量和应力的不变量表示。基于塑性应变增量的不变量定义内变量,本构关系归结为确定内变量的演化。使用张量函数表示定理,给出了内变量演化方程的一般表达式,它取决于应力不变量的增量,因而与主轴旋转无关。讨论了如何根据试验资料和引入适当的假定,确定具体的演化方程。通过与塑性势理论和多重屈服面理论进行比较,表明所建模型是这些理论的最一般表示,且简捷直观、使用方便。     

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.     

Discrete element modelling of material non‐coaxiality in simple shear flows

We investigate the quasi‐static simple shear flow of a two‐dimensional assembly of cohesionless particles using discrete element method (DEM) simulations. We focus on the unsteady flow regime where the solid would experience significant evolution of stresses, mobilised shear strength and dilation. We construct the DEM model using a discretised‐wall confined granular cell where the apparent boundary is allowed to dilate or contract synchronously with the confined solid. A rather uniform simple shear field is achieved across the whole assembly, which benefits rheological studies in generalising constitutive laws for continuum methods. We examine two aspects of the simple shear behaviour: macroscopic stress and strain rate evolution, particularly the non‐coaxiality between the principal directions of the two; and micromechanics such as evolution of fabric. For an initially anisotropic specimen sheared under constant normal pressure, the direction of principal stress rotates towards that of the principal strain rate, gradually reducing the degree of non‐coaxiality from about 45° to fluctuating around 0°. The rate in approaching coaxiality is slower in samples with larger initial porosity, stress ratio and mean stress. Generally, a faster rate in approaching coaxiality in simple shear is observed in a more dilatant sample, which often shows a larger degree of mobilised fabric anisotropy, suggesting the possible important role of instantaneous internal friction angle. The evolution of principal fabric direction resembles that of the principal stress direction. © 2013 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons, Ltd.     

The role of non‐coaxiality in the simulation of strain localization based on classical and Cosserat continua

It is normally accepted that materials inside the shear band undergo severe rotation of the principal stress direction, which causes non‐coaxiality between the principal stress and principal plastic strain rate. However, classical plasticity flow theory implicitly assumes that the principal stress and the principal plastic strain rate are coaxial; thus, it may not correctly predict the onset of the shear band. In addition, classical continuum does not contain any internal length scales; as a result, it cannot provide a reasonable shear band thickness. In this study, the original vertex non‐coaxial plastic model based on the classical continuum is extended to the Cosserat continuum. The corresponding codes are implemented via the interface of the user defined element subroutine in ABAQUS. Through a simple shear test, the effectiveness of the user's codes is verified. Through a uniaxial compression test, the influence of non‐coaxiality on the onset, the orientation, and the thickness of the shear band is investigated. Results show that the onset of the shear localization is delayed, and the thickness of the shear band is widened when the non‐coaxial degree increases, while the orientation of the shear band is little affected by the non‐coaxial degree. In addition, it is found that the non‐coaxiality can weaken the micro‐polar effect to some extent; nonetheless, the Cosserat non‐coaxial model still has its advantage over the classical non‐coaxial model in capturing the pre‐bifurcation as well as the post‐bifurcation behaviors of strain localization. Copyright © 2016 John Wiley & Sons, Ltd.     

基于广义位势理论的土的非共轴特性研究

传统的塑性位势理论隐含了应力主方向和塑性应变增量主方向共轴的假定,无法客观地描述主应力轴旋转过程中的非共轴现象。基于广义位势理论提出的拟弹性弹塑性本构模型,把总的塑性应变分解为满足弹性分解准则的拟弹性部分和符合传统塑性理论假设的纯塑性部分,分解后建立的模型更为合理和简便,同时又可以解决土的非共轴问题。通过单剪试验结果的验证表明,基于广义位势理论的拟弹性弹塑性模型的模拟效果较好,传统的弹塑性模型（共轴模型）模拟得到的主应力方向和塑性主应变增量方向保持共轴,而拟弹性弹塑性模型（非共轴模型）的模拟结果则能够合理地描述主应力轴旋转过程中的非共轴特性,结果更符合实际,从而为解决土的非共轴特性问题提供了一种有效的方法。     

Modeling of elastic anisotropy due to one-dimensional plastic consolidation of clays

Experiments indicate that in one-dimensionally consolidated natural clays the elastic anisotropy is much stronger than the plastic strain anisotropy. Moreover, the elastic anisotropy appears to be dependent on the pre-consolidation strain. Coupled elasto-plastic constitutive law is shown to be able to simulate these anisotropy effects of natural clay deposits. In this law the elastic potential is not only a function of stress, but additionly of the plastic strain. The plastic strain comprises the geological process of pre-consolidation idealized as an one-dimensional plastic straining as well as a mechanically induced strain due to engineering activity. Calibration of the model and simulation of some stress paths are presented and related to the classical experimental results by Mitchell (1972).     

A micromechanics-based model for estimating localized failure with effects of fabric anisotropy

This paper presents a micromechanics-based approach to investigate the effects of fabric anisotropy on the behavior of localized failure in granular materials. Based on a micromechanical analysis, the origin of deviatoric stress is decomposed into two components: contact force anisotropy and fabric anisotropy. Using a micro–macro approach, the back stress is interpreted as an contribution to the change of the fabric’s principal direction. The evolution of the back stress is deduced from the stress–fabric relationship and determined with reference to the deviation of the principal directions between the rate of the reduced stress tensor and the actual reduced stress tensor. With this micro–macro framework, a mixed (isotropic–kinematic) hardening model is developed based on the classical isotropic hardening theory. A laboratory simple shear test is first analyzed to validate the proposed model and illustrate the kinematic-hardening mechanism which is usually displayed under non-proportional loading. The analysis further focuses on the anisotropic aspect of localized failure. It has been discovered that the fabric anisotropy can play an important role in the occurrence of shear banding. An increasing degree of fabric anisotropy tends to delay the initiation of the strain localization and result in higher failure strength. The effects of fabric anisotropy have also been illustrated by comparing the theoretical predictions and measured results on the shear band inclination angle, shear strain level and dilatancy at bifurcation.     

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.     

Anisotropic UH model for soils based on a simple transformed stress method

A simple method called anisotropic transformed stress (ATS) method is proposed to develop failure criteria and constitutive models for anisotropic soils. In this method, stress components in different directions are modified differently in order to reflect the effect of anisotropy. It includes two steps of mapping of stress. First, a modified stress tensor is introduced, which is a symmetric multiplication of stress tensor and fabric tensor. In the modified stress space, anisotropic soils can be treated to be isotropic. Second, a TS tensor is derived from the modified stress tensor for the convenience of developing anisotropic constitutive models to account for the effect of intermediate principal stress. By replacing the ordinary stress tensor with the TS tensor directly, the unified hardening model is extended to model the anisotropic deformation of soils. Anisotropic Lade's criterion is adopted for shear yield and shear failure in the model. The form of the original model formulations remains unchanged, and the model parameters are independent of the loading direction. Good agreement between the experimental results and predictions of the anisotropic unified hardening model is observed. Copyright © 2016 John Wiley & Sons, Ltd.     

Constitutive analysis of the mechanical anisotropy of Opalinus Clay

This paper aims to analyse the anisotropic features of behaviour of Opalinus Clay using the theory of plastic multi-mechanisms. The results of triaxial tests conducted with different load levels and directions showed that the mechanical behaviour of this shale is cross-anisotropic. The stiffer samples are those in which the loading direction is parallel to the bedding plane. This indicates that the preconsolidation stress depends on the orientation of the load with respect to the fabric of Opalinus Clay. It is proposed to interpret the observed cross-anisotropy with an elastoplastic model based on four plastic strain mechanisms that may be successively mobilised depending on the loading direction. The predicted stress–strain responses vary according to the directions of the space as a result of the hardening process, depending on the number of plastic strain mechanisms that have been mobilised. The numerical predictions show overall good agreement with the experimental data in terms of deviatoric stress versus axial strain, demonstrating that multi-mechanism plasticity is a suitable constitutive tool for the interpretation of the mechanical anisotropy of this shale.     

REV and its properties on fracture system and mechanical properties,and an orthotropic constitutive model for a jointed rock mass in a dam site in China

Fracture data available for one of the rock masses (limestone) in the dam site of Yujian River Reservoir were used to build and validate a stochastic 3-D fracture network model, and to perform a REV and equivalent continuum study in 3-D. A number of relations are developed in the paper between the rock mass mechanical parameters and fracture tensor components in 3-D. Based on the mechanical parameter values obtained in every 45° direction in 3-D, the principal parameter values, principal directions and tensors are developed for rock mass mechanical parameters to represent the REV block size properties. An incrementally linear elastic, orthotropic constitutive model is suggested to represent the equivalent continuum pre-failure mechanical behavior of the jointed rock mass by incorporating the effect of joint geometry network by the fracture tensor components.     

Non‐coaxiality and stress–dilatancy rule in granular materials: FE investigation within micro‐polar hypoplasticity

This paper deals with FE investigations of shear localization in dilatant granular bodies. The calculations were carried out with a hypoplastic constitutive law enhanced by micro‐polar terms to properly model the shear zone evolution. The behaviour of an initially medium dense sand specimen with very smooth and very rough horizontal boundaries was analyzed during a plane strain compression test. A stochastic distribution of the initial void ratio was assumed to be spatially correlated. Attention was focused on the non‐coaxiality of the directions of the principal strain increments and principal stresses in the shear zone and on the stress–dilatancy rule. Copyright © 2008 John Wiley & Sons, Ltd.     

Constitutive modelling of granular materials using a contact normal-based fabric tensor

This paper presents a fabric tensor-based bounding surface model accounting for anisotropic behaviour (e.g. the dependency of peak strength on loading direction and non-coaxial deformation) of granular materials. This model is developed based on a well-calibrated isotropic bounding surface model. The yield surface is modified by incorporating the back stress which is proportional to a contact normal-based fabric tensor for characterising fabric anisotropy. The evolution law of the fabric tensor, which is dependent on both rates of the stress ratio and the plastic strain, rules that the material fabric tends to align with the loading direction and evolves towards a unique critical state fabric tensor under monotonic shearing. The incorporation of the evolution law leads to a rotational hardening of the yield surface. The anisotropic critical state is assumed to be independent of the initial values of void ratio and fabric tensor. The critical state fabric tensor has the same intermediate stress ratio (i.e. b value) and principal directions as the critical state stress tensor. A non-associated flow rule in the deviatoric plane is adopted, which is able to predict the non-coaxial flow naturally. The stress–strain relation and fabric evolution of model predictions show a satisfactory agreement with DEM simulation results under monotonic shearing with different loading directions. The model is also validated by comparing with laboratory test results of Leighton Buzzard sand and Toyoura sand under various loading paths. The comparison results demonstrate encouraging applicability of the model for predicting the anisotropic behaviour of granular materials.

     

Comment on the postulate of three plane strain mechanisms

This note discusses the inconsistencies that are inherent in the postulate of three plane strain mechanisms. It is shown that this postulate violates the principle of invariance and one obtains different results depending on the choice of the reference axes. If formulated in the principal stress space, this postulate requires that the principal stress and principal plastic strain increment directions be coaxial. Constitutive models based on this postulate cannot be used for general loading situations involving principal stress rotation where significant non-coaxiality is obtained.     

Pure cross-anisotropy for geotechnical elastic potentials

The pure cross-anisotropy is understood as a special scaling of strain (or stress). The scaled tensor is used as an argument in the elastic stiffness (or compliance). Such anisotropy can be overlaid on the top of any elastic stiffness, in particular on one obtained from an elastic potential with its own stress-induced anisotropy. This superposition does not violate the Second Law. The method can be also applied to other functions like plastic potentials or yield surfaces, wherever some cross-anisotropy is desired. The pure cross-anisotropy is described by the sedimentation vector and at most two constants. Scaling with more than two purely anisotropic constants is shown impossible. The formulation was compared with experiments and alternative approaches. Static and dynamic calibration of the pure anisotropy is also discussed. Graphic representation of stiffness with the popular response envelopes requires some enhancement for anisotropy. Several examples are presented. All derivations and examples were accomplished using the algebra program Mathematica.

     

Effects of principal stress rotation on wave-induced soil response in a poro-elastoplastic sandy seabed

In this study, a constitutive model is developed in order to investigate wave–seabed interactions. This model takes into account the impact of principal stress rotation (PSR) and is based on the generalized plasticity theory, in which plastic strain generated by PSR is considered an additional item in the constitutive relationship of soil. The normalized loading direction and plastic flow direction were determined based on the stress tensor invariant. Comparisons between the present model and previous Hollow Cylinder Apparatus tests and geotechnical centrifugal wave tests show good agreement. Numerical results show the effects of PSR on predictions of liquefaction potential due to: (a) the cumulative impact of plastic strain in the seafloor and (b) the buildup of pore pressure. Parametric study shows that the model parameters, including the wave and seabed parameters, have significant effects on the wave-induced soil liquefaction.

     

基于临界状态模型的砂土非共轴本构模拟

传统的砂土本构理论隐含了应力和塑性应变率的共轴条件,无法客观描述主应力轴旋转试验中的非共轴现象,并且当密度和围压变化较大时也不适用。基于材料状态相关砂土临界状态概念,将Pietruszczak和Stolle所提出的砂土本构模型进行了改进,并在模型中引入非共轴塑性流动理论来描述非共轴现象。通过对单剪试验和空心圆柱试验进行数值模拟,表明基于临界状态理论的非共轴模型能够合理描述主应力轴旋转过程中砂土的非共轴变形特性     

The influence of anisotropy on compaction bands: The case of coaxiality between stress and fabric anisotropy tensors

Compaction bands are localized failure patterns that appear in highly porous rock material under the effect of relatively high confining pressure. Being affected mainly by volumetric compression, these bands appear to be almost perpendicular to the most compressive principal stress of a stress state at the so-called “cap” of the yield surface (YS). In this study, we focus on the mechanism that leads to the onset of compaction bands by using a viscoplasticity model able to describe the post-localization response of these materials. The proposed constitutive framework is based on the overstress theory of Perzyna (1966) and the anisotropic clay plasticity model of Dafalias (1986), which provides not only the necessary “cap” of the YS, but introduces a rotational hardening (RH) mechanism, thus, accounting for the effect of fabric anisotropy. Following the analysis of Veveakis and Regenauer-Lieb (2015), we identify the compaction bands as “static” cnoidal wave formations in the medium that occur at a post-yield regime, and we study the effect of rotational and isotropic hardening on their onset. Moreover, we determine a theoretical range of confining pressures in triaxial compression tests for the compaction bands to develop. Under the assumption of coaxiality between stress and anisotropy tensors, the results show that the isotropic hardening promotes compaction localization, whereas the RH has a slightly negative effect on the onset of compaction localization.