颗粒材料三维应力路径下的接触组构特性

颗粒材料的宏观应力变形特征与其微观接触力、组构等紧密相关.一般而言,强接触系统属于颗粒内部体系的传力结构,其对应的组构张量是影响宏观应力性质的重要因素.细观数值方法(如离散单元法)能够反映物理试验的基本规律,并且可以方便地提取宏微观数据来研究颗粒体系的应力变形机制.采用离散单元法(discrete element method,DEM)进行一系列等$p$等$b$应力路径下颗粒材料的真三轴试验,在此基础上研究了三维应力路径下颗粒材料的宏微观力学参数的演化过程、三维组构张量与应力张量多重联系以及强接触体系反映的宏观应力特征.研究表明:颗粒体系偏应力峰值状态和临界状态均存在与加载路径无关的宏微观特征;三维应力路径下组构张量与应力张量存在非共轴性,但其联合不变量演化过程表现出加载路径无关的特征;与弱接触系统的组构张量相比,强接触系统的组构张量更能反映宏观应力张量的特征;强弱接触体系的组构张量对颗粒体系宏观响应的贡献不同,其分界点存在一定取值范围,但采用平均接触力较为简单合理.      

弹脆性材料的损伤本构关系及应用

本文根据连续损伤力学方法,对弹脆性材料损伤的力学响应进行一般分析。理论分析中,材料与损伤都是各向异性的。还导出了计算损伤张量、有效弹性张量、真实应力张量以及损伤能耗率张量的实用表达式。     

基于微面有效应力矢量的各向异性屈服准则

基于微面模型，定义损伤变量为微面上有效承载面积的减少. 将Kachanov的一维有效 应力概念推广到三维，提出微面有效应力矢量的概念. 根据微面的有效应力矢量，将无损材 料的宏观应力张量及不变量与微面应力矢量的积分关系拓展到有损材料，得到了有损材料的 宏观有效应力张量及其不变量与宏观名义应力张量、微面面积损伤组构张量之间的关系. 将 无损材料的以应力张量不变量表示的Drucker-Prager准则推广到有损材料，建立了含缺陷 材料的各向异性屈服准则. 对有损材料，宏观有效应力张量与Murakami的有效应力张量具 有相同的形式，各向异性强度准则与Liu等提出的扩展Hill准则有相同的形式，当不考虑 静水应力对屈服的影响时，它与Hill准则具有相同的形式.     

Derivation of the general form of elasticity tensor of the transverse isotropic material by tensor derivate

SymbolsU--FunchonofstrainenergyQ--OrthonormaltensorE--StraintensorEar--ComponentsofthestraintensorE,i,j=l,2,3n--VectorofthesymmetricaamsofthetransverseisotropicmaterialU*,E.,n*--FormsofU,EandninanothercoordinatesystemJf--MaininvariantsofstraintensorE,i=l,2,3Jf'n--InvariantsofstraintensorEconnectingwithvectorn,i=4,5Ji--TheabbreviatedformsofJf,Jf,Jf,Jf,",Jf,",i=l,2,3,4,5fi--ConstantsindependentonE,n,i=l,2,3,4,5el,e"--Thecovariantandcontravariantofthonormalbasisoftheusedcoordinatesyste…     

Surface Impedance Tensor and Green’s Function for Weakly Anisotropic Elastic Materials

Recently Fu and Mielke uncovered a new identity that the surface impedance tensor of any anisotropic elastic material has to satisfy. By solving algebraically a matrix equation that follows from the new identity, we derive an explicit expression for the surface impedance tensor, which is correct up to terms linear in the components of the anisotropic part of the elasticity tensor of the material in question. From the well-known relationship between the surface impedance tensor and the Green’s function for infinite space, we obtain an explicit expression for the Green’s function, which is correct up to terms linear in the components of the anisotropic part of the elasticity tensor.      

Macro–micro relations in granular mechanics

In granular mechanics, macroscopic approaches treat a granular material as an equivalent continuum at macro-scale, and study its constitutive relationship between macro-quantities, such as stresses and strains. On the other hand, microscopic approaches consider a granular material as an assembly of individual particles interacting with each other at micro-scale (i.e., particle-scale), and the physical quantities under study are forces and displacements. This paper aims at linking up the findings from these two scales and to establish the macro–micro relations in granular mechanics.Three aspects of the macro–micro relations are investigated. They are about the internal structure, the stress tensor and the strain tensor. The internal structure is described with geometrical systems at the particle scale. Micro-structural definitions of the stress and strain tensors are derived, which link the macro-stress tensor with the contact forces, and the macro-strain tensor with the relative displacements at contact. In addition to a brief review of the past research work on these topics, further generalizations are made in this paper. In particular, the two cell systems proposed by Li and Li (2009), namely the solid cell system and the void cell system, are introduced and used for the derivation of the macro-structural expressions. The stress expression is derived based on Newton’s second law of motion. The result is valid for both static and dynamic cases. The strain expression is derived based on the compatibility requirement. And the expression is valid for any tessellation subdividing the granular assembly into polyhedral elements.The homogenization for deriving a macroscopic constitutive relationship from microscopic behaviour is discussed. Attention is placed on the macroscopic quantification of the internal structure in terms of a second rank tensor, known as the fabric tensor. Existing definitions of the fabric tensors have been reviewed. The correlations among different fabric tensors and their relations with the stress–strain behaviours have been investigated.     

On the definitions of failure and critical states in hypoplasticity

A granular body is said to be at failure or in a critical state if the stress state does not change while the body is continuously deformed. Within the framework of hypoplasticity, such states, generally called stationary states,are conventionally defined by the condition that an objective (the Jaumann) stress rate vanishes. However, not all stationary states attained under monotonic deformation lie within the scope of this definition. Simple shear is an example. In fact, stationary states are characterized by zero material time derivative of the stress tensor rather than zero Jaumann rate. In the present paper, we give a generalized definition of stationarity by the condition of zero material time derivative of the stress tensor. The new definition extends the set of possible stationary states and includes those which are not covered by the previous definition. Stationary states are analysed quantitatively using calibrated hypoplastic equations. It is shown numerically that, if the norm of the spin tensor is of the same order as, or smaller than, the norm of the stretching tensor, the old definition approximates all possible sationary states with sufficient accuracy.      

A Solution for the Stress Distribution in a Granular Medium

We propose an example of an incremental elastic problem for a granular material. The mechanical behavior of this material is sensitive to the confining pressure that typically is applied before any loading. Experiments, numerical simulations, and theoretical models show that the effective elastic moduli of a granular medium are function of the confining pressure in non-linear way. Therefore, if we consider a reference state where the pressure is not constant the material behaves differently in each point and, for example, the stress associated to a subsequently loading should be obtained as solution of a non-homogeneous material. We focus our attention on this kind of problem for a granular material that fills an hollow cylinder first isotropically compressed and then sheared under a rotatory motion. Next a small perturbation is applied on the boundaries of the specimen and we evaluate the corresponding stress distribution in the plane perpendicular to the axis of the cylinder.     

THEORETICAL MODEL OF EFFECTIVE STRESS COEFFICIENT FOR ROCK/SOIL-LIKE POROUS MATERIALS

Physical mechanisms and influencing factors on the effective stress coefficient for rock/soil-like porous materials are investigated, based on which equivalent connectivity index is proposed. The equivalent connectivity index, relying on the meso-scale structure of porous material and the property of liquid, denotes the connectivity of pores in Representative Element Area （REA）. If the conductivity of the porous material is anisotropic, the equivalent connectivity index is a second order tensor. Based on the basic theories of continuous mechanics and tensor analysis, relationship between area porosity and volumetric porosity of porous materials is deduced. Then a generalized expression, describing the relation between effective stress coefficient tensor and equivalent connectivity tensor of pores, is proposed, and the expression can be applied to isotropic media and also to anisotropic materials. Furthermore, evolution of porosity and equivalent connectivity index of the pore are studied in the strain space, and the method to determine the corresponding functions in expressions above is proposed using genetic algorithm and genetic programming. Two applications show that the results obtained by the method in this paper perfectly agree with the test data. This paper provides an important theoretical support to the coupled hydro-mechanical research.     

裂隙岩体损伤局部化破坏分岔模型及其应用

采用概率统计方法分析节理裂隙岩体的几何特征，定义了反映裂隙岩体几何特征的组构张量．基于不可逆热力学理论，通过裂纹扩展的细观分析，得出了损伤的发展机理和演化方程，把损伤演化和裂隙的几何特征的变化联系起来，建立了弹塑性损伤本构关系．为分析含有节理裂隙岩体在发生局部化破坏时的特征，通过对发生局部化时的裂隙岩体的分析。构造了适用于节理裂隙岩体局部化分析的不连续分岔模型．利用非线性规划数值解法，可以得出局部化破坏的方向特征．在有限元方法中，根据该模型给出了节理裂隙岩体相关的算例，分析表明该模型用于分析裂隙岩体的局部化破坏是有效的．     

有限变形时材料性质张量的选取

在有限变形时导出了用欧拉观点描述的与用拉格朗日观点描述的材料性质张量之间用矩阵形式表达的转换关系，指出正确应用这个关系时将到唯一的计算结果，与计算方法无关，建立在连续体力学中应将材料性质实验结果与欧拉观点描述的材料性质张量相对应。     

Dissipation consistent fabric tensor definition from DEM to continuum for granular media

In elastoplastic soil models aimed at capturing the impact of fabric anisotropy, a necessary ingredient is a measure of anisotropic fabric in the form of an evolving tensor. While it is possible to formulate such a fabric tensor based on indirect phenomenological observations at the continuum level, it is more effective and insightful to have the tensor defined first based on direct particle level microstructural observations and subsequently deduce a corresponding continuum definition. A practical means able to provide such observations, at least in the context of fabric evolution mechanisms, is the discrete element method (DEM). Some DEM defined fabric tensors such as the one based on the statistics of interparticle contact normals have already gained widespread acceptance as a quantitative measure of fabric anisotropy among researchers of granular material behavior. On the other hand, a fabric tensor in continuum elastoplastic modeling has been treated as a tensor-valued internal variable whose evolution must be properly linked to physical dissipation. Accordingly, the adaptation of a DEM fabric tensor definition to a continuum constitutive modeling theory must be thermodynamically consistent in regards to dissipation mechanisms. The present paper addresses this issue in detail, brings up possible pitfalls if such consistency is violated and proposes remedies and guidelines for such adaptation within a recently developed Anisotropic Critical State Theory (ACST) for granular materials.     

A novel internal dissipation inequality by isotropy and its implication for inelastic constitutive characterization

In the present work a novel inelastic deformation caused internal dissipation inequality by isotropy is revealed. This inequality has the most concise form among a variety of internal dissipation inequalities, including the one widely used in constitutive characterization of isotropic finite strain elastoplasticity and viscoelasticiy. Further, the evolution term describing the difference between the rate of deformation tensor and the “principal rate” of the elastic logarithmic strain tensor is set, according to the standard practice by isotropy, to equal a rank-two isotropic tensor function of the corresponding branch stress, with the tensor function having an eigenspace identical to the eigenspace of the branch stress tensor. Through that a general form of evolution equation for the elastic logarithmic strain is formulated and some interesting and important results are derived. Namely, by isotropy the evolution of the elastic logarithmic strain tensor is embodied separately by the evolutions of its eigenvalues and eigenprojections, with the evolution of the eigenprojections driven by the rate of deformation tensor and the evolution of the eigenvalues connected to specific material behavior. It can be proved that by isotropy the evolution term in the present dissipation inequality stands for the essential form of the evolution term in the extensively applied dissipation inequality.     

MICROMECHANICS ANALYSIS FOR UNSATURATED GRANULAR SOILS

This paper aims at establishing an anisotropic stress expression for unsaturated pendular-state granular soils. Using the second-order fabric tensor, we formulate a micromechanics scheme of soils with statistically averaging method, and reveal that the macroscopic average stress of unsaturated granular soils in pendular-state is not isotropic. Not only is the stress from contact forces anisotropic due to the fabric, but also the capillary stress is directional dependent, which is different from the common point that the capillary stress is isotropic. The capillary stress of unsaturated pendular-state granular soils is determined by the orientation distribution of contact normals, so it is closely related to the initial and induced anisotropy of soils. Finally, DEM numerical simulations of triaxial compression tests of pendular-state soils at different degrees of saturation are used to verify the existence of above anisotropy of stresses.     

基于正则结构的各向异性损伤演化律

在Rice的正则结构框架下，推导出基于共轭力的各向异性损伤演化律。其中损伤变量采用二阶裂隙张量，它是固体内微裂纹的一个宏观测度。推导过程不涉及自由能的具体形式，主要结果包括损伤势函数及演化方程的解析表达式。在唯象的损伤力学模型里，损伤演化方程经常以唯象方程的形式出现。研究了唯象方程成立的条件及损伤特征张量的解析表达式。引入了广义裂隙张量及脆性指数的概念，并介绍了它们的作用和意义。     

Statistics of particle interactions in dense granular material under uniaxial compression

Stress evolution in a dense granular material is closely related to interactions of contacting particles. We investigate statistics related to particle interactions and the relationship between the averaged local relative motion and the macroscopic motion. The validity of the Voigt and Reuss assumptions is examined, and extensions to these assumptions are proposed. Effects of history in the dense granular material are investigated. Statistical samples used in this paper are obtained using three-dimensional numerical simulations of dense granular media under uniaxial cyclical compression. The results show that stresses arise mostly from normal forces between particles, and direct contributions from frictional tangential forces between particles are small. Tangential friction, however, significantly increases the particle contact time, and thus reduces the rate of contact breakage. The contact breakage rate is demonstrated to be a stress relaxation rate. Therefore, stress increases significantly with friction between particles as a result of prolonged relaxation time.     

An elasto-plastic model for granular materials with microstructural consideration

In this paper, we have extended the granular mechanics approach to derive an elasto-plastic stress–strain relationship. The deformation of a representative volume of the material is generated by mobilizing particle contacts in all orientations. Thus, the stress–strain relationship can be derived as an average of the mobilization behavior of these local contact planes. The local behavior is assumed to follow a Hertz–Mindlin’s elastic law and a Mohr–Coulomb’s plastic law. Essential features such as continuous displacement field, inter-particle stiffness, and fabric tensor are discussed. The predictions of the derived stress–strain model are compared to experimental results for sand under both drained and undrained triaxial loading conditions. The comparisons demonstrate the ability of this model to reproduce accurately the overall mechanical behavior of granular media and to account for the influence of key parameters such as void ratio and mean stress. A part of this paper is devoted to the study of anisotropic specimens loaded in different directions, which shows the model capability of considering the influence of inherent anisotropy on the stress–strain response under a drained triaxial loading condition.     

Mean stress in a granular medium in dynamics

In this paper we propose to construct a mean stress tensor for a granular medium, valid in static and in dynamics, which takes into account the contact reactions and the body forces acting at the grain level. A simple analytical example shows that taking account of inertia forces is essential to insure the symmetry of the mean stress tensor in dynamics. Finally, numerical simulations illustrating this definition of the mean stress tensor are presented for a granular medium ensiled.     

Elastoplastic deformation of elements of an isotropic solid along paths of small curvature: Constitutive equations incorporating the stress mode

Constitutive equations relating the components of the stress tensor in a Eulerian coordinate system and the linear components of the finite-strain tensor are derived. These stress and strain measures are energy-consistent. It is assumed that the stress deviator is coaxial with the plastic-strain differential deviator and that the first invariants of the stress and strain tensors are in a nonlinear relationship. In the case of combined elastoplastic deformation of elements of the body, this relationship, as well as the relationship between the second invariants of the stress and strain deviators, is determined from fundamental tests on a tubular specimen subjected to proportional loading at several values of stress mode angle (the third invariant of the stress deviator). Methods to individualize these relationships are proposed. The initial assumptions are experimentally validated. The constitutive equations derived underlie an algorithm for solving boundary-value problems __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 43–55, June 2007.