Modeling of textural anisotropy in granular materials with stochastic micro-polar hypoplasticity

The paper deals with a numerical analysis of the effect of textural anisotropy on the behaviour of cohesionless granular materials with consideration of shear localization. For a simulation of the mechanical behaviour of a granular material during a monotonic deformation path, an isotropic micro-polar hypoplastic constitutive model was used. To describe textural effects, spatially correlated random fields of the initial void ratio were subject to rotation against the horizontal axis. The 2D random fields were generated using a conditional rejection method. The results were compared with those obtained with an anisotropic micro-polar constitutive model for a uniform distribution of the initial void ratio. The calculations were carried out with an initially dense granular specimen during plane strain compression under constant lateral pressure.     

基于深度强化学习算法的颗粒材料应力?应变关系数据驱动模拟研究

颗粒材料的宏观力学行为受颗粒组分等材料参数, 孔隙率、配位数等状态参数的影响, 同时又具备复杂的加载路径和加载历史相关性, 建立包含多个内变量以及各变量间相互关联的颗粒材料本构模型是一个重要的科学难题. 不同于传统的基于屈服面、流动法则和硬化函数框架下的唯象本构模型, 本文基于颗粒物质力学的研究基础, 以颗粒材料平均孔隙率、细观组构参数和弹性刚度参数作为内变量, 结合深度学习方法建立以有向图表征的数据本构模型. 有向图中以不同的链接网络表示不同的内变量信息流动方向, 各个内变量间的映射关系采用循环神经网络来建立, 将各个神经网络相互组合, 形成包含不同内变量且具有不同预测能力的本构模型. 该本构模型的建立过程等价于在众多可能的内变量链接关系空间中寻找最能描述实际材料宏观应力应变行为的优化问题. 因此, 可将有向图本构模型的建立过程看作“马尔可夫决策过程”, 采用深度强化学习算法构建有向图的内变量链接组合优化过程, 具体采用AlphaGo Zero算法自动寻找最优的颗粒材料数据驱动本构模型建模路径. 研究结果表明, 采用有向图和深度强化学习算法可建立起完全依靠“数据驱动”的颗粒材料应力?应变关系. 此外, 本方法提供了一种将不同理论模型从数据角度统一起来, 且基于人工智能算法发展更优模型的研究思路, 可为相似问题的研究提供借鉴.      

Micromechanical analysis of kinematic hardening in natural clay

This paper presents a micromechanical analysis of the macroscopic behaviour of natural clay. A microstructural stress–strain model for clayey material has been developed which considers clay as a collection of clusters. The deformation of a representative volume of the material is generated by mobilizing and compressing all the clusters along their contact planes. Numerical simulations of multistage drained triaxial stress paths on Otaniemi clay have been performed and compared the numerical results to the experimental ones in order to validate the modelling approach. Then, the numerical results obtained at the microscopic level were analysed in order to explain the induced anisotropy observed in the clay behaviour at the macroscopic level. The evolution of the state variables at each contact plane during loading can explain the changes in shape and position in the stress space of the yield surface at the macroscopic level, as well as the rotation of the axes of anisotropy of the material.     

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.     

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.      

Incorporating the effects of fabric in the dilatant double shearing model for planar deformation of granular materials

The objective of this paper is to incorporate the effects of fabric and its evolution into the Dilatant Double Shearing Model [Mehrabadi, M.M., Cowin, S.C., 1978. Initial planar deformation of dilatant granular materials. J. Mech. Phys. Solids 26, 269–284] for granular materials in order to capture the anisotropic behavior and the complex response of granular materials in cyclic shear loading. An important consequence of considering the fabric is that one can have unequal shearing rates along the two slip directions. This property leads to the non-coaxiality of the principal axes of stress and strain rate, which is more appropriate for a material that exhibits initial and induced anisotropy. In addition, we employ a fabric-dependent elasticity tensor with orthotropic symmetry. The model developed in this paper also predicts one of the experimentally observed characteristics of granular materials: the gradual concentration of the contact normals towards the maximum principal stress direction.We implement the constitutive equations into ABAQUS/Explicit by writing a user material subroutine in order to predict the strength anisotropy of granular materials in a plane strain biaxial compression test and investigate the mechanical behavior of granular materials under the cyclic shear loading conditions. The predictions from this model show good quantitative agreement with the experiments of [Park, C.S., 1990. Anisotropy in deformation and strength properties of sands in plane strain compression, Masters Thesis, University of Tokyo; Park, C.S., Tatsuoka, F., 1994. Anisotropic strength and deformation of sands in plane strain compression. In: XIII ICSMFE, New Delhi, India; Okada, N., 1992. Energy dissipation in inelastic flow of cohesionless granular media. Ph.D. Thesis, University of California, San Diego].     

粒料固有各向异性的离散元模拟与细观分析

料在摊铺后形成的颗粒定向排列将导致其力学性质的固有各向异性. 依据粒料的实际不规则形状, 构建了可模拟粒间咬合嵌挤作用的三维离散元复杂形状颗粒; 生成了5 种不同沉积方向的各向异性试件和1种各向同性试件, 对比了各试件在三轴压缩试验中的宏观力学特性差异; 引入组构张量以量化各向异性程度, 利用玫瑰图表达接触法向与接触力的分布特征, 探究了粒料各向异性的细观发展规律. 结果表明: 颗粒长轴愈趋向水平排布, 峰值应力比愈大, 剪缩与剪胀程度愈明显; 相较于各向同性试件, 沉积角$\theta$为料在摊铺后形成的颗粒定向排列将导致其力学性质的固有各向异性. 依据粒料的实际不规则形状, 构建了可模拟粒间咬合嵌挤作用的三维离散元复杂形状颗粒; 生成了5 种不同沉积方向的各向异性试件和1种各向同性试件, 对比了各试件在三轴压缩试验中的宏观力学特性差异; 引入组构张量以量化各向异性程度, 利用玫瑰图表达接触法向与接触力的分布特征, 探究了粒料各向异性的细观发展规律. 结果表明: 颗粒长轴愈趋向水平排布, 峰值应力比愈大, 剪缩与剪胀程度愈明显; 相较于各向同性试件, 沉积角$\theta$为$0^\circ$时, 峰值应力比和最大体积压缩应变分别提高了12.6\%和18.8\%, 其原因在于加载过程中颗粒旋转和滑动百分比更小, 内部调整时间更短、更易被剪密; 固有各向异性对颗粒法向接触力分布的影响不大, 但显著影响接触法向分布特征; 剪切过程中, $\theta$为$90^\circ$时的接触法向各向异性系数先快速减小后逐渐增大, 而$\theta$为$0^\circ$到$60^\circ$时则呈现出增大后稍有回落或趋于稳定的趋势, 且$\theta$ 愈小的试件各向异性系数增大愈快.     

Micro-scale behavior of granular materials during cyclic loading

This study presents the micro-scale behavior of granular materials under biaxial cyclic loading for differ- ent confining pressures using the two-dimensional （2D） discrete element method （DEM）. Initially, 8450 ovals were generated in a rectangular frame without any overlap. Four dense samples having confining pressures of 15, 25, 50, and 100 kPa were prepared from the initially generated sparse sample. Numeri- cal simulations were performed under biaxial cyclic loading using these isotropically compressed dense samples. The numerical results depict stress-strain-dilatancy behavior that was similar to that observed in experimental studies. The relationship between the stress ratio and dilatancy rate is almost indepen- dent of confining pressures during loading but significantly dependent on the confining pressures during unloading. The evolution of the coordination number, effective coordination number and slip coordina- tion number depends on both the confining pressures and cyclic loading. The cyclic loading significantly affects the microtopology of the granular assembly. The contact fabric and the fabric-related anisotropy are reported, as well. A strong correlation between the stress ratio and the fabric related to contact normals is observed during cyclic loading, irrespective of confining pressures.     

A microstructural elastoplastic model for unsaturated granular materials

The homogenization technique is used to obtain an elastoplastic stress–strain relationship for dry, saturated and unsaturated granular materials. Deformation of a representative volume of material is generated by mobilizing particle contacts in all orientations. In this way, 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. For the non-saturated state, capillary forces at the grain contacts are added to the contact forces created by an external load. They are calculated as a function of the degree of saturation, depending on the grain size distribution and on the void ratio of the granular assembly. Numerical simulations show that the model is capable of reproducing the major trends of a partially saturated granular assembly under various stress and water content conditions. The model predictions are compared to experimental results on saturated and unsaturated samples of silty sands under undrained triaxial loading condition. This comparison shows that the model is able to account for the influence of capillary forces on the stress–strain response of the granular materials and therefore, to reproduce the overall mechanical behavior of unsaturated granular materials.     

Effect of particle size distribution on micro- and macromechanical response of granular packings under compression

The role of particle size heterogeneity on micro- and macromechanical properties of assemblies of spherical particles was studied using DEM simulations. The response to an imposed load of a granular material composed of non-uniformly sized spheres subjected to uniaxial confined compression was investigated. A range of geometrical and micro-mechanical properties of granular packings (e.g., void fraction, contact force distribution, average coordination number and degree of mobilisation of friction at contacts between particles) were examined, and provided a more accurate interpretation of the macroscopic behaviour of mixtures than has previously been available. The macromechanical study included stress transmission, stiffness and angle of internal friction of the granular assemblies.The degree of polydispersity showed slight effect on both, the void fraction and the elastic properties of the system. The tendency for increase in the lateral-to-vertical pressure ratios was observed with an increasing degree of particle size heterogeneity; however, the different pressure ratios calculated for samples with various degrees of polydispersity lay within the range of data scatter.     

Three-dimensional constitutive relations for granular materials based on the dilatant double shearing mechanism and the concept of fabric

An extension of a three-dimensional model proposed by Anand and Gu (2000) for amorphous granular materials to include the effects of initial and induced anisotropy is presented in this paper. The proposed model can also be considered as a three-dimensional generalization of a model recently developed by Zhu et al. (2005) for the planar deformation of granular materials. The main ingredients of the model include the dilatant double shearing mechanism (Spencer, 1964, Mehrabadi and Cowin, 1978), the concept of fabric (Oda, 1972), and an extension of the Mohr–Coulomb yield criterion (Shield, 1955, Spencer, 1982) to three dimensions.The constitutive equations are implemented in the finite element program ABAQUS/Explicit (ABAQUS, 2001) by developing a user-material subroutine to conduct numerical triaxial compression tests for samples of granular materials with different initial anisotropy. The numerical results agree with the observed behavior and show that the extended constitutive model is capable of capturing the strength anisotropy of granular materials. Employing the anisotropic model developed here, we have also repeated the numerical simulation of the stress state in a static conical sand pile conducted earlier by Anand and Gu (2000). We find that fabric has little or no influence on the vertical stress distribution except at the base of the sand pile where the peak value of this stress is slightly higher than that predicted by the model of Anand and Gu (2000) which does not include the effects of fabric. We also find that the direction of the principal compressive stress changes from vertical at points away from the center of the pile to almost horizontal at points close to the center of the pile. This result provides a possible explanation for the observed dip in the vertical stress distribution in sand piles.     

Micro-scale behavior of granular materials during cyclic loading

This study presents the micro-scale behavior of granular materials under biaxial cyclic loading for different confining pressures using the two-dimensional (2D) discrete element method (DEM). Initially, 8450 ovals were generated in a rectangular frame without any overlap. Four dense samples having confining pressures of 15, 25, 50, and 100 kPa were prepared from the initially generated sparse sample. Numerical simulations were performed under biaxial cyclic loading using these isotropically compressed dense samples. The numerical results depict stress–strain–dilatancy behavior that was similar to that observed in experimental studies. The relationship between the stress ratio and dilatancy rate is almost independent of confining pressures during loading but significantly dependent on the confining pressures during unloading. The evolution of the coordination number, effective coordination number and slip coordination number depends on both the confining pressures and cyclic loading. The cyclic loading significantly affects the microtopology of the granular assembly. The contact fabric and the fabric-related anisotropy are reported, as well. A strong correlation between the stress ratio and the fabric related to contact normals is observed during cyclic loading, irrespective of confining pressures.     

土的各向同性化变换应力方法

在不同方向的力学参数、结构特性及应力应变关系的不同为材料的各向异性, 建立能够反映这种复杂特性的强度准则、本构模型, 对于材料本构关系的研究具有重要的理论意义. 但材料的各向异性一直是其力学特性描述的难点, 对此, 郑泉水院士提出了各向同性化定理, 为后续研究解决材料的各向异性问题提供了方向及思路. 作者等针对土的应力诱导各向异性提出了变换应力方法, 这种方法同样遵循对材料进行各向异性问题各向同性化处理的思路, 与郑泉水院士的各向同性化定理是一脉相承的, 也是对各向同性化定理的发展. 本文旨在通过分析各向同性化定理与变换应力方法明确两者间的内在联系, 并以土材料的应力诱导各向异性处理为例, 说明在具体材料的各向异性处理过程中面临的现实问题以及变换应力方法是如何解决这些问题的. 分析并给出了变换应力方法应用时的三个合理假设, 推导出了具体的变换应力数学公式, 阐明了在考虑土的应力诱导各向异性的具体函数已经给出的情况下, 在构造土的弹塑性本构模型中采用变换应力方法的必要性.      

椭球颗粒体系剪切过程中自由体积的分布与演化

颗粒材料是一种复杂的多体相互作用体系, 由大量离散的颗粒和其周围的自由体积组成. 虽然颗粒的自由体积与颗粒材料的力学性能和变形特征的相关性已得到证实, 但是由于表征上的困难, 目前对非球形颗粒体系的局部自由体积的认识还不够充分. 本文采用连续离散耦合分析方法进行了不同主轴长度的椭球颗粒试样的三轴剪切数值模拟, 基于Set Voronoi算法对剪切过程中的颗粒试样进行了Voronoi元胞分割, 分析了颗粒试验在剪切过程中自由体积的统计分布特性和演化规律, 研究了颗粒形态对自由体积的影响. 剪切过程中Voronoi元胞的各向异性逐渐增强, 且各项异性增强程度随颗粒非球度的增加而增大, 表明非球颗粒在剪切过程中经历更加强烈的重排列. 具有不同非球度的椭球颗粒体系的局部孔隙比均服从k?Γ分布, 且这个分布仅与颗粒体系的全局孔隙比相关, 不受颗粒形态和剪切状态的影响. 局部孔隙比的波动呈现非对称拉普拉斯分布, 非对称参数刻画了局部自由体积收缩和膨胀的博弈, 其与全局孔隙比呈线性关系.      

A finite strain constitutive model for metal powder compaction using a unique and convex single surface yield function

Constitutive modelling of metal powder compaction processes is a challenge in view of realistic simulations. To this end, the article under consideration has two objectives: the first goal is to present a new unique and convex single surface yield function for pressure dependent materials, which is also applicable to other areas of granular materials such as soils or concrete. The flexibility is shown at various materials. The yield function is based on a log-interpolation of two known simple yield functions. A convexity proof of the new yield function is provided. The second objective is to propose a new rate-independent finite strain plasticity model for metal powder compaction, which is based on the multiplicative decomposition of the deformation gradient into an elastic and a plastic part with evolution equations for internal variables representing the basic behaviour of powder materials under compaction conditions. These variables are used for the evolution of the yield function in order to represent the compressible hardening behaviour of powder materials. On the basis of the constitutive model, the material parameters are identified at experimental data of copper powder.     

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.     

基于深度学习和细观力学的颗粒材料本构关系研究

颗粒材料的本构关系对岩土工程等众多领域至关重要. 不同于传统的唯象本构理论, 本文基于机器学习模型探索了一种细观力学理论指导下的数据驱动型颗粒材料本构关系预测方法. 根据Vogit均质化假设, 建立了小应变条件下颗粒材料应力?应变解析关系, 此关系唯一地确定了一组与颗粒材料本构行为相关的细观组构变量. 这些变量与反应颗粒材料宏观性质的主应力和主应变信息通过一系列离散元三轴压缩数值试验获得. 考虑到细观组构变量为内变量, 不能直接作为本构模型的输入. 本文基于有向图方法将颗粒材料微观结构信息隐式地包含在应力?应变的预测当中, 并采用门控循环单元(GRU)循环神经网络作为基础深度学习模型描述有向图中结点之间的映射关系. 通过将有向图从目标节点沿源节点展开, 整个应力?应变预测模型可由两个神经网络分别训练并组装而成. 将训练后的深度学习模型在全新的数据集上进行测试, 结果表明该训练策略能有效捕捉到颗粒材料在常规三轴任意加卸载, 等中主应力系数b的真三轴加载, 和等平均有效应力p的真三轴加卸载等复杂多轴加载工况下的应力?应变响应关系, 模型具有良好的内插和外推预测能力. 考虑到深度学习模型捕捉颗粒材料力学响应的能力及其开放式学习的特点, 充分结合数据驱动方法和理论本构模型可能是颗粒材料本构研究的一个重要方向.      

Fabric and connectivity as field descriptors for deformations in granular media

Granular materials involve microphysics across the various scales giving rise to distinct behaviours of geomaterials, such as steady states, plastic limit states, non-associativity of plastic and yield flow, as well as instability of homogeneous deformations through strain localization. Incorporating such micro-scale characteristics is one of the biggest challenges in the constitutive modelling of granular materials, especially when micro-variables may be interdependent. With this motivation, we use two micro-variables such as coordination number and fabric anisotropy computed from tessellation of the granular material to describe its state at the macroscopic level. In order to capture functional dependencies between micro-variables, the correlation between coordination number and fabric anisotropy limits is herein formulated at the particle level rather than on an average sense. This is the essence of the proposed work which investigates the evolutions of coordination number distribution (connectivity) and anisotropy (contact normal) distribution curves with deformation history and their inter-dependencies through discrete element modelling in two dimensions. These results enter as probability distribution functions into homogenization expressions during upscaling to a continuum constitutive model using tessellation as an abstract representation of the granular system. The end product is a micro-mechanically inspired continuum model with both coordination number and fabric anisotropy as underlying micro-variables incorporated into a plasticity flow rule. The derived plastic potential bears striking resemblance to cam–clay or stress–dilatancy-type yield surfaces used in soil mechanics.     

Representation of the glass-transition in mechanical and thermal properties of glass-forming materials: A three-dimensional theory based on thermodynamics with internal state variables

In the vicinity of the glass transition, glass-forming materials exhibit pronounced frequency-dependent changes in the mechanical material properties, the thermal expansion behaviour and the specific heat. The frequency dependence becomes apparent under harmonic stress, strain or temperature excitations. The Prigogine-Defay ratio is a characteristic number which connects the changes in magnitude of these quantities at the glass transition. In order to represent the thermoviscoelastic properties of glass-forming materials in continuum mechanics, a three-dimensional approach which is based on the Gibbs free energy as thermodynamic potential is developed in this article. The Gibbs free energy depends on the stress tensor, the temperature and a set of internal variables which is introduced to take history-dependent phenomena into account. In the vicinity of an equilibrium reference state, the specific Gibbs free energy is approximated up to second order terms. Evaluating the Clausius-Duhem inequality, the constitutive relations for the strain tensor, the entropy and the internal variables are derived. In comparison with other approaches, the entropy, the strain tensor and the internal variables are functionals not only of the stress tensor but also of the temperature. Applying harmonic temperature- or stress-controlled excitations, complex frequency-dependent relations for the specific heat under constant stress, for the thermal expansion coefficients as well as for the dynamic mechanical compliance are obtained. The frequency-dependence of these quantities depicts the experimentally observed behaviour of glass-forming materials as published in literature. Under the assumption of isotropic material behaviour, it is shown that the developed theory is compatible with the Prigogine-Defay inequality for arbitrary values of the material parameters.     

Structural changes in granular materials: The case of irregular polygonal particles

This paper presents a series of numerical simulations of biaxial tests performed on assemblies of two-dimensional irregular polygonal particles. Each sample is prepared with a technique similar to dry pluviation. Different aspect ratios (1–3) are considered and the behavior of granular samples is analyzed from both a global and a local point of view. More precisely, the influence of the particle aspect ratio on both inherent (initial) and induced anisotropy is investigated. New internal variables which are related to the orientation of particles are proposed. They give new insight into the specific mechanisms that control the behavior of irregular polygonal materials. Associated to global variables, they demonstrate the existence of a critical state irrespective of the investigated aspect ratios. However, for materials with higher aspect ratios (2 and 3), their inherent anisotropy prevents any extensive reorganization, this means that, within the range of usual strains considered in civil engineering, the particle reorientation remains in progress and considerable deformations are required to reach the critical state.