基于平均场理论的颗粒材料离散颗粒集合-Cosserat连续体模型多尺度模拟
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摘要
颗粒材料广泛存在于自然界和工程实际中,如岩石、土壤、粮食、药品等。一般说来颗粒材料由随机分布的离散颗粒及其间孔隙流体组成,具高度非均质性。颗粒材料力学行为的理论研究与数值模拟是岩土力学、陶瓷烧制等工程科学中的重要课题,引起了众多研究者的关注。
颗粒材料宏观上通常模型化为连续体,数值模拟一般采用有限单元法(或无网格方法)。连续体途径的优势在于主要未知变量数目依赖于宏观有限单元网格密度,较为有限,因而在工程实际问题中得到广泛应用；缺点在于唯象本构关系包含很多没有物理意义、很难确定的参数。颗粒材料在细观上模型化为离散颗粒集合体,数值模拟采用离散单元法。其优势在于体现材料的离散颗粒本质,能够模拟各种破坏现象；缺点在于应用于实际问题时所需颗粒数目巨大(上亿甚至更多的颗粒),计算量难以接受。颗粒材料的多尺度方法研究则提供了一条能够充分利用二者优势避免各自缺点的途径。
本论文致力于发展基于平均场理论的细观离散颗粒集合-宏观Cosserat连续体模型的多尺度模拟方法,内容包括：(1)推导非均质Cosserat连续体平均场理论的Hill定理,给定合适的表征元(representative volume element, RVE)边界条件；(2)建立细观离散颗粒集合-宏观Cosserat连续体模型的多尺度计算均匀化方法；(3)发展基于细观方向平均模型的颗粒材料宏观Cosserat连续体本构关系。
在平均场理论框架内实施非均质材料的均匀化模拟首先需要提出Hill定理,它为正确给定RVE边界条件提供了依据。基于Cosserat连续介质控制方程,参考经典Cauchy连续体Hill定理,本文推导了非均质Cosserat连介质平均场理论的Hill定理。从该Hill定理出发,详细分析和讨论了各种形式的RVE边界条件,并据此给出了满足细-宏观Hill-Mandel能量等价条件、符合平均场理论基本假定的RVE边界条件,为基于Cosserat连续体模型的细-宏观均匀化模拟提供了坚实基础。
基于经典连续体计算均匀化方法和所推导Cosserat连续介质平均场理论基本公式,本文发展了颗粒材料多尺度计算均匀化方法,细观尺度上采用基于离散颗粒模型的离散单元法,宏观尺度上采用基于Cosserat连续体模型的有限元法。该方法在宏观尺度有限单元内的每个积分点处设定一个反映材料细观离散颗粒微结构的RVE,通过RVE边值问题的计算求解获得材料在该点处的局部宏观响应,包括积分点的应力和切线模量矩阵。详细分析和推导了该计算均匀化方案实施过程中的关键环节,包括RVE边界条件的给定和转换、离散颗粒集合体RVE刚度阵的分块与凝聚、宏观率形式应力和微曲率表达式及其一致性切线模量张量的推导,以及宏观积分点处切线模量矩阵的组集和形成。该方法能够有效计及离散颗粒微结构及其演变信息,而不需要提供复杂的、包含许多没有明确物理意义且不易识别的材料参数的唯象宏观本构关系,并且在模拟材料宏观响应的同时,还能够显示相应的微结构响应特征。数值算例结果显示出所发展方法的有效性和可应用性。
考虑到细观颗粒不仅具有平移自由度,还具有独立的旋转自由度；颗粒间不仅存在接触力还能够传递力矩,基于经典连续体细观微-方向模型(micro-directional model),发展了基于细观方向平均模型的颗粒材料宏观Cosserat连续体本构关系。在细观尺度上考虑了颗粒的旋转自由度及颗粒间接触力矩,RVE内接触分布的概率密度函数体现了材料微结构对宏观各向异性和非均质性的影响,运动学分量的细宏观联系通过Cosserat连续体平均场理论的Hill-Mandel能量等价条件建立。具体给出了均质、各向同性Cosserat连续体弹性常数的细观参数表达式。分别给出了基于离散单元法数值模拟和本文理论公式预测的典型离散颗粒集合体的宏观响应量,二者的一致性说明了所发展模型的有效性。此外,本文所发展理论公式也为合理解释离散单元法数值模拟结果提供了参考依据。
Granular materials exist widely in nature and engineering practice, such as rock, soil, food supplies and medicament etc. Generally speaking, granular materials are composed of randomly packed discrete particles partially or fully filled with void fluids and are characterized with highly heterogeneity. Theoretical and numerical studies of mechanical behavior of granular materials are challenging tasks in engineering science and attract the interests of the researchers from a variety of disciplines.
Granular materials are modeled in the macroscopic view as continuum, which is discretized by means of the finite element method (FEM) (and/or its alternatives such as the mesh-free method) for the numerical simulation of macro-mechanical behaviors in the material domain. The advantage of the continuum approach lies in that the number of primary unknown variables to be determined is rather limited depending on the FE mesh density so that the approach has been widely and effectively applied to engineering problems. However, the disadvantage of the approach is that phenomenological constitutive relations based on the continuum model include many parameters with no physical meaning and difficult to be identified.
On the other hand, granular materials are modeled in the microscopic view as a discrete particle assembly, which is consistent with the physical characteristics of granular materials. The numerical simulation of mechanical behaviors for the particle assembly is performed by using the discrete element method (DEM). The advantages of the approach lie not only in the rationality and simplicity of constitutive relations defined at particle contacts so that the realistic constitutive behaviors of granular materials can be effectively described and predicted, but also in the detection of the mechanisms of different types of material failure processes. However, the motions of a huge number of particles (even billions of particles) need to be quantitatively determined in its applications to engineering problems, which is far beyond of the computer capability even nowadays.
Many efforts have been devoted to develop the multi-scale methods for granular materials to fully exert the advantages of both macro and micro approaches and to avoid their respective disadvantages.
The present work is focused on the multi-scale modeling of discrete particle assembly Cosserat continuum model for granular materials in the frame of average-field theory, including:(1) Derivation of Hill's lemma of the average-field theory for heterogeneous Cosserat continuum, with which proper boundary conditions for the representative volume element (RVE) are presented; (2) Development of the computational homogenization method for granular materials modeled as discrete particle assembly and Cosserat continuum in micro- and macro- scales respectively; (3) Development of a multi-scale constitutive relation of macro Cosserat continuum description for granular materials in light of the micro-directional model proposed for classical Cauchy continuum in the literature.
The presentation of Hill's lemma is first required to properly specify RVE boundary conditions for micro-macro homogenization modeling of heterogeneous materials in the frame of average-field theory. On the basis of Hill's lemma for classical Cauchy continuum, a version of Hill's lemma is systematically derived for micro-macro homogenization modeling of heterogeneous Cosserat continuum. According to the derived Hill's lemma, different types of statically admissible stress boundary conditions and/or kinematically admissible displacement boundary conditions to be imposed on the RVE are extracted and discussed. Then proper RVE boundary conditions are determined to satisfy the Hill-Mandel energy condition and fundamental assumptions of the first-order average-field theory. This work provides the foundation of the following studies of micro-macro homogenization methods for granular materials.
In light of the computational homogenization approach and the derived formulae of average-field theory for heterogeneous Cosserat continuum, a micro-macro computational homogenization scheme of discrete particle assembly-Cosserat continuum model for granular materials is developed in the present work.
DEM is employed for the microscopic analysis and FEM is adopted for the macroscopic computation. The detailed procedure of the developed computational homogenization scheme is given and discussed. With the link between the discrete particle assembly and its Cosserat continuum equivalent in an individual RVE, the boundary conditions prescribed on the RVE modeled as Cosserat continuum are transformed into those prescribed to the peripheral particles of the RVE modeled as the discrete particle assembly. The average stresses and strains and their variations over the RVE defined for Cosserat continuum equivalent are then determined by the physical quantities of the discrete particle assembly.. The consistent macroscopic modular tensors and the macroscopic constitutive relations defined at the integration point are formulated in terms of the averaged behavior of associated microstructures.
As the proposed micro-macro computational homogenization procedure is used within the finite element framework, there is no need to specify the macroscopic constitutive relation at the macroscopic integration points. The results of typical numerical examples for granular materials demonstrate the validity of developed method and its advantages in comparison to existing computational homogenization methods of granular materials in the literature.
In accordance with the nature of the granular medium in the sense that each discrete particle at the micro-scale possesses rotational degrees of freedom in addition to translational degrees of freedom in kinematics and is capable of bearing and transmitting couples from one particle to the other in contact in kinetics, a micromechanically based constitutive model is developed for macro Cosserat continuum description of granular materials in light of the micro-directional model proposed to describe the constitutive behavior of classical Cauchy continuum. The effects of micro structures/properties on the anisotropy and heterogeneity of macroscopic constitutive behaviors are embodied with the probability density function of the contact distribution. The micro-macro transitions of kinematical quantities are fulfilled by means of Hill-Mandel condition of the average-field theory for heterogeneous Cosserat continuum. To analyze the asymptotic trend and to validate the proposed model, the micromechanically-based expressions of macroscopic elastic constants for the materials under homogeneous and isotropic assumptions are particularly derived. The validity of the developed model is verified through the comparisons of the theoretical predictions given by the derived formulae and numerical results obtained by using the discrete element method on overall behavior of a regularly packed granular assembly. Moreover, the derived formulae also provide theoretical interpretations to the numerical results obtained by the discrete element method.
颗粒材料宏观上通常模型化为连续体,数值模拟一般采用有限单元法(或无网格方法)。连续体途径的优势在于主要未知变量数目依赖于宏观有限单元网格密度,较为有限,因而在工程实际问题中得到广泛应用；缺点在于唯象本构关系包含很多没有物理意义、很难确定的参数。颗粒材料在细观上模型化为离散颗粒集合体,数值模拟采用离散单元法。其优势在于体现材料的离散颗粒本质,能够模拟各种破坏现象；缺点在于应用于实际问题时所需颗粒数目巨大(上亿甚至更多的颗粒),计算量难以接受。颗粒材料的多尺度方法研究则提供了一条能够充分利用二者优势避免各自缺点的途径。
本论文致力于发展基于平均场理论的细观离散颗粒集合-宏观Cosserat连续体模型的多尺度模拟方法,内容包括：(1)推导非均质Cosserat连续体平均场理论的Hill定理,给定合适的表征元(representative volume element, RVE)边界条件；(2)建立细观离散颗粒集合-宏观Cosserat连续体模型的多尺度计算均匀化方法；(3)发展基于细观方向平均模型的颗粒材料宏观Cosserat连续体本构关系。
在平均场理论框架内实施非均质材料的均匀化模拟首先需要提出Hill定理,它为正确给定RVE边界条件提供了依据。基于Cosserat连续介质控制方程,参考经典Cauchy连续体Hill定理,本文推导了非均质Cosserat连介质平均场理论的Hill定理。从该Hill定理出发,详细分析和讨论了各种形式的RVE边界条件,并据此给出了满足细-宏观Hill-Mandel能量等价条件、符合平均场理论基本假定的RVE边界条件,为基于Cosserat连续体模型的细-宏观均匀化模拟提供了坚实基础。
基于经典连续体计算均匀化方法和所推导Cosserat连续介质平均场理论基本公式,本文发展了颗粒材料多尺度计算均匀化方法,细观尺度上采用基于离散颗粒模型的离散单元法,宏观尺度上采用基于Cosserat连续体模型的有限元法。该方法在宏观尺度有限单元内的每个积分点处设定一个反映材料细观离散颗粒微结构的RVE,通过RVE边值问题的计算求解获得材料在该点处的局部宏观响应,包括积分点的应力和切线模量矩阵。详细分析和推导了该计算均匀化方案实施过程中的关键环节,包括RVE边界条件的给定和转换、离散颗粒集合体RVE刚度阵的分块与凝聚、宏观率形式应力和微曲率表达式及其一致性切线模量张量的推导,以及宏观积分点处切线模量矩阵的组集和形成。该方法能够有效计及离散颗粒微结构及其演变信息,而不需要提供复杂的、包含许多没有明确物理意义且不易识别的材料参数的唯象宏观本构关系,并且在模拟材料宏观响应的同时,还能够显示相应的微结构响应特征。数值算例结果显示出所发展方法的有效性和可应用性。
考虑到细观颗粒不仅具有平移自由度,还具有独立的旋转自由度；颗粒间不仅存在接触力还能够传递力矩,基于经典连续体细观微-方向模型(micro-directional model),发展了基于细观方向平均模型的颗粒材料宏观Cosserat连续体本构关系。在细观尺度上考虑了颗粒的旋转自由度及颗粒间接触力矩,RVE内接触分布的概率密度函数体现了材料微结构对宏观各向异性和非均质性的影响,运动学分量的细宏观联系通过Cosserat连续体平均场理论的Hill-Mandel能量等价条件建立。具体给出了均质、各向同性Cosserat连续体弹性常数的细观参数表达式。分别给出了基于离散单元法数值模拟和本文理论公式预测的典型离散颗粒集合体的宏观响应量,二者的一致性说明了所发展模型的有效性。此外,本文所发展理论公式也为合理解释离散单元法数值模拟结果提供了参考依据。
Granular materials exist widely in nature and engineering practice, such as rock, soil, food supplies and medicament etc. Generally speaking, granular materials are composed of randomly packed discrete particles partially or fully filled with void fluids and are characterized with highly heterogeneity. Theoretical and numerical studies of mechanical behavior of granular materials are challenging tasks in engineering science and attract the interests of the researchers from a variety of disciplines.
Granular materials are modeled in the macroscopic view as continuum, which is discretized by means of the finite element method (FEM) (and/or its alternatives such as the mesh-free method) for the numerical simulation of macro-mechanical behaviors in the material domain. The advantage of the continuum approach lies in that the number of primary unknown variables to be determined is rather limited depending on the FE mesh density so that the approach has been widely and effectively applied to engineering problems. However, the disadvantage of the approach is that phenomenological constitutive relations based on the continuum model include many parameters with no physical meaning and difficult to be identified.
On the other hand, granular materials are modeled in the microscopic view as a discrete particle assembly, which is consistent with the physical characteristics of granular materials. The numerical simulation of mechanical behaviors for the particle assembly is performed by using the discrete element method (DEM). The advantages of the approach lie not only in the rationality and simplicity of constitutive relations defined at particle contacts so that the realistic constitutive behaviors of granular materials can be effectively described and predicted, but also in the detection of the mechanisms of different types of material failure processes. However, the motions of a huge number of particles (even billions of particles) need to be quantitatively determined in its applications to engineering problems, which is far beyond of the computer capability even nowadays.
Many efforts have been devoted to develop the multi-scale methods for granular materials to fully exert the advantages of both macro and micro approaches and to avoid their respective disadvantages.
The present work is focused on the multi-scale modeling of discrete particle assembly Cosserat continuum model for granular materials in the frame of average-field theory, including:(1) Derivation of Hill's lemma of the average-field theory for heterogeneous Cosserat continuum, with which proper boundary conditions for the representative volume element (RVE) are presented; (2) Development of the computational homogenization method for granular materials modeled as discrete particle assembly and Cosserat continuum in micro- and macro- scales respectively; (3) Development of a multi-scale constitutive relation of macro Cosserat continuum description for granular materials in light of the micro-directional model proposed for classical Cauchy continuum in the literature.
The presentation of Hill's lemma is first required to properly specify RVE boundary conditions for micro-macro homogenization modeling of heterogeneous materials in the frame of average-field theory. On the basis of Hill's lemma for classical Cauchy continuum, a version of Hill's lemma is systematically derived for micro-macro homogenization modeling of heterogeneous Cosserat continuum. According to the derived Hill's lemma, different types of statically admissible stress boundary conditions and/or kinematically admissible displacement boundary conditions to be imposed on the RVE are extracted and discussed. Then proper RVE boundary conditions are determined to satisfy the Hill-Mandel energy condition and fundamental assumptions of the first-order average-field theory. This work provides the foundation of the following studies of micro-macro homogenization methods for granular materials.
In light of the computational homogenization approach and the derived formulae of average-field theory for heterogeneous Cosserat continuum, a micro-macro computational homogenization scheme of discrete particle assembly-Cosserat continuum model for granular materials is developed in the present work.
DEM is employed for the microscopic analysis and FEM is adopted for the macroscopic computation. The detailed procedure of the developed computational homogenization scheme is given and discussed. With the link between the discrete particle assembly and its Cosserat continuum equivalent in an individual RVE, the boundary conditions prescribed on the RVE modeled as Cosserat continuum are transformed into those prescribed to the peripheral particles of the RVE modeled as the discrete particle assembly. The average stresses and strains and their variations over the RVE defined for Cosserat continuum equivalent are then determined by the physical quantities of the discrete particle assembly.. The consistent macroscopic modular tensors and the macroscopic constitutive relations defined at the integration point are formulated in terms of the averaged behavior of associated microstructures.
As the proposed micro-macro computational homogenization procedure is used within the finite element framework, there is no need to specify the macroscopic constitutive relation at the macroscopic integration points. The results of typical numerical examples for granular materials demonstrate the validity of developed method and its advantages in comparison to existing computational homogenization methods of granular materials in the literature.
In accordance with the nature of the granular medium in the sense that each discrete particle at the micro-scale possesses rotational degrees of freedom in addition to translational degrees of freedom in kinematics and is capable of bearing and transmitting couples from one particle to the other in contact in kinetics, a micromechanically based constitutive model is developed for macro Cosserat continuum description of granular materials in light of the micro-directional model proposed to describe the constitutive behavior of classical Cauchy continuum. The effects of micro structures/properties on the anisotropy and heterogeneity of macroscopic constitutive behaviors are embodied with the probability density function of the contact distribution. The micro-macro transitions of kinematical quantities are fulfilled by means of Hill-Mandel condition of the average-field theory for heterogeneous Cosserat continuum. To analyze the asymptotic trend and to validate the proposed model, the micromechanically-based expressions of macroscopic elastic constants for the materials under homogeneous and isotropic assumptions are particularly derived. The validity of the developed model is verified through the comparisons of the theoretical predictions given by the derived formulae and numerical results obtained by using the discrete element method on overall behavior of a regularly packed granular assembly. Moreover, the derived formulae also provide theoretical interpretations to the numerical results obtained by the discrete element method.
引文
离散元求解模块。应用该程序分析了颗粒材料中的若干典型例题,数值算例结果显示出
所发展方法的有效性和可应用性。
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Fr表征了颗粒在空间的排列,如对椭圆形颗粒集合,颗粒的长轴方向的统计分布可以通
过Fr的体现。同理也可以用类似的方式定义颗粒方向密度函数Er(θ)来表征长轴方向为
θ的颗粒的分布密度。上述各种描述颗粒材料的几何量中,基于颗粒的几何度量多用于描述构成材料的颗
粒的基本几何性质,比如颗粒形状、尺寸、粒度等,以及相应的颗粒材料的分类,如本
章2.2-2.3节所述；接触法向及分支向量多用于颗粒材料等效应力的定义；微单元以及
各种几何系统是颗粒材料等效应变定义的基础,如本论文第三章3.2节所述；接触方向
密度函数以及接触结构张量则为建立颗粒材料的解析形式多尺度本构关系提供了基础,
参见本论文第三章3.3节内容。综上所述,建立对颗粒类材料在颗粒尺度上的物理、几何性质的认识,并作出恰当
的度量,既有助于对颗粒材料宏观本构属性(比如抗剪强度)的理解和描述,也是在多
尺度均匀化方法中建立颗粒材料细-宏观物理量衔接及相应多尺度本构关系的基础。
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此外,Tsutsumi和Kaneko[38]将该方法推广到三维情况,并且分析了颗粒材料的各向异性和非共轴性等本构行为。Miehe和Dettmar[36]讨论了将基于平均场理论Hill-Mandel能量等价条件的计算均匀化过程推广到颗粒材料中的一些基本问题。Andrade和Tu[39]在连续体有限元法计算塑性框架内,通过离散单元法计算宏观本构更新算法所需的内状态变量参数,发展了一个颗粒材料力学行为分析的准-协同(semi-concurrent)多尺度计算方法,有效地模拟了颗粒材料的分岔现象。
一方面,随着固体力学多尺度分析方法的广泛发展与应用以及人们对颗粒材料的固有多尺度本质特征的深入认识,针对颗粒材料的多尺度计算方法也越来越多地受到人们的关注与重视。其中基于平均场理论的多尺度计算均匀化方法以其独有的优势成为人们广泛关注与大力发展的有效途径之一。另一方面,以上所述方法宏观上均采用经典连续体,相应地细观没有考虑颗粒的转角自由度以及颗粒间的接触力矩作用,因而未能很好地反映颗粒材料的物理实际。因此发展能够充分考虑颗粒材料细观物理实际、并能有效俘获颗粒材料宏、细观复杂力学行为的多尺度计算方法成为了近年来人们研究的热点问题。
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(1)推导了一个用于非均质Cosserat连续体细-宏观均匀化模拟过程的Hill定理形式。
(2)从Hill定理出发,系统分析和讨论了各种类型的RVE边界条件,并据此确定了恰当的强形式RVE边界条件,使得细-宏观联系同时满足Hill-Mandel能量等价条件和一阶平均场理论基本假定。
(3)经典Cauchy连续体平均场理论中常用的RVE周期边界条件仅对于变量ui和σki成立,而对Cosserat连续体模型中的特有变量ωi和μki,RVE周期边界条件不能够保证同时满足一阶平均场理论基本假定和Hill-Mandel能量等价条件,有待进一步研究。
(4)所提出的RVE边界条件对于在平均场理论框架内建立Cosserat类非均质材料的细宏观分量之间的正确关联是非常重要的,它为推导相应的细-宏观本构关系,以及在多尺度计算均匀化途径中推导宏观一致性切线模量张量奠定了基础,为正确求解非均质微结构RVE的力学响应、以及恰当地建立跨尺度均匀化模拟过程提供了理论指导与实施保障。
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通过对一系列典型数值算例结果的分析,显示出本章所发展方法的有效性和可应用性。通过不同RVE尺寸对材料宏观行为影响的多尺度模拟与分析,对颗粒材料多尺度计算均匀化方法中RVE尺寸的确定和选择给出了建议,为后续研究工作奠定基础。
本章工作致力于基本理论框架、算法公式的构建和实现,以及基本数值例题的考核与验证。作为对该多尺度计算均匀化方法的初步探索和尝试,本章工作充分显示了该方法的有效性和可发展潜力。继续深入发展基于平均场理论的、高效的颗粒材料多尺度计算均匀化方法,并将其应用于工程实际问题,具有重要的理论意义与应用价值。
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本章发展了颗粒材料基于宏观经典连续体描述的多尺度微-方向本构模型,其中宏观采用Cosserat连续体,细观同时考虑颗粒间的接触力和力矩。离散颗粒微结构对宏观行为的影响通过接触分布的概率密度函数体现,运动学分量的细宏观联系根据平均场理论能量等价条件确定。给出了宏观均质、各向同性Cosserat连续体弹性常数的细观力学表达式。
从本章所发展的多尺度本构模型出发,结合宏观Cosserat连续体弹性常数间关系的基本假定,建议了二维Cosserat连续体内尺度参数表达式。与文献中已有工作相比,其特点在于该表达式通过多尺度本构关系的严格推导得到并且完全由颗粒的细观参数表不。
对一颗粒集合体采用离散单元法数值模拟其宏观力学行为。数值模拟结果与理论预·测结果的一致性显示了(定性地)本章所发展本构模型的有效性。另一方面,本章所推导颗粒材料宏观本构参数的细观力学表达式(式(6.63)-(6.64)、(6.67)、(6.70)-(6.71))也为合理解释离散单元法模拟得到的数值结果提供了参考依据。
后续研究工作是：细观颗粒间各向异性接触分布和非均匀颗粒尺寸分布因素的考虑,以及与之对应的颗粒材料宏观弹塑性行为的表征。采用离散单元法对各种不同尺寸、随机分布的离散颗粒集合体宏观行为进行数值模拟,将数值模拟结果与所发展理论公式预测的结果对比,以验证所发展模型的有效性,并深入研究颗粒材料的宏观非线性本构行为与细观微结构参数之间的关系。
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以上每种类型数据的成员内容还可以根据需要进行扩展。另外,类似于积分点下的内状态变量,在每一增量步计算完毕后,微结构信息也需要存储并更新,以作为下一个载荷增量步的参考状态以及备后处理使用。设置如下四个数组完成此功能：颗粒单元信息数组：ELEM_DEM (NELEM, NINMAX, NGRAMAX)颗粒邻居信息数组：NEIB_DEM (NELEM, NINMAX, NGRAMAX)颗粒接触信息数组：ICONT_DEM (NELEM, NINMAX, NGRAMAX) CONT_DEM (NELEM, NINMAX, NGRAMAX, NCLMAX)
其中NGRAMAX是微结构颗粒的数目上限；NCLMAX是每个颗粒的接触点个数上限；NINMAX, NELEM分别是单元的积分点个数上限和单元总数。这样就完成了多尺度信息数据的存储和传递工作。
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所发展方法的有效性和可应用性。
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Fr表征了颗粒在空间的排列,如对椭圆形颗粒集合,颗粒的长轴方向的统计分布可以通
过Fr的体现。同理也可以用类似的方式定义颗粒方向密度函数Er(θ)来表征长轴方向为
θ的颗粒的分布密度。上述各种描述颗粒材料的几何量中,基于颗粒的几何度量多用于描述构成材料的颗
粒的基本几何性质,比如颗粒形状、尺寸、粒度等,以及相应的颗粒材料的分类,如本
章2.2-2.3节所述；接触法向及分支向量多用于颗粒材料等效应力的定义；微单元以及
各种几何系统是颗粒材料等效应变定义的基础,如本论文第三章3.2节所述；接触方向
密度函数以及接触结构张量则为建立颗粒材料的解析形式多尺度本构关系提供了基础,
参见本论文第三章3.3节内容。综上所述,建立对颗粒类材料在颗粒尺度上的物理、几何性质的认识,并作出恰当
的度量,既有助于对颗粒材料宏观本构属性(比如抗剪强度)的理解和描述,也是在多
尺度均匀化方法中建立颗粒材料细-宏观物理量衔接及相应多尺度本构关系的基础。
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此外,Tsutsumi和Kaneko[38]将该方法推广到三维情况,并且分析了颗粒材料的各向异性和非共轴性等本构行为。Miehe和Dettmar[36]讨论了将基于平均场理论Hill-Mandel能量等价条件的计算均匀化过程推广到颗粒材料中的一些基本问题。Andrade和Tu[39]在连续体有限元法计算塑性框架内,通过离散单元法计算宏观本构更新算法所需的内状态变量参数,发展了一个颗粒材料力学行为分析的准-协同(semi-concurrent)多尺度计算方法,有效地模拟了颗粒材料的分岔现象。
一方面,随着固体力学多尺度分析方法的广泛发展与应用以及人们对颗粒材料的固有多尺度本质特征的深入认识,针对颗粒材料的多尺度计算方法也越来越多地受到人们的关注与重视。其中基于平均场理论的多尺度计算均匀化方法以其独有的优势成为人们广泛关注与大力发展的有效途径之一。另一方面,以上所述方法宏观上均采用经典连续体,相应地细观没有考虑颗粒的转角自由度以及颗粒间的接触力矩作用,因而未能很好地反映颗粒材料的物理实际。因此发展能够充分考虑颗粒材料细观物理实际、并能有效俘获颗粒材料宏、细观复杂力学行为的多尺度计算方法成为了近年来人们研究的热点问题。
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(1)推导了一个用于非均质Cosserat连续体细-宏观均匀化模拟过程的Hill定理形式。
(2)从Hill定理出发,系统分析和讨论了各种类型的RVE边界条件,并据此确定了恰当的强形式RVE边界条件,使得细-宏观联系同时满足Hill-Mandel能量等价条件和一阶平均场理论基本假定。
(3)经典Cauchy连续体平均场理论中常用的RVE周期边界条件仅对于变量ui和σki成立,而对Cosserat连续体模型中的特有变量ωi和μki,RVE周期边界条件不能够保证同时满足一阶平均场理论基本假定和Hill-Mandel能量等价条件,有待进一步研究。
(4)所提出的RVE边界条件对于在平均场理论框架内建立Cosserat类非均质材料的细宏观分量之间的正确关联是非常重要的,它为推导相应的细-宏观本构关系,以及在多尺度计算均匀化途径中推导宏观一致性切线模量张量奠定了基础,为正确求解非均质微结构RVE的力学响应、以及恰当地建立跨尺度均匀化模拟过程提供了理论指导与实施保障。
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通过对一系列典型数值算例结果的分析,显示出本章所发展方法的有效性和可应用性。通过不同RVE尺寸对材料宏观行为影响的多尺度模拟与分析,对颗粒材料多尺度计算均匀化方法中RVE尺寸的确定和选择给出了建议,为后续研究工作奠定基础。
本章工作致力于基本理论框架、算法公式的构建和实现,以及基本数值例题的考核与验证。作为对该多尺度计算均匀化方法的初步探索和尝试,本章工作充分显示了该方法的有效性和可发展潜力。继续深入发展基于平均场理论的、高效的颗粒材料多尺度计算均匀化方法,并将其应用于工程实际问题,具有重要的理论意义与应用价值。
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本章发展了颗粒材料基于宏观经典连续体描述的多尺度微-方向本构模型,其中宏观采用Cosserat连续体,细观同时考虑颗粒间的接触力和力矩。离散颗粒微结构对宏观行为的影响通过接触分布的概率密度函数体现,运动学分量的细宏观联系根据平均场理论能量等价条件确定。给出了宏观均质、各向同性Cosserat连续体弹性常数的细观力学表达式。
从本章所发展的多尺度本构模型出发,结合宏观Cosserat连续体弹性常数间关系的基本假定,建议了二维Cosserat连续体内尺度参数表达式。与文献中已有工作相比,其特点在于该表达式通过多尺度本构关系的严格推导得到并且完全由颗粒的细观参数表不。
对一颗粒集合体采用离散单元法数值模拟其宏观力学行为。数值模拟结果与理论预·测结果的一致性显示了(定性地)本章所发展本构模型的有效性。另一方面,本章所推导颗粒材料宏观本构参数的细观力学表达式(式(6.63)-(6.64)、(6.67)、(6.70)-(6.71))也为合理解释离散单元法模拟得到的数值结果提供了参考依据。
后续研究工作是：细观颗粒间各向异性接触分布和非均匀颗粒尺寸分布因素的考虑,以及与之对应的颗粒材料宏观弹塑性行为的表征。采用离散单元法对各种不同尺寸、随机分布的离散颗粒集合体宏观行为进行数值模拟,将数值模拟结果与所发展理论公式预测的结果对比,以验证所发展模型的有效性,并深入研究颗粒材料的宏观非线性本构行为与细观微结构参数之间的关系。
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以上每种类型数据的成员内容还可以根据需要进行扩展。另外,类似于积分点下的内状态变量,在每一增量步计算完毕后,微结构信息也需要存储并更新,以作为下一个载荷增量步的参考状态以及备后处理使用。设置如下四个数组完成此功能：颗粒单元信息数组：ELEM_DEM (NELEM, NINMAX, NGRAMAX)颗粒邻居信息数组：NEIB_DEM (NELEM, NINMAX, NGRAMAX)颗粒接触信息数组：ICONT_DEM (NELEM, NINMAX, NGRAMAX) CONT_DEM (NELEM, NINMAX, NGRAMAX, NCLMAX)
其中NGRAMAX是微结构颗粒的数目上限；NCLMAX是每个颗粒的接触点个数上限；NINMAX, NELEM分别是单元的积分点个数上限和单元总数。这样就完成了多尺度信息数据的存储和传递工作。
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