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.      

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.     

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.     

非均匀颗粒材料的类固-液相变行为及本构方程

以非均匀颗粒介质为研究对象，采用三维离散元方法对其在不同密集度和剪切速率下的动 力过程进行了数值模拟，分析了其在由瞬时接触的快速流动向持续接触的准静态流动的转变 过程及其行为特点. 通过对不同材料性质下相变过渡区内颗粒材料的宏观应力、接触时间数、 配位数、团聚颗粒数量、有效摩擦系数等参量的计算，更加全面地描述了非均匀颗粒材料在 类固-液相变过程中的基本特征. 基于以上数值计算结果，建立了一个适用于颗粒材料 类固态、类液态以及其相变过程的本构方程，并通过剪切室实验结果验证了它的合理性.     

DISCRETE AND CONTINUUM MODELLING OF GRANULAR FLOW

This paper analyses three popular methods simulating granular flow at different time and length scales： discrete element method （DEM）, averaging method and viscous, elastic-plastic continuum model. The theoretical models of these methods and their applications to hopper flows are discussed. It is shown that DEM is an effective method to study the fundamentals of granular flow at a particle or microscopic scale. By use of the continuum approach, granular flow can also be described at a continuum or macroscopic scale. Macroscopic quantities such as velocity and stress can be obtained by use of such computational method as FEM. However, this approach depends on the constitutive relationship of materials and ignores the effect of microscopic structure of granular flow. The combined approach of DEM and averaging method can overcome this problem. The approach takes into account the discrete nature of granular materials and does not require any global assumption and thus allows a better understanding of the fundamental mechanisms of granular flow. However, it is difficult to adapt this approach to process modelling because of the limited number of particles which can be handled with the present computational capacity, and the difficulty in handling non-spherical particles. Further work is needed to develoo an aoorooriate aooroach to overcome these problems.     

Multi-mechanism anisotropic model for granular materials

By representing the assembly by a simplified column model, a constitutive theory, referred to as sliding–rolling theory, was recently developed for a two-dimensional assembly of rods subjected to biaxial loading, and then extended to a three-dimensional assembly of spheres subjected to triaxial (equibiaxial) loading. The sliding–rolling theory provides a framework for developing a phenomenological constitutive law for granular materials, which is the objective of the present work. The sliding–rolling theory provides information concerning yield and flow directions during radial and non-radial loading. In addition, the theory provides information on the role of fabric anisotropy on the stress–strain behavior and critical state shear strength. In the present paper, a multi-axial phenomenological model is developed within the sliding–rolling framework by utilizing the concepts of critical state, classical elasto-plasticity and bounding surface. The resulting theory involves two yield surfaces and falls within the definition of the multi-mechanism models. Computational issues concerning the solution uniqueness for stress states at the corner of yield surfaces are addressed. The effect of initial and induced fabric anisotropy on the constitutive behavior is incorporated. It is shown that the model is capable of simulating the effect of anisotropy, and the behavior of loose and dense sands under drained and undrained loading.     

A micromechanics-based elastoplastic damage model for granular materials at low confining pressure

This paper is devoted to the formulation of a micromechanics-based constitutive model for granular materials under relatively low confining pressure. The constitutive formulation is performed within the general framework of homogenization for granular materials. However, new rigorous stress localization laws are proposed. Some local constitutive relations are established under the consideration of irreversible thermodynamics. Macroscopic plastic deformation is obtained by considering local plastic sliding in a limit number of families of contact planes. The plastic sliding at each contact plane is described by a non-associated plastic flow rule, taking into account pressure sensitivity and normal dilatancy. Nonlinear elastic deformation related to progressive compaction of contacts is also taken into account. Material softening is described by involving damage process related to degradation of microstructure fabric. The proposed model is applied to some typical granular materials (sands). The numerical predictions are compared with experimental data.     

The constitutive equations for mixed hardening orthotropic material

IntroductionAyieldingcriterionandtheassociatedflowtheorywereproposedbyHill[1]forinitiallyorthotropicmetalsin 1 948,whichareusedbroadly .AquadraticformofstressesisusedastheplasticpotentialthatisindependentofhydrostaticstressinHillplasticitytheory .Butcompressedbyhydrostaticstress,considerabledeformationwillbeproducedinorthotropicmaterials.Inthecaseofcyclicloading ,duetoBauschingereffect,thekinematichardeningcannotbeneglected .Inthispaper,kinematichardeningandproportionalhardeningareconsidered…     

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.     

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.     

Basic features of plastic strains: From micro-mechanics to incrementally nonlinear models

Most rate-independent constitutive relations for granular materials are based on the existence of a regular flow rule. This assumption states that once the mechanical state of a material point belongs to the yield surface, then the direction of the plastic strains is independent of the loading direction. In this paper, the notion of a regular flow rule is shown to exist only for two-dimensional and axisymmetric loading conditions. By considering our incrementally nonlinear constitutive model, it is established that this notion disappears as soon as more general loading conditions are applied, as also predicted from discrete element simulations. Moreover, a sound micro-mechanical interpretation of the vanishing of a regular flow rule in three-dimensional loading conditions is given from a multi-scale perspective using the micro-directional model. This model highlights the great influence of the loading history on the shape of the plastic Gudehus response-envelope.     

Plastic potentials for anisotropic porous solids

The aim of this paper is to incorporate plastic anisotropy into constitutive equations of porous ductile metals. It is shown that plastic anisotropy of the matrix surrounding the voids in a ductile material could have an influence on both effective stress–strain relation and damage evolution. Two theoretical frameworks are envisageable to study the influence of plastic flow anisotropy: continuum thermodynamics and micromechanics. By going through the Rousselier thermodynamical formulation, one can account for the overall plastic anisotropy, in a very simple manner. However, since this model is based on a weak coupling between plasticity and damage dissipative processes, it does not predict any influence of plastic anisotropy on cavity growth, unless a more suitable choice of the thermodynamical force associated with the damage parameter is made. Micromechanically-based models are then proposed. They consist of extending the famous Gurson model for spherical and cylindrical voids to the case of an orthotropic material. We derive an upper bound of the yield surface of a hollow sphere, or a hollow cylinder, made of a perfectly plastic matrix obeying the Hill criterion. The main findings are related to the so-called ‘scalar effect’ and ‘directional effect’. First, the effect of plastic flow anisotropy on the spherical term of the plastic potential is quantified. This allows a classification of sheet materials with regard to the anisotropy factor h; this is the scalar effect. A second feature of the model is the plasticity-induced damage anisotropy. This results in directionality of fracture properties (‘directional effect’). The latter is mainly due to the principal Hill coefficients whilst the scalar effect is enhanced by ‘shear’ Hill coefficients. Results are compared to some micromechanical calculations using the finite element method.     

Micromechanical model for viscoelastic materials undergoing damage

We have derived a stress–strain relationship for viscoelastic materials undergoing damage using a granular micromechanics approach. This approach assumes the material to possess a granular meso-structure such that the material is treated as a discrete or a particulate system. By considering the particle kinematics in terms of Taylor series expansion, a continuum model of the discrete system is obtained. The material rate-dependence and damage are modeled by assuming appropriate inter-granular force–displacement relationships that satisfy thermodynamic constraints. The advantage of this micromechanical approach is that the resultant continuum model retains the discrete nature by incorporating the effect of nearest neighbor grain interactions through the inter-granular force–displacement relationship and orientation vector. As a result, the derived model has the ability to predict a number of material phenomena, such as loading-induced anisotropy, dilation or pressure sensitivity, and secondary creep, which often manifest due to material granularity.     

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.      

SPH-based simulation of granular collapse on an inclined bed

The paper presents a numerical model for simulating a granular flow and its deposition on an inclined bed. A granular material is described as an elastic–plastic continuum and its constitutive law, namely Hooke's law, is discretized on the basis of the Smoothed Particle Hydrodynamics (SPH) method. In the equation of motion, however, the artificial viscosity, which is widely used in SPH, is not applied. The diffusive term derived from Hooke's law is introduced with a diffusion coefficient that varies depending on the stress and strain rate based on the Drucker–Prager yield function. The model is verified and validated through two numerical tests. It is shown that the basic elastic–perfectly plastic characteristics are reproduced with a simple shearing test. The effects of the diffusion coefficient and spatial resolution are investigated to show the validity of the model. In the simulation of the gravitational collapse of a granular column on an inclined bed, the performance of the model from the final deposition profile, the time history of the front position of the granular flow, the maximum runout distance, and the velocity profile are investigated for several cases of basal inclinations. The calculated results show good agreement with the experimental results.     

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.     

Dynamics of Viscous Entrapped Saturated Zones in Partially Wetted Porous Media

As a typical multiphase fluid flow process, drainage in porous media is of fundamental interest both in nature and in industrial applications. During drainage processes in unsaturated soils and porous media in general, saturated regions, or clusters, in which a liquid phase fully occupies the pore space between solid grains, affect the relative permeability and effective stress of the system. Here, we experimentally study drainage processes in unsaturated granular media as a model porous system. The distribution of saturated clusters is analysed by optical imaging under different drainage conditions, with pore-scale information from Voronoi and Delaunay tessellation used to characterise the topology of saturated cluster distributions. By employing statistical analyses, we describe the observed spatial and temporal evolution of multiphase flow and fluid entrapment in granular media. Results indicate that the distributions of both the crystallised cell size and pore size are positively correlated to the spatial and temporal distribution of saturated cluster sizes. The saturated cluster size is found to follow a lognormal distribution, in which the generalised Bond number (\( Bo^{*} \)) correlates negatively to the scale parameter (μ) and positively to the shape parameter (σ). With further consideration of the total surface energy obtained based on liquid–air interfaces, we were able to include additional grain-scale information in the constitutive modelling of unsaturated soils using both the degree of saturation and generalised Bond number. These findings can be used to connect pore-scale behaviour with overall hydro-mechanical characteristics in granular systems.     

Micromechanical study of plasticity of granular materials

Plastic deformation of granular materials is investigated from the micromechanical viewpoint, in which the assembly of particles and interparticle contacts is considered as a mechanical structure. This is done in three ways. Firstly, by investigating the degree of redundancy of the system by comparing the number of force degrees of freedom at contacts with the number of governing equilibrium equations; Secondly, by determining the spectrum of eigenvalues of the stiffness matrix for the structure that is represented by the particles and their contacts; Thirdly, by investigating the evolution with imposed strain of the continuum elastic stiffness tensor of the system. It is found that, with increasing imposed strain, the degree of redundancy rapidly evolves towards a state with small redundancy, i.e. the system becomes nearly statically determinate. The spectrum of the system shows many singular and near-singular modes at peak shear strength and at large strain. The continuum elastic stiffness tensor becomes strongly anisotropic with increasing imposed strain and shows strong non-affinity of deformation. The assumption of a constant and isotropic elastic stiffness tensor in elasto-plastic continuum constitutive relations for granular materials is generally incorrect. Overall, the plastic continuum behaviour of granular materials originates from the plastic frictional behaviour at contacts and from damage in the form of changes in the contact network.