颗粒材料的离散颗粒模型与离散—连续耦合模型及数值方法
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摘要
颗粒材料是自然界分布与工程应用最广泛的物质之一。但颗粒材料物理性质非常复杂，人们对颗粒材料的认识还处于比较初级的阶段。近年来颗粒材料力学行为的研究及颗粒材料力学的发展引起了众多科研工作者的注意。
颗粒材料力学最终目的之一是建立颗粒材料宏观行为与微观量之间的联系，基于离散颗粒模型的离散单元法由于方便处理颗粒层次的物理量，成为颗粒材料力学行为数值研究的主要工具。颗粒材料离散单元法的核心是颗粒之间接触力的计算，但是接触颗粒之间的相对运动机制，特别是切向滑动与滚动仍没有达成共识，为此本文对不同半径的圆形颗粒在接触点处相对运动作了系统地分析，定义了相对运动的测度，区分了单个颗粒的角速度、相对滚动速度、相对滑动速度、相对滚动位移、相对滑动位移等概念。在此基础上应用弹簧-粘壶-滑片模型给出了滚动阻矩、滚动阻力、滑动阻力、法向力及切向力的计算公式，形成一个适用于颗粒材料动力模拟的离散颗粒模型。考虑到目前文献中对滚动机制的物理及数值研究不足，该模型强调了滚动机制在颗粒材料破坏中重要性。此外为了表征每个颗粒相对于周围颗粒的运动，基于连续介质理论定义了颗粒的名义有效应变及名义体积应变。
可破碎土具有特殊的力学和工程特性。对土体颗粒破碎的力学机理和工程效应的研究关系到各种大型工程建筑的安全和正常工作。同时细观尺度上颗粒破碎和材料宏观非线性行为有直接的联系，研究颗粒破碎对土力学的完善有重要意义。并且破碎机理的研究可以促进制粒技术的发展。本文在所建议传统离散颗粒模型基础上引入了多尺度分级模型，定义了不同尺度级别的基本颗粒和颗粒簇。在统计断裂力学Weibull分布公式基础上应用名义应力给出了多尺度分级模型意义下颗粒破碎概率；结合颗粒簇的解簇模型建议了一个可模拟颗粒破碎的颗粒分级离散颗粒模型。
目前应用离散单元法对颗粒材料的研究主要集中在单相颗粒材料或干颗粒材料领域。但一方面少量液体即可引起干土壤颗粒粘性及抗剪性能的很大变化，另一方面由于对干颗粒材料研究的深入，近年来基于离散单元法对湿相颗粒材料的研究逐渐得到了重视。结合固相离散颗粒模型间隙水存在两种数学模型，即液-桥模型与连续介质模型。液．桥模型较为简单，通常以粘性体现湿相影响，但是一方面液桥模型只适合较少量液体的情况，另一方面液桥模型无法模拟间隙水流动。本文基于连续途径的颗粒材料间隙水N-S方程描述及Darcy定律描述，借鉴特征线方案发展了用于求解颗粒材料液相的特征线SPH方案。形成了基于离散颗粒模型及特征线SPH方案的饱和含液颗粒材料数值求解体系。在引入被动空气假设的前提下，进一步模拟了降雨条件下非饱和颗粒材料含液量的变化。同时基于固相连续介质概念，讨论了颗粒材料孔隙度变化与体积应变之间的关系，给出了孔隙水压力与体积应变及颗粒材料胀缩之间的关系。
概言之，论文的主要工作包括：发展了一个强调滚动机制的颗粒材料离散颗粒模型，并在此基础上引入基本颗粒与颗粒簇的概念，形成了一个能够模拟颗粒破碎的颗粒分级离散颗粒模型；为含液颗粒材料的液固耦合分析发展了离散．连续耦合模型，即应用离散颗粒模型模拟固相，而应用基于平均N-S方程或Darcy定律的连续介质模型模拟液相。进一步发展了基于离散单元方案与特征线SPH的数值求解过程。
为了论文的完整性，本文还简单介绍了颗粒材料及颗粒材料力学的一些基本概念，讨论了离散单元法刚性球模型与软球模型的区别与联系。此外，还综述了常用的描述接触点力-位移关系的本构模型及颗粒材料的常用边界条件提法。
Granular material is one kind of matter, which exists most widely in nature and is comprehensively applied to engineering practices. However, the physical properties of granular material are so complicated that it still remains as one among a few most less-understood materials. In the recent years, the researches on mechanical behavior of granular materials and the developments in mechanics of granular materials have attracted comprehensive attentions in many scientific and engineering fields.
A final goal in the study of the mechanics of granular materials is to formulate their macro-behavior in terms of micro-quantities. The discrete element method (DEM) based on the discrete particle model (DPM) has been widely accepted as a useful tool in numerical simulation of mechanical behavior of granular materials owing to its easy to determine physical quantities at grain level. The calculation of contact forces between particles is the core part of DEM, however, a universally accepted view on the mechanism of relative movement between two typical particles in contact, especially for the sliding and rolling movement in tangential direction, has not been achieved. One of the objectives of this thesis is to propose a discrete particle model for granular materials. Each grain in the particulate system under consideration is assumed as rigid and circular. Starting with kinematical analysis of relative movements of two typical grains with different radius in contact, measurements to describe both the relative rolling and sliding motions, including the translational and angular velocities (displacements), are defined. Further, Both the rolling and sliding friction tangential forces, and the rolling friction resistance moment, which are constitutively related to the defined relative motion measurements respectively, are formulated and integrated into the framework of dynamic model of DEM. In view of the fact that the rolling resistance mechanism described in currently available models in the literature is inadequate in both physical and numerical aspects, the proposed model is thus focused on an adequate description of the rolling resistance mechanism. Moreover, the definitions of nominal effective and volumetric strains are proposed on the formalism of the continuum theory to illustrate the movements of each particle relating to its surrounding particles in a particulate system.
The crushable soil possesses special characters in mechanical and engineering properties. The studies on crushing mechanism of soil granules and its engineering effects are strongly related to safety and work in order to be ensured for engineering constructions with large scales. Meanwhile, the particle crushing at micro-scale directly influences upon macroscopic nonlinear behavior of granular materials, studies on particle crushing possess a scientific significance in advancing the theory of soil mechanics. In addition, the studies will be also beneficial to improve the granulation technology. Based on the proposed conventional discrete particle model, a hierarchical multi-scale model for granular materials, in which basic particles and particle clusters are defined for simulation of particle crushing in different scales, is presented in this thesis. The particle crushing probability in the model is derived on the basis of the Weibull distribution formula in the frame of statistical fracture mechanics with the nominal particle stress concept. With the de-cluster mechanism of the particle clusters the proposed model can be used to simulate particle crushing phenomenon in failure analysis of granular materials.
At present, the discrete element methods are mostly concerned with single-phase granular materials or dry granular materials. However, it is well known that the addition of liquid to a dry granular medium, even in small quantity, can modify in a significant way its behavior when compared to the dry state. Moreover, with deep and intensive investigations on dry granular medium by means of DEM, the numerical simulation for wet granular materials based on DEM has been paid more attentions in recent years. In the frame of discrete particle model for modeling of granular materials, there exist two main types of mathematical models for modelling interstitial liquid, i.e. the discrete (liquid-bridge)and continuum models. The liquid-bridge model is mainly used for low moisture content, i.e. the pendular and funicular state of interstitial liquid, which exists as liquid-bridges around the contacts between solid particles and exerts the adhesion and suction effects at liquid-air interfaces. Although the fluid phase in a porous medium is inherently discrete at the microscopic level and the discrete (i.e. the liquid-bridge) model takes into account the effect of capillary liquid surrounding the contacts between solid particles on inter-particle adhesion, it is noticed that the discrete models for interstitial liquid are not able to model the fluid transport within a porous medium and can be only used in the case of low liquid content. On the other hand, with an increasing degree of liquid saturation in the medium, the capillary and droplet states of interstitial liquid develop and the voids are fully or nearly saturated with liquid. In such cases, the liquid-bridge model is no longer valid. To model fluid flow through a porous medium and its interaction with the solid grains of the medium, a continuum model should be employed. In this thesis, a coupled discrete particle-continuum model for wet granular materials is developed. The motion of the interstitial fluid phase and its interaction with the solid phase are described by means of the two parallel continuum schemes govemed by the averaged incompressible N-S equations and Darcy's law respectively. In light of SPH approach, the characteristic based SPH method is presented for numerical modelling of pore fluid flow. A numerical Solution procedure based on DEM and characteristic based SPH for saturated granular materials is formulated. Furthermore, as the passive air phase assumption is introduced, the change of water content in unsaturated granular materials can be simulated. In addition, based on the concepts of continuum theory the relation between the voidage variation and volumetric stain is discussed, the equation to determine the pore water pressure depending on volumetric strain and dilation (or contraction) of particulate assemblage are given.
The main work of the present thesis can be summarized as follows. A discrete particle model for granular materials in which the rolling mechanism is particularly emphasized is first developed. With introduction of the concepts of basic grain and the cluster into the model, a hierarchical discrete particle model capable of modeling particle crushing phenomenon is formulated. To model the coupled response of discrete solid particles with pore fluid a discrete particle-continuum model for wet granular materials is further developed, in which the solid phase is modelled by the discrete particle model while the interstitial liquid is modelled by the two continuum schemes based on the averaged incompressible N-S equations or the Darcy's law respectively. Finally a numerical solution procedure on the basis of the DEM and characteristic based SPH is developed.
For "self-completeness" of present thesis, fundamental concepts of both granular material and mechanics of granular material are briefly introduced; differences and relations between rigid-particle models and soft-particle models are discussed. In addition, most comprehensively used constitutive models for displacement-force relation at contacts and the boundary condition presentations in common use for granular assemblages are summarized.
颗粒材料力学最终目的之一是建立颗粒材料宏观行为与微观量之间的联系，基于离散颗粒模型的离散单元法由于方便处理颗粒层次的物理量，成为颗粒材料力学行为数值研究的主要工具。颗粒材料离散单元法的核心是颗粒之间接触力的计算，但是接触颗粒之间的相对运动机制，特别是切向滑动与滚动仍没有达成共识，为此本文对不同半径的圆形颗粒在接触点处相对运动作了系统地分析，定义了相对运动的测度，区分了单个颗粒的角速度、相对滚动速度、相对滑动速度、相对滚动位移、相对滑动位移等概念。在此基础上应用弹簧-粘壶-滑片模型给出了滚动阻矩、滚动阻力、滑动阻力、法向力及切向力的计算公式，形成一个适用于颗粒材料动力模拟的离散颗粒模型。考虑到目前文献中对滚动机制的物理及数值研究不足，该模型强调了滚动机制在颗粒材料破坏中重要性。此外为了表征每个颗粒相对于周围颗粒的运动，基于连续介质理论定义了颗粒的名义有效应变及名义体积应变。
可破碎土具有特殊的力学和工程特性。对土体颗粒破碎的力学机理和工程效应的研究关系到各种大型工程建筑的安全和正常工作。同时细观尺度上颗粒破碎和材料宏观非线性行为有直接的联系，研究颗粒破碎对土力学的完善有重要意义。并且破碎机理的研究可以促进制粒技术的发展。本文在所建议传统离散颗粒模型基础上引入了多尺度分级模型，定义了不同尺度级别的基本颗粒和颗粒簇。在统计断裂力学Weibull分布公式基础上应用名义应力给出了多尺度分级模型意义下颗粒破碎概率；结合颗粒簇的解簇模型建议了一个可模拟颗粒破碎的颗粒分级离散颗粒模型。
目前应用离散单元法对颗粒材料的研究主要集中在单相颗粒材料或干颗粒材料领域。但一方面少量液体即可引起干土壤颗粒粘性及抗剪性能的很大变化，另一方面由于对干颗粒材料研究的深入，近年来基于离散单元法对湿相颗粒材料的研究逐渐得到了重视。结合固相离散颗粒模型间隙水存在两种数学模型，即液-桥模型与连续介质模型。液．桥模型较为简单，通常以粘性体现湿相影响，但是一方面液桥模型只适合较少量液体的情况，另一方面液桥模型无法模拟间隙水流动。本文基于连续途径的颗粒材料间隙水N-S方程描述及Darcy定律描述，借鉴特征线方案发展了用于求解颗粒材料液相的特征线SPH方案。形成了基于离散颗粒模型及特征线SPH方案的饱和含液颗粒材料数值求解体系。在引入被动空气假设的前提下，进一步模拟了降雨条件下非饱和颗粒材料含液量的变化。同时基于固相连续介质概念，讨论了颗粒材料孔隙度变化与体积应变之间的关系，给出了孔隙水压力与体积应变及颗粒材料胀缩之间的关系。
概言之，论文的主要工作包括：发展了一个强调滚动机制的颗粒材料离散颗粒模型，并在此基础上引入基本颗粒与颗粒簇的概念，形成了一个能够模拟颗粒破碎的颗粒分级离散颗粒模型；为含液颗粒材料的液固耦合分析发展了离散．连续耦合模型，即应用离散颗粒模型模拟固相，而应用基于平均N-S方程或Darcy定律的连续介质模型模拟液相。进一步发展了基于离散单元方案与特征线SPH的数值求解过程。
为了论文的完整性，本文还简单介绍了颗粒材料及颗粒材料力学的一些基本概念，讨论了离散单元法刚性球模型与软球模型的区别与联系。此外，还综述了常用的描述接触点力-位移关系的本构模型及颗粒材料的常用边界条件提法。
Granular material is one kind of matter, which exists most widely in nature and is comprehensively applied to engineering practices. However, the physical properties of granular material are so complicated that it still remains as one among a few most less-understood materials. In the recent years, the researches on mechanical behavior of granular materials and the developments in mechanics of granular materials have attracted comprehensive attentions in many scientific and engineering fields.
A final goal in the study of the mechanics of granular materials is to formulate their macro-behavior in terms of micro-quantities. The discrete element method (DEM) based on the discrete particle model (DPM) has been widely accepted as a useful tool in numerical simulation of mechanical behavior of granular materials owing to its easy to determine physical quantities at grain level. The calculation of contact forces between particles is the core part of DEM, however, a universally accepted view on the mechanism of relative movement between two typical particles in contact, especially for the sliding and rolling movement in tangential direction, has not been achieved. One of the objectives of this thesis is to propose a discrete particle model for granular materials. Each grain in the particulate system under consideration is assumed as rigid and circular. Starting with kinematical analysis of relative movements of two typical grains with different radius in contact, measurements to describe both the relative rolling and sliding motions, including the translational and angular velocities (displacements), are defined. Further, Both the rolling and sliding friction tangential forces, and the rolling friction resistance moment, which are constitutively related to the defined relative motion measurements respectively, are formulated and integrated into the framework of dynamic model of DEM. In view of the fact that the rolling resistance mechanism described in currently available models in the literature is inadequate in both physical and numerical aspects, the proposed model is thus focused on an adequate description of the rolling resistance mechanism. Moreover, the definitions of nominal effective and volumetric strains are proposed on the formalism of the continuum theory to illustrate the movements of each particle relating to its surrounding particles in a particulate system.
The crushable soil possesses special characters in mechanical and engineering properties. The studies on crushing mechanism of soil granules and its engineering effects are strongly related to safety and work in order to be ensured for engineering constructions with large scales. Meanwhile, the particle crushing at micro-scale directly influences upon macroscopic nonlinear behavior of granular materials, studies on particle crushing possess a scientific significance in advancing the theory of soil mechanics. In addition, the studies will be also beneficial to improve the granulation technology. Based on the proposed conventional discrete particle model, a hierarchical multi-scale model for granular materials, in which basic particles and particle clusters are defined for simulation of particle crushing in different scales, is presented in this thesis. The particle crushing probability in the model is derived on the basis of the Weibull distribution formula in the frame of statistical fracture mechanics with the nominal particle stress concept. With the de-cluster mechanism of the particle clusters the proposed model can be used to simulate particle crushing phenomenon in failure analysis of granular materials.
At present, the discrete element methods are mostly concerned with single-phase granular materials or dry granular materials. However, it is well known that the addition of liquid to a dry granular medium, even in small quantity, can modify in a significant way its behavior when compared to the dry state. Moreover, with deep and intensive investigations on dry granular medium by means of DEM, the numerical simulation for wet granular materials based on DEM has been paid more attentions in recent years. In the frame of discrete particle model for modeling of granular materials, there exist two main types of mathematical models for modelling interstitial liquid, i.e. the discrete (liquid-bridge)and continuum models. The liquid-bridge model is mainly used for low moisture content, i.e. the pendular and funicular state of interstitial liquid, which exists as liquid-bridges around the contacts between solid particles and exerts the adhesion and suction effects at liquid-air interfaces. Although the fluid phase in a porous medium is inherently discrete at the microscopic level and the discrete (i.e. the liquid-bridge) model takes into account the effect of capillary liquid surrounding the contacts between solid particles on inter-particle adhesion, it is noticed that the discrete models for interstitial liquid are not able to model the fluid transport within a porous medium and can be only used in the case of low liquid content. On the other hand, with an increasing degree of liquid saturation in the medium, the capillary and droplet states of interstitial liquid develop and the voids are fully or nearly saturated with liquid. In such cases, the liquid-bridge model is no longer valid. To model fluid flow through a porous medium and its interaction with the solid grains of the medium, a continuum model should be employed. In this thesis, a coupled discrete particle-continuum model for wet granular materials is developed. The motion of the interstitial fluid phase and its interaction with the solid phase are described by means of the two parallel continuum schemes govemed by the averaged incompressible N-S equations and Darcy's law respectively. In light of SPH approach, the characteristic based SPH method is presented for numerical modelling of pore fluid flow. A numerical Solution procedure based on DEM and characteristic based SPH for saturated granular materials is formulated. Furthermore, as the passive air phase assumption is introduced, the change of water content in unsaturated granular materials can be simulated. In addition, based on the concepts of continuum theory the relation between the voidage variation and volumetric stain is discussed, the equation to determine the pore water pressure depending on volumetric strain and dilation (or contraction) of particulate assemblage are given.
The main work of the present thesis can be summarized as follows. A discrete particle model for granular materials in which the rolling mechanism is particularly emphasized is first developed. With introduction of the concepts of basic grain and the cluster into the model, a hierarchical discrete particle model capable of modeling particle crushing phenomenon is formulated. To model the coupled response of discrete solid particles with pore fluid a discrete particle-continuum model for wet granular materials is further developed, in which the solid phase is modelled by the discrete particle model while the interstitial liquid is modelled by the two continuum schemes based on the averaged incompressible N-S equations or the Darcy's law respectively. Finally a numerical solution procedure on the basis of the DEM and characteristic based SPH is developed.
For "self-completeness" of present thesis, fundamental concepts of both granular material and mechanics of granular material are briefly introduced; differences and relations between rigid-particle models and soft-particle models are discussed. In addition, most comprehensively used constitutive models for displacement-force relation at contacts and the boundary condition presentations in common use for granular assemblages are summarized.
引文
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